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Dictionnaire anglais - français

complex number

nombre complexe

informatique et traitement des données / sciences - acta.es iate.europa.eu
absolute value of a complex number

valeur absolue d'un nombre complexe

électronique et électrotechnique - iate.europa.eu
The converter output is a complex number.

La sortie du convertisseur est un nombre complexe.

électronique et électrotechnique - wipo.int
The complex number x − iy is by definition the complex conjugate of the complex number x iy.

Le nombre complexe x – iy, noté , est appelé conjugué du nombre complexe z.

général - CCMatrix (Wikipedia + CommonCrawl)
Is the sum of a complex number and its complex conjugate a real number?

Est-ce que la somme d’un nombre complexe et de son conjugué est toujours un nombre réel ?

général - CCMatrix (Wikipedia + CommonCrawl)
A complex number correlator for calculating the square |Z|2 of the absolute value of the correlation value Z=(a+jb)* x (c+jd) of a first complex number a+jb and a second complex number c+jd by the following method.

Un corrélateur de nombre complexe destiné à calculer le carré Z2 de la valeur absolue de la valeur de corrélation Z = (a+jb)* x (c+jd) d'un premier nombre complexe a+bj et d'un second nombre complexe c+jd au moyen de ce procédé.

informatique et traitement des données - wipo.int
▷
La hiérarchie de szegő quadratique
...the following Hamiltonian PDE : $i \partial_t u = 2J\Pi(|u|^2)+\bar{J}u^2$, $u(0, \cdot)=u_0$, where $\Pi$ is the Szegő projector onto nonnegative modes, and $J = J(u)$ is the complex number given by $J=\int_\mathbb{T}|u|^2u$....
...Le but de cet article est d'approfondir l'étude de l'équation de Szegő quadratique, qui est l'EDP hamiltonienne suivante : $i \partial_t u = 2J\Pi(|u|^2)+\bar{J}u^2$, $u(0, \cdot)=u_0$, où $\Pi$ est le projecteur de Szegő sur les modes positifs, et $J = J(u)$ est un nombre complexe donné par $J=\int_\mathbb{T}|u|^2u$....
général - core.ac.uk -
▷
Les fractales : une nouvelle source d’inspiration pédagogique, musicale et scientifique
... I then converted each equation, derived from a complex number, into a series of frequencies or audible musical notes....
... Par la suite, j’ai converti chaque itéra­tion provenant d’un nombre complexe quel­conque en une série de fréquences ou notes de musique pouvant être perçues par l’oreille hu­maine....
général - core.ac.uk - PDF: journal.fsst.ca
▷
Sur la structure cellulaire et la théorie de la représentation des algèbres de temperley-lieb à couture
... Those unitary and associative algebras of finite dimension are parametrised by two positive integers n, k and a complex number β, expressed with q ∈ ℂˣ as β = q + q−1....
... Elles sont paramétrisées par deux nombres entiers positifs n,k et un nombre complexe β qui est exprimé souvent à l’aide d’un autre paramètre q ∈ ℂˣ comme β = q + q−1....
général - core.ac.uk -
▷
Arbitrarily substantial number representation for complex numberResearchers are often perplexed when their machine learning algorithms are required to deal with complex number....
▷
Quatre étapes dans l'histoire des nombres complexes : "quelques commentaires épistémologiques et didactiques".... L'analyse met en particulier en évidence le développement dialectique des dimensions outil et objet du concept de nombre complexe, le rôle joué dans le développement de ce concept par l'articulation des cadres algébrique et géométrique, le rôle joué par l'analogie dans la recherche mathématique....
général - core.ac.uk - PDF: journal.utem.edu.mygénéral - core.ac.uk -
An orthogonal estimation algorithm for complex number systemsAn orthogonal estimation algorithm for complex number systems is derived....
 PDF: eprints.whiterose.ac.uk
Abstract arithmetic unit for complex number processingThis paper presents development of a complex number arithmetic unit (CAU), based on the single-component representation of complex numbers by positional binary codes with complex radix....
 PDF: citeseerx.ist.psu.edu
On-line algorithms for complex number arithmeticA class of on-line algorithms for complex number arith-metic is presented....
 PDF: arith.cs.ucla.edu
The fixed point of fuzzy complex number-valued mappingIn this paper, we introduce the concepts of fuzzy complex number, the operations of fuzzy complex number, the concepts of convergence for fuzzy complex number-valued sequence and the fuzzy complex number-valued mapping....
 PDF: www.m-hikari.com

Synonymes et termes associés anglais

complex number

Exemples anglais - français

complex number

nombre complexe
sciences naturelles et appliquées - techdico

Traductions en contexte anglais - français

A complex number coprocessor for performing predefined calculation on complex number elements.

L'invention se rapporte à un coprocesseur de nombres complexes permettant d'effectuer des calculs prédéfinis sur des éléments de nombres complexes.

informatique et traitement des données - wipo.int
Is the sum of a complex number and its complex conjugate a real number?

Le produit d’un complexe et de son conjugué est-il toujours un nombre réel ?

général - CCMatrix (Wikipedia + CommonCrawl)
To construct the complex number x + iy, you use complex:

Si z désigne le nombre complexe x + iy, l'intégrale généralisée :

général - CCMatrix (Wikipedia + CommonCrawl)
To construct the complex number x + iy, you use complex:

Pour créer dans Ruby le complexe x+iy, on utilise Complex(x,y):

général - CCMatrix (Wikipedia + CommonCrawl)
COMPLEX Converts real and imaginary coefficients into a complex number.

Convertit des coefficients réels et imaginaires en un nombre complexe.

général - CCMatrix (Wikipedia + CommonCrawl)
COMPLEX Converts real and imaginary coefficients into a complex number.

COMPLEXE Ingénierie Convertit des coefficients réel et imaginaire en un nombre complexe.

général - CCMatrix (Wikipedia + CommonCrawl)
complex An immutable complex number with real and imaginary parts.

complexe d'un nombre complexe immuable avec des parties réelle et imaginaire.

général - CCMatrix (Wikipedia + CommonCrawl)
A complex number is its own complex conjugate if and only if it is a real number.

Un complexe est égal à son conjugué si et seulement si c'est un réel.

général - CCMatrix (Wikipedia + CommonCrawl)
The argument of the complex number determines hue, and the magnitude of the complex number determines saturation.

L’argument du nombre complexe détermine la teinte, et la valeur absolue détermine la saturation.

général - CCMatrix (Wikipedia + CommonCrawl)
The complex conjugate of a complex number is It is found by changing the sign of the imaginary part of the complex number.

Conjuguer un nombre complexe revient à changer le signe de la partie imaginaire.

général - CCMatrix (Wikipedia + CommonCrawl)
We cannot say any complex number is also a real number, since a complex number has a real and an imaginary part.

Nous ne pouvons pas dire que tout nombre complexe est aussi un nombre réel, puisqu’un nombre complexe a une partie réelle et une partie imaginaire.

général - CCMatrix (Wikipedia + CommonCrawl)
The Hermitian conjugate of a complex number is its complex conjugate.

la symétrie qui à un nombre complexe associe son conjugué.

général - CCMatrix (Wikipedia + CommonCrawl)
Hence the product of two complex numbers is a complex number.

c'est-à-dire le produit de deux nombres complexes est un nombre complexe dont

général - CCMatrix (Wikipedia + CommonCrawl)
The magnitude of the complex number is normalized (106) and processed (108, 110) through a closed loop to produce an output (112) proportional to the phase of the complex number.

En l'occurrence, on normalise (106) la grandeur du nombre complexe, et on la traite au moyen d'une boucle fermée de façon produire une sortie (112) proportionnelle à la phase du nombre complexe.

électronique et électrotechnique - wipo.int
imaginary-number is the imaginary part of the complex number.

imaginary-number est la partie imaginaire du nombre complexe.

général - CCMatrix (Wikipedia + CommonCrawl)


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Nouveau : Traduire du texte avec ChatDico – 172 langues

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Requêtes fréquentes français :1-200, -1k, -2k, -3k, -4k, -5k, -7k, -10k, -20k, -40k, -100k, -200k, -500k, -1000k,

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s must be zero. As a consequence, we characterize growing solutions which can be written as the sum of two solitons.","abstr2":"Le but de cet article est d'approfondir l'étude de l'équation de Szegő quadratique, qui est l'EDP hamiltonienne suivante : $i \\partial_t u = 2J\\Pi(|u|^2)+\\bar{J}u^2$, $u(0, \\cdot)=u_0$, où $\\Pi$ est le projecteur de Szegő sur les modes positifs, et $J = J(u)$ est un nombre complexe donné par $J=\\int_\\mathbb{T}|u|^2u$. Nous mettons en évidence une suite infinie de nouvelles lois de conservation $\\{\\ell_k \\}$, qui sont en involution. Ces lois nous permettent de mieux comprendre le comportement \"turbulent\" de certaines solutions rationnelles de l'équation : nous montrons que, si l'orbite d'une solution rationnelle n'est pas bornée dans un espace $H^s$, $s > 1/2$, alors l'un des $\\ell_k$ vaut zéro. En conséquence, nous donnons une caractérisation de telles solutions croissantes qui s'écrivent comme une somme de deux solitons","authors":["Thirouin, Joseph"],"coreId":"153980464","datePublished":"2018-04-03T00:00:00","lg-ab1":"en","lg-ab2":"fr","lg-original":"fr","lg-t":"fr","title":"La hiérarchie de Szegő quadratique"},{"abstr1":"Math teachers often make use of graphs to visu­ally represent equations and concepts based on the expected curriculum. Thus, I decided to inves­tigate the possibility of converting an equation, normally interpreted using a graph, into auditory form, by converting geometric shapes into sound. To do so, I decided to use the celebrated Man­delbrot fractal, which uses the general equation z = z2 + c as a basic geometric concept. I then converted each equation, derived from a complex number, into a series of frequencies or audible musical notes. Each series was played on a com­puter and represented in graph form, so that the mathematical self-similarity could be observed. The results obtained show that one can hear this self-similarity, and suggest that it would be pos­sible to use auditory methods in conjunction with traditional pedagogical methods to teach math­ematics.","abstr2":"Lorsqu’on étudie les mathématiques, nos ensei­gnants font appel à l’utilisation de graphiques afin de représenter les équations et concepts que nous devons apprendre. C’est dans cette op­tique de pensée que j’ai entrepris l’investigation la possibilité de convertir une équation, nor­malement interprétée graphiquement, et de l’interpréter de manière auditive, c’est-à-dire, de transformer la forme géométrique en forme so­nore. Pour ce faire, j’ai décidé d’utiliser, comme forme géométrique de base, la célèbre fractale de Mandelbrot, dont l’équation itérative est z = z2 + c. Par la suite, j’ai converti chaque itéra­tion provenant d’un nombre complexe quel­conque en une série de fréquences ou notes de musique pouvant être perçues par l’oreille hu­maine. Cette série est alors jouée à l’ordinateur et représentée graphiquement afin d’en observer l’autosimilarité. Les résultats obtenus démon­trent que l’on puisse entendre cette autosimi­larité et suggèrent qu’il serait possible d’utiliser l’audio comme moyen pédagogique complémen­taire dans l’apprentissage des mathématiques","authors":["Roewer-Després, François"],"coreId":"267828103","datePublished":"2014-03-01","lg-ab1":"en","lg-ab2":"fr","lg-original":"fr","lg-t":"fr","pdfURL":"https://journal.fsst.ca/index.php/jsst/article/download/16/13","title":"Les fractales : une nouvelle source d’inspiration pédagogique, musicale et scientifique"},{"abstr1":"This master’s thesis considers the representation theory of boundary seam algebras Bn,k (β), introduced by Morin-Duchesne, Rasmussen and Ridout [21]. Those unitary and associative algebras of finite dimension are parametrised by two positive integers n, k and a complex number β, expressed with q ∈ ℂˣ as β = q + q−1. They admit a diagrammatic definition and a realization of type Pk(ᵏ)TL n+k Pk(ᵏ) using the Temperley-Lieb algebra TLn+k and a family of idempotents Pk(ᵏ): the Wenzl-Jones projectors. We prove the cellularity in the sense of Graham and Lehrer [10] of this family of algebras for most values of β and use it to construct the simple, projective and cellular modules. Expressing β = q + q−1 for q ∈ ℂˣ, we get that the behaviour depends on the value of q : when it is generic, Bn,k (β) is semisimple; the case of q being a root of unity is richer and is studied in depth in chapter 4.Chapter 1 pedagogically presents the theory of Temperley-Lieb algebras. Chapter 2 uses Temperley-Lieb algebras as examples to introduce cellular algebras. In chapter 3, boundary seam algebras are introduced in three different ways : by generators and relations, diagrammatically and by Wenzl-Jones proj","abstr2":"Ce mémoire étudie la théorie de la représentation des algèbres à couture Bn,k (β), introduites par Morin-Duchesne, Rasmussen et Ridout [21]. Elles sont paramétrisées par deux nombres entiers positifs n,k et un nombre complexe β qui est exprimé souvent à l’aide d’un autre paramètre q ∈ ℂˣ comme β = q + q−1. Ces algèbres unifères, associatives et de dimension finie admettent une définition diagrammatique ainsi qu’une réalisation de type Pk(ᵏ)TLn+k Pk(ᵏ) pour une famille d’idempotents Pk(ᴷ) issue des algèbres de Temperley-Lieb originales : les projecteurs de Wenzl-Jones. L’objectif est de caractériser les modules simples, projectifs et cellulaires. Un résultat majeur à cet effet est la preuve de la cellularité de cette famille d’algèbres pour la plupart des valeurs de β. Ce résultat est nouveau et facilite grandement l’étude. En résumé, le comportement dépend de la valeur de q : lorsque q est générique, l’algèbre Bn,k (β) est connue semi-simple ; le cas q racine de l’unité est plus riche et son étude occupe la grande partie de ce mémoire.Le chapitre 1 présente la théorie de base des algèbres de Temperley-Lieb dans une optique pédagogique. Le chapitre 2 utilise l’exemple des algèbres de Temperley-Lieb pourintroduire les algèbres cellulaires. Dans le chapitre 3, les algèbres à couture sont introduites de trois façons : par générateurs et relations, diagrammatiquement et par les projecteurs de Wenzl-Jones ; l’isomorphisme entre les trois présentations est ensuite montré, sauf dans le cas où q 2ᵏ = 1. La cellularité et la théorie de la représentation sont ensuite étudiées pour q générique. Finalement, le chapitre 4 présente en trois étapes les résultats nouveaux en q racine de l’unité : la dimension des radicaux et des modules simples, la construction des morphismes non-triviaux entre les modules cellulaires et la caractérisation des modules indécomposables principaux.","authors":["Langlois-Rémillard, Alexis"],"coreId":"211061604","datePublished":"2018-12-01T00:00:00","lg-ab1":"en","lg-ab2":"fr","lg-original":"fr","lg-t":"fr","title":"Sur la structure cellulaire et la théorie de la représentation des algèbres de Temperley-Lieb à couture"},{"abstr1":"Researchers are often perplexed when their machine learning algorithms are required to deal with complex number. Various strategies are commonly employed to project complex number into real number, although it is frequently sacrificing the information contained in the complex number. This paper proposes a new method and four techniques to represent complex number as real number, without having to sacrifice the information contained. The proposed techniques are also capable of retrieving the original complex number from the representing real number, with little to none of information loss. The promising applicability of the proposed techniques has been demonstrated and worth to receive further exploration in representing the complex number","authors":["Pratama, Satrya Fajri","Muda, Azah Kamilah","Choo, Yun-Huoy"],"coreId":"229273549","datePublished":"2018-02-12","lg-ab1":"en","lg-original":"en","lg-t":"en","pdfURL":"http://journal.utem.edu.my/index.php/jtec/article/download/3590/2483","title":"Arbitrarily Substantial Number Representation for Complex Number"},{"abstr1":"In this paper, we introduce the concepts of fuzzy complex number, the operations of fuzzy complex number, the concepts of convergence for fuzzy complex number-valued sequence and the fuzzy complex number-valued mapping. Then the fixed point of fuzzy complex number-valued mapping is discussed, some existent theorems of this mapping are given. It will establish a foundation for researching fuzzy complex analysis","authors":["Shenquan Ma","Dejun Peng"],"coreId":"101329923","datePublished":"2015-02-03","lg-ab1":"en","lg-original":"en","lg-t":"en","pdfURL":"http://www.m-hikari.com/ams/ams-password-2007/ams-password13-16-2007/maAMS13-16-2007.pdf","title":"The Fixed Point of Fuzzy Complex Number-Valued Mapping"},{"abstr1":"A class of on-line algorithms for complex number arith-metic is presented. These algorithms adopt a redundant complex number system (RCNS) to represent complex num-bers as a single number. Such a scheme simplifies the specifi-cation of the design, and has the additionaleffect that single-precision complex arithmetic can be easily reconfigured for double-precision real arithmetic. We present cost delay comparisons with the more conventional approach to show a significant improvement, demonstrating that the presented algorithms are attractive for VLSI systems demanding com-plex number operations. 1","authors":["Robert Mcilhenny","Milos ̌ D. Ercegovac"],"coreId":"105761598","datePublished":"1998","lg-ab1":"en","lg-original":"en","lg-t":"en","pdfURL":"http://arith.cs.ucla.edu/publications/mcilhenny_asilomar98.pdf","title":"On-Line Algorithms for Complex Number Arithmetic"},{"abstr1":"This paper presents development of a complex number arithmetic unit (CAU), based on the single-component representation of complex numbers by positional binary codes with complex radix. Algorithms of the basic arithmetic operations in this representation are described and analyzed. The results of CAU design, simulation and synthesis and comparison of CAU characteristics to those of traditional arithmetic units (TAU) are presented. It is shown that implementation of CAU algorithms in math processors achieves significant (5-10 times) speed-up of complex number processing over TAU equivalents. 1","authors":["Dr. Solomon Khmelnik","Dr. Sergey Selyutin","R Viduetsky","Inna Doubson"],"coreId":"20709155","datePublished":"2008-04-03","lg-ab1":"en","lg-original":"en","lg-t":"en","pdfURL":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.102.8292","title":"Abstract Arithmetic Unit for Complex Number Processing"},{"abstr1":"An orthogonal estimation algorithm for complex number systems is derived. It is shown that a modified Error Reduction Ratio (eRR) test together with the orthogonal estimation algorithm provides an efficient way of identifying both the structure and the unknown parameters of complex number systems. A forward regression procedure is proposed as an optimal search algorithm for this problem and simulated examples which show the application of the method to the parameterisation of linear frequency response functions are included","authors":["Tsang, K.M.","Billings, S.A."],"coreId":"19967850","datePublished":"1990-08","lg-ab1":"en","lg-original":"en","lg-t":"en","pdfURL":"http://eprints.whiterose.ac.uk/78400/1/acse%20research%20report%20405.pdf","title":"An Orthogonal Estimation Algorithm for Complex Number Systems"}]},"fr":{"nombre complexe":[{"abstr1":"International audienceCette brochure est issue de l'enseignement optionnel de maîtrise à l'université Paris7 intitulé \"Approche historique et didactique des mathématiques\". En s'appuyant sur des textes historiques, les auteurs pointent quatre moments décisifs de l'histoire des nombres complexes et les exploitent pour une analyse épistémologique et didactique. L'analyse met en particulier en évidence le développement dialectique des dimensions outil et objet du concept de nombre complexe, le rôle joué dans le développement de ce concept par l'articulation des cadres algébrique et géométrique, le rôle joué par l'analogie dans la recherche mathématique. Elle conduit aussi à interroger certaines interprétations didactiques de la notion d'obstacle épistémologiqu","authors":["Artigue, Michèle","Deledicq, André"],"coreId":"200974942","datePublished":"1992","lg-ab1":"fr","lg-original":"fr","lg-t":"fr","title":"Quatre étapes dans l'histoire des nombres complexes : \"quelques commentaires épistémologiques et didactiques\"."}]},"fromlang":"en","tolang":"fr"},"synonymes":["complex number "]},"data":{"fr":[{"glossary":{"fromRaw":"complex number","toRaw":"nombre complexe","from":[{"text":"","type":0},{"text":"complex number","type":1}],"to":[{"text":"","type":0},{"text":"nombre complexe","type":1}],"d2":["3236","36"],"d0":"4","source":["acta.es","iate.europa.eu"]},"collapsed":false,"sentences":[{"fromRaw":"absolute value of a complex number","toRaw":"valeur absolue d'un nombre complexe","from":[{"text":"absolute value of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"valeur absolue d'un ","type":0},{"text":"nombre complexe","type":1}],"d2":"6826","d0":"13","source":"iate.europa.eu"},{"fromRaw":"The converter output is a complex number.","toRaw":"La sortie du convertisseur est un nombre complexe.","from":[{"text":"The converter output is a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"La sortie du convertisseur est un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"6826","d0":"13","source":"wipo.int"},{"fromRaw":"The complex number x − iy is by definition the complex conjugate of the complex number x iy.","toRaw":"Le nombre complexe x – iy, noté , est appelé conjugué du nombre complexe z.","from":[{"text":"The ","type":0},{"text":"complex number","type":1},{"text":" x − iy is by definition the ","type":0},{"text":"complex","type":2},{"text":" conjugate of the ","type":0},{"text":"complex number","type":1},{"text":" x iy.","type":0}],"to":[{"text":"Le ","type":0},{"text":"nombre complexe","type":1},{"text":" x – iy, noté , est appelé conjugué du ","type":0},{"text":"nombre complexe","type":1},{"text":" z.","type":0}],"d2":"0","d0":"11","source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Is the sum of a complex number and its complex conjugate a real number?","toRaw":"Est-ce que la somme d’un nombre complexe et de son conjugué est toujours un nombre réel ?","from":[{"text":"Is the sum of a ","type":0},{"text":"complex number","type":1},{"text":" and its ","type":0},{"text":"complex","type":2},{"text":" conjugate a real ","type":0},{"text":"number","type":2},{"text":"?","type":0}],"to":[{"text":"Est-ce que la somme d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" et de son conjugué est toujours un ","type":0},{"text":"nombre","type":2},{"text":" réel ?","type":0}],"d2":"0","d0":"11","source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number correlator for calculating the square |Z|2 of the absolute value of the correlation value Z=(a+jb)* x (c+jd) of a first complex number a+jb and a second complex number c+jd by the following method.","toRaw":"Un corrélateur de nombre complexe destiné à calculer le carré Z2 de la valeur absolue de la valeur de corrélation Z = (a+jb)* x (c+jd) d'un premier nombre complexe a+bj et d'un second nombre complexe c+jd au moyen de ce procédé.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" correlator for calculating the square |Z|2 of the absolute value of the correlation value Z=(a+jb)* x (c+jd) of a first ","type":0},{"text":"complex number","type":1},{"text":" a+jb and a second ","type":0},{"text":"complex number","type":1},{"text":" c+jd by the following method.","type":0}],"to":[{"text":"Un corrélateur de ","type":0},{"text":"nombre complexe","type":1},{"text":" destiné à calculer le carré Z2 de la valeur absolue de la valeur de corrélation Z = (a+jb)* x (c+jd) d'un premier ","type":0},{"text":"nombre complexe","type":1},{"text":" a+bj et d'un second ","type":0},{"text":"nombre complexe","type":1},{"text":" c+jd au moyen de ce procédé.","type":0}],"d2":"3236","d0":"4","source":"wipo.int"},{"fromRaw":"Thus, modulus of any complex number is equal to the positive square root of the product of the complex number and its conjugate complex number.","toRaw":"La valeur absolue d'un nombre complexe est égale à son module, ou la racine carrée du produit du nombre et de son complexe conjugué.","from":[{"text":"Thus, modulus of any ","type":0},{"text":"complex number","type":1},{"text":" is equal to the positive square root of the product of the ","type":0},{"text":"complex number","type":1},{"text":" and its conjugate ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"La valeur absolue d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" est égale à son module, ou la racine carrée du produit du ","type":0},{"text":"nombre","type":2},{"text":" et de son ","type":0},{"text":"complexe","type":2},{"text":" conjugué.","type":0}],"d2":"0","d0":"11","source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"product of the complex number and its complex conjugate.","toRaw":"Produit d’un nombre complexe et de son conjugué","from":[{"text":"product of the ","type":0},{"text":"complex number","type":1},{"text":" and its ","type":0},{"text":"complex","type":2},{"text":" conjugate.","type":0}],"to":[{"text":"Produit d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" et de son conjugué","type":0}],"d2":"0","d0":"11","source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"See also ARGUMENT, MODULUS OF A COMPLEX NUMBER and POLAR FORM OF A COMPLEX NUMBER.","toRaw":"Je te rappelle ce qu’est la forme algébrique d’un nombre complexe ainsi que le module d’un nombre complexe.","from":[{"text":"See also ARGUMENT, MODULUS OF A ","type":0},{"text":"COMPLEX NUMBER","type":1},{"text":" and POLAR FORM OF A ","type":0},{"text":"COMPLEX NUMBER","type":1},{"text":".","type":0}],"to":[{"text":"Je te rappelle ce qu’est la forme algébrique d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" ainsi que le module d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Every non-zero number x, real or complex, has n different complex number nth roots.","toRaw":"Chaque nombre non nul x, réel ou complexe, a n différentes racines de nombre complexe nième.","from":[{"text":"Every non-zero ","type":0},{"text":"number","type":2},{"text":" x, real or ","type":0},{"text":"complex","type":2},{"text":", has n different ","type":0},{"text":"complex number","type":1},{"text":" nth roots.","type":0}],"to":[{"text":"Chaque ","type":0},{"text":"nombre","type":2},{"text":" non nul x, réel ou ","type":0},{"text":"complexe","type":2},{"text":", a n différentes racines de ","type":0},{"text":"nombre complexe","type":1},{"text":" nième.","type":0}],"d2":"0","d0":"11","source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Learners examine the complex number function to convert real and imaginary coefficients into a complex number.","toRaw":"Utilisez la fonction COMPLEXE pour convertir des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"Learners examine the ","type":0},{"text":"complex number","type":1},{"text":" function to convert real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Utilisez la fonction ","type":0},{"text":"COMPLEXE","type":2},{"text":" pour convertir des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","source":"CCMatrix (Wikipedia + CommonCrawl)"}]},{"glossary":{},"sentences":[{"fromRaw":"A complex number coprocessor for performing predefined calculation on complex number elements.","toRaw":"L'invention se rapporte à un coprocesseur de nombres complexes permettant d'effectuer des calculs prédéfinis sur des éléments de nombres complexes.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" coprocessor for performing predefined calculation on ","type":0},{"text":"complex number","type":1},{"text":" elements.","type":0}],"to":[{"text":"L'invention se rapporte à un coprocesseur de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s permettant d'effectuer des calculs prédéfinis sur des éléments de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"3236","d0":"4","collapsed":false,"source":"wipo.int"},{"fromRaw":"Is the sum of a complex number and its complex conjugate a real number?","toRaw":"Le produit d’un complexe et de son conjugué est-il toujours un nombre réel ?","from":[{"text":"Is the sum of a ","type":0},{"text":"complex number","type":1},{"text":" and its ","type":0},{"text":"complex","type":2},{"text":" conjugate a real ","type":0},{"text":"number","type":2},{"text":"?","type":0}],"to":[{"text":"Le produit d’un ","type":0},{"text":"complexe","type":2},{"text":" et de son conjugué est-il toujours un ","type":0},{"text":"nombre","type":2},{"text":" réel ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"To construct the complex number x + iy, you use complex:","toRaw":"Si z désigne le nombre complexe x + iy, l'intégrale généralisée :","from":[{"text":"To construct the ","type":0},{"text":"complex number","type":1},{"text":" x + iy, you use ","type":0},{"text":"complex","type":2},{"text":":","type":0}],"to":[{"text":"Si z désigne le ","type":0},{"text":"nombre complexe","type":1},{"text":" x + iy, l'intégrale généralisée :","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"To construct the complex number x + iy, you use complex:","toRaw":"Pour créer dans Ruby le complexe x+iy, on utilise Complex(x,y):","from":[{"text":"To construct the ","type":0},{"text":"complex number","type":1},{"text":" x + iy, you use ","type":0},{"text":"complex","type":2},{"text":":","type":0}],"to":[{"text":"Pour créer dans Ruby le ","type":0},{"text":"complexe","type":2},{"text":" x+iy, on utilise Complex(x,y):","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"COMPLEX Converts real and imaginary coefficients into a complex number.","toRaw":"Convertit des coefficients réels et imaginaires en un nombre complexe.","from":[{"text":"","type":0},{"text":"COMPLEX","type":2},{"text":" Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Convertit des coefficients réels et imaginaires en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"COMPLEX Converts real and imaginary coefficients into a complex number.","toRaw":"COMPLEXE Ingénierie Convertit des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"","type":0},{"text":"COMPLEX","type":2},{"text":" Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"","type":0},{"text":"COMPLEXE","type":2},{"text":" Ingénierie Convertit des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"complex An immutable complex number with real and imaginary parts.","toRaw":"complexe d'un nombre complexe immuable avec des parties réelle et imaginaire.","from":[{"text":"","type":0},{"text":"complex","type":2},{"text":" An immutable ","type":0},{"text":"complex number","type":1},{"text":" with real and imaginary parts.","type":0}],"to":[{"text":"","type":0},{"text":"complexe","type":2},{"text":" d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" immuable avec des parties réelle et imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is its own complex conjugate if and only if it is a real number.","toRaw":"Un complexe est égal à son conjugué si et seulement si c'est un réel.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is its own ","type":0},{"text":"complex","type":2},{"text":" conjugate if and only if it is a real ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Un ","type":0},{"text":"complexe","type":2},{"text":" est égal à son conjugué si et seulement si c'est un réel.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The argument of the complex number determines hue, and the magnitude of the complex number determines saturation.","toRaw":"L’argument du nombre complexe détermine la teinte, et la valeur absolue détermine la saturation.","from":[{"text":"The argument of the ","type":0},{"text":"complex number","type":1},{"text":" determines hue, and the magnitude of the ","type":0},{"text":"complex number","type":1},{"text":" determines saturation.","type":0}],"to":[{"text":"L’argument du ","type":0},{"text":"nombre complexe","type":1},{"text":" détermine la teinte, et la valeur absolue détermine la saturation.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex conjugate of a complex number is It is found by changing the sign of the imaginary part of the complex number.","toRaw":"Conjuguer un nombre complexe revient à changer le signe de la partie imaginaire.","from":[{"text":"The ","type":0},{"text":"complex","type":2},{"text":" conjugate of a ","type":0},{"text":"complex number","type":1},{"text":" is It is found by changing the sign of the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Conjuguer un ","type":0},{"text":"nombre complexe","type":1},{"text":" revient à changer le signe de la partie imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"We cannot say any complex number is also a real number, since a complex number has a real and an imaginary part.","toRaw":"Nous ne pouvons pas dire que tout nombre complexe est aussi un nombre réel, puisqu’un nombre complexe a une partie réelle et une partie imaginaire.","from":[{"text":"We cannot say any ","type":0},{"text":"complex number","type":1},{"text":" is also a real ","type":0},{"text":"number","type":2},{"text":", since a ","type":0},{"text":"complex number","type":1},{"text":" has a real and an imaginary part.","type":0}],"to":[{"text":"Nous ne pouvons pas dire que tout ","type":0},{"text":"nombre complexe","type":1},{"text":" est aussi un ","type":0},{"text":"nombre","type":2},{"text":" réel, puisqu’un ","type":0},{"text":"nombre complexe","type":1},{"text":" a une partie réelle et une partie imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The Hermitian conjugate of a complex number is its complex conjugate.","toRaw":"la symétrie qui à un nombre complexe associe son conjugué.","from":[{"text":"The Hermitian conjugate of a ","type":0},{"text":"complex number","type":1},{"text":" is its ","type":0},{"text":"complex","type":2},{"text":" conjugate.","type":0}],"to":[{"text":"la symétrie qui à un ","type":0},{"text":"nombre complexe","type":1},{"text":" associe son conjugué.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Hence the product of two complex numbers is a complex number.","toRaw":"c'est-à-dire le produit de deux nombres complexes est un nombre complexe dont","from":[{"text":"Hence the product of two ","type":0},{"text":"complex number","type":1},{"text":"s is a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"c'est-à-dire le produit de deux ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s est un ","type":0},{"text":"nombre complexe","type":1},{"text":" dont","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The magnitude of the complex number is normalized (106) and processed (108, 110) through a closed loop to produce an output (112) proportional to the phase of the complex number.","toRaw":"En l'occurrence, on normalise (106) la grandeur du nombre complexe, et on la traite au moyen d'une boucle fermée de façon produire une sortie (112) proportionnelle à la phase du nombre complexe.","from":[{"text":"The magnitude of the ","type":0},{"text":"complex number","type":1},{"text":" is normalized (106) and processed (108, 110) through a closed loop to produce an output (112) proportional to the phase of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"En l'occurrence, on normalise (106) la grandeur du ","type":0},{"text":"nombre complexe","type":1},{"text":", et on la traite au moyen d'une boucle fermée de façon produire une sortie (112) proportionnelle à la phase du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"6826","d0":"13","collapsed":false,"source":"wipo.int"},{"fromRaw":"imaginary-number is the imaginary part of the complex number.","toRaw":"imaginary-number est la partie imaginaire du nombre complexe.","from":[{"text":"imaginary-","type":0},{"text":"number","type":2},{"text":" is the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"imaginary-number est la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Examples of Multiplying a Complex Number by a Real Number","toRaw":"f) Multiplication d’un nombre complexe par un nombre réel","from":[{"text":"Examples of Multiplying a ","type":0},{"text":"Complex Number","type":1},{"text":" by a Real ","type":0},{"text":"Number","type":2}],"to":[{"text":"f) Multiplication d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" par un ","type":0},{"text":"nombre","type":2},{"text":" réel","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"That is, every real number is also a complex number.","toRaw":"Nous avons donc montré que tout nombre réel est aussi un nombre complexe.","from":[{"text":"That is, every real ","type":0},{"text":"number","type":2},{"text":" is also a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Nous avons donc montré que tout ","type":0},{"text":"nombre","type":2},{"text":" réel est aussi un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"real-number is the real part of the complex number.","toRaw":"real-number est la partie réelle du nombre complexe.","from":[{"text":"real-","type":0},{"text":"number","type":2},{"text":" is the real part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"real-number est la partie réelle du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Notice that the number 2 is a complex number and a real number.","toRaw":"Proposition 2 : Le nombre complexe est un nombre réel.","from":[{"text":"Notice that the ","type":0},{"text":"number","type":2},{"text":" 2 is a ","type":0},{"text":"complex number","type":1},{"text":" and a real ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Proposition 2 : Le ","type":0},{"text":"nombre complexe","type":1},{"text":" est un ","type":0},{"text":"nombre","type":2},{"text":" réel.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The invention concerns a complex number multiplier receiving the binary number A, B, C and D complementarily coded in pairs so as to perform the complex multiplication (A+jB)*(C+jD).","toRaw":"Multiplieur de nombres complexes recevant les nombres binaires A, B, C et D, codés en complément à deux afin de réaliser la multiplication complexe (A+jB)*(C+jD).","from":[{"text":"The invention concerns a ","type":0},{"text":"complex number","type":1},{"text":" multiplier receiving the binary ","type":0},{"text":"number","type":2},{"text":" A, B, C and D complementarily coded in pairs so as to perform the ","type":0},{"text":"complex","type":2},{"text":" multiplication (A+jB)*(C+jD).","type":0}],"to":[{"text":"Multiplieur de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s recevant les ","type":0},{"text":"nombre","type":2},{"text":"s binaires A, B, C et D, codés en complément à deux afin de réaliser la multiplication ","type":0},{"text":"complexe","type":2},{"text":" (A+jB)*(C+jD).","type":0}],"d2":"3236","d0":"4","collapsed":false,"source":"wipo.int"},{"fromRaw":"Represent complex numbers on the complex plane in rectangular form, and explain why the rectangular form of a given complex number represents the same number.","toRaw":"Représenter les nombres complexes sur le plan complexe de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un nombre complexe représentent le même nombre.","from":[{"text":"Represent ","type":0},{"text":"complex number","type":1},{"text":"s on the ","type":0},{"text":"complex","type":2},{"text":" plane in rectangular form, and explain why the rectangular form of a given ","type":0},{"text":"complex number","type":1},{"text":" represents the same ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Représenter les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sur le plan ","type":0},{"text":"complexe","type":2},{"text":" de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" représentent le même ","type":0},{"text":"nombre","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"There are no complex literals (complex numbers can be formed by adding a real number and an imaginary number).","toRaw":"Il n'y a pas de littéraux complexes (les nombres complexes peuvent être construits en ajoutant un nombre réel et un nombre imaginaire).","from":[{"text":"There are no ","type":0},{"text":"complex","type":2},{"text":" literals (","type":0},{"text":"complex number","type":1},{"text":"s can be formed by adding a real ","type":0},{"text":"number","type":2},{"text":" and an imaginary ","type":0},{"text":"number","type":2},{"text":").","type":0}],"to":[{"text":"Il n'y a pas de littéraux ","type":0},{"text":"complexe","type":2},{"text":"s (les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s peuvent être construits en ajoutant un ","type":0},{"text":"nombre","type":2},{"text":" réel et un ","type":0},{"text":"nombre","type":2},{"text":" imaginaire).","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Utilize COMPLEX to convert imaginary and real coefficients within the complex number.","toRaw":"Convertit des coefficients réels et imaginaires en un nombre complexe.","from":[{"text":"Utilize ","type":0},{"text":"COMPLEX","type":2},{"text":" to convert imaginary and real coefficients within the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Convertit des coefficients réels et imaginaires en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Utilize COMPLEX to convert imaginary and real coefficients within the complex number.","toRaw":"Utilisez la fonction COMPLEXE pour convertir des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"Utilize ","type":0},{"text":"COMPLEX","type":2},{"text":" to convert imaginary and real coefficients within the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Utilisez la fonction ","type":0},{"text":"COMPLEXE","type":2},{"text":" pour convertir des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"An illustration of a complex number z plotted on the complex plane","toRaw":"Une illustration du nombre complexe z placé dans le plan complexe","from":[{"text":"An illustration of a ","type":0},{"text":"complex number","type":1},{"text":" z plotted on the ","type":0},{"text":"complex","type":2},{"text":" plane","type":0}],"to":[{"text":"Une illustration du ","type":0},{"text":"nombre complexe","type":1},{"text":" z placé dans le plan ","type":0},{"text":"complexe","type":2}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"There are no complex literals (complex numbers can be formed by adding a real number and an imaginary number).","toRaw":"Il n’y a pas de littéraux complexes (les nombres complexes peuvent être construits en ajoutant un nombre réel et un nombre imaginaire).","from":[{"text":"There are no ","type":0},{"text":"complex","type":2},{"text":" literals (","type":0},{"text":"complex number","type":1},{"text":"s can be formed by adding a real ","type":0},{"text":"number","type":2},{"text":" and an imaginary ","type":0},{"text":"number","type":2},{"text":").","type":0}],"to":[{"text":"Il n’y a pas de littéraux ","type":0},{"text":"complexe","type":2},{"text":"s (les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s peuvent être construits en ajoutant un ","type":0},{"text":"nombre","type":2},{"text":" réel et un ","type":0},{"text":"nombre","type":2},{"text":" imaginaire).","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Closure Property: The sum of two complex numbers is a complex number.","toRaw":"Propriétés : la somme de deux nombres constructibles est un nombre constructible.","from":[{"text":"Closure Property: The sum of two ","type":0},{"text":"complex number","type":1},{"text":"s is a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Propriétés : la somme de deux ","type":0},{"text":"nombre","type":2},{"text":"s constructibles est un ","type":0},{"text":"nombre","type":2},{"text":" constructible.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"In mathematics, a complex number is a number of the form","toRaw":"En mathématiques, un nombre complexe est un nombre qui a une partie","from":[{"text":"In mathematics, a ","type":0},{"text":"complex number","type":1},{"text":" is a ","type":0},{"text":"number","type":2},{"text":" of the form","type":0}],"to":[{"text":"En mathématiques, un ","type":0},{"text":"nombre complexe","type":1},{"text":" est un ","type":0},{"text":"nombre","type":2},{"text":" qui a une partie","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Multiplication or division of a complex number with a real number","toRaw":"Produit ou quotient d'un nombre complexe par un nombre réel","from":[{"text":"Multiplication or division of a ","type":0},{"text":"complex number","type":1},{"text":" with a real ","type":0},{"text":"number","type":2}],"to":[{"text":"Produit ou quotient d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" par un ","type":0},{"text":"nombre","type":2},{"text":" réel","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Addition or subtraction of a complex number with a real number","toRaw":"Produit ou quotient d'un nombre complexe par un nombre réel","from":[{"text":"Addition or subtraction of a ","type":0},{"text":"complex number","type":1},{"text":" with a real ","type":0},{"text":"number","type":2}],"to":[{"text":"Produit ou quotient d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" par un ","type":0},{"text":"nombre","type":2},{"text":" réel","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Infinity is certainly not a real number, nor a complex number.","toRaw":"L'infini est bel et bien un nombre, pas réel cependant, ni complexe.","from":[{"text":"Infinity is certainly not a real ","type":0},{"text":"number","type":2},{"text":", nor a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"L'infini est bel et bien un ","type":0},{"text":"nombre","type":2},{"text":", pas réel cependant, ni ","type":0},{"text":"complexe","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is the sum of a real number and an imaginary number.","toRaw":"L'adresse d'un nombre complexe est la somme d'un nombre réel et d'un nombre imaginaire.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is the sum of a real ","type":0},{"text":"number","type":2},{"text":" and an imaginary ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"L'adresse d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" est la somme d'un ","type":0},{"text":"nombre","type":2},{"text":" réel et d'un ","type":0},{"text":"nombre","type":2},{"text":" imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is a number with real and imaginary components.","toRaw":"Un nombre complexe est composé de composants réels et imaginaires.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is a ","type":0},{"text":"number","type":2},{"text":" with real and imaginary components.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" est composé de composants réels et imaginaires.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex number calculator is also called an imaginary number calculator.","toRaw":"La calculatrice de nombre complexe est également appelée calculatrice de nombre imaginaire ou encore calculateur de nombre complexe.","from":[{"text":"The ","type":0},{"text":"complex number","type":1},{"text":" calculator is also called an imaginary ","type":0},{"text":"number","type":2},{"text":" calculator.","type":0}],"to":[{"text":"La calculatrice de ","type":0},{"text":"nombre complexe","type":1},{"text":" est également appelée calculatrice de ","type":0},{"text":"nombre","type":2},{"text":" imaginaire ou encore calculateur de ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A conjugate of a complex number is the other complex number with the same real part and opposite imaginary part","toRaw":"Le conjugué d'un nombre complexe est le nombre complexe qui a la même partie réelle et la partie imaginaire opposée.","from":[{"text":"A conjugate of a ","type":0},{"text":"complex number","type":1},{"text":" is the other ","type":0},{"text":"complex number","type":1},{"text":" with the same real part and opposite imaginary part","type":0}],"to":[{"text":"Le conjugué d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" est le ","type":0},{"text":"nombre complexe","type":1},{"text":" qui a la même partie réelle et la partie imaginaire opposée.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Convert the complex number into trigonometric form.","toRaw":"Écrire un nombre complexe sous forme trigonométrique.","from":[{"text":"Convert the ","type":0},{"text":"complex number","type":1},{"text":" into trigonometric form.","type":0}],"to":[{"text":"Écrire un ","type":0},{"text":"nombre complexe","type":1},{"text":" sous forme trigonométrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Convert the complex number into trigonometric form.","toRaw":"Écrire un nombre complexe sous forme trigonométrique.","from":[{"text":"Convert the ","type":0},{"text":"complex number","type":1},{"text":" into trigonometric form.","type":0}],"to":[{"text":"Écrire un ","type":0},{"text":"nombre complexe","type":1},{"text":" sous forme trigonométrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Convert the complex number into trigonometric form.","toRaw":"c. Ecrire le nombre complexe sous forme trigonométrique.","from":[{"text":"Convert the ","type":0},{"text":"complex number","type":1},{"text":" into trigonometric form.","type":0}],"to":[{"text":"c. Ecrire le ","type":0},{"text":"nombre complexe","type":1},{"text":" sous forme trigonométrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Convert the complex number into trigonometric form.","toRaw":"c) Écrire le nombre complexe sous forme trigonométrique.","from":[{"text":"Convert the ","type":0},{"text":"complex number","type":1},{"text":" into trigonometric form.","type":0}],"to":[{"text":"c) Écrire le ","type":0},{"text":"nombre complexe","type":1},{"text":" sous forme trigonométrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"COMPLEX - used to turn the real and imaginary coefficients into a complex number","toRaw":"Convertit des coefficients réels et imaginaires en un nombre complexe.","from":[{"text":"","type":0},{"text":"COMPLEX","type":2},{"text":" - used to turn the real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Convertit des coefficients réels et imaginaires en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Their Complex Number Computer, completed 8 January , was able to calculate complex numbers.","toRaw":"Leur complexe Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des nombres complexes.","from":[{"text":"Their ","type":0},{"text":"Complex Number","type":1},{"text":" Computer, completed 8 January , was able to calculate ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Leur ","type":0},{"text":"complexe","type":2},{"text":" Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"If either argument is a complex number, the other is converted to complex;","toRaw":"Si l’un des deux arguments est du type nombre complexe, l’autre est converti en nombre complexe ;","from":[{"text":"If either argument is a ","type":0},{"text":"complex number","type":1},{"text":", the other is converted to ","type":0},{"text":"complex","type":2},{"text":";","type":0}],"to":[{"text":"Si l’un des deux arguments est du type ","type":0},{"text":"nombre complexe","type":1},{"text":", l’autre est converti en ","type":0},{"text":"nombre complexe","type":1},{"text":" ;","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"If either argument is a complex number, the other is converted to complex;","toRaw":"Si l'un des deux arguments est du type nombre complexe, l'autre est converti en nombre complexe ;","from":[{"text":"If either argument is a ","type":0},{"text":"complex number","type":1},{"text":", the other is converted to ","type":0},{"text":"complex","type":2},{"text":";","type":0}],"to":[{"text":"Si l'un des deux arguments est du type ","type":0},{"text":"nombre complexe","type":1},{"text":", l'autre est converti en ","type":0},{"text":"nombre complexe","type":1},{"text":" ;","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"COMPLEX - used to turn the real and imaginary coefficients into a complex number","toRaw":"Ingénierie : convertit des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"","type":0},{"text":"COMPLEX","type":2},{"text":" - used to turn the real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Ingénierie : convertit des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is merely the sum of a real number and an imaginary number.","toRaw":"L'adresse d'un nombre complexe est la somme d'un nombre réel et d'un nombre imaginaire.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is merely the sum of a real ","type":0},{"text":"number","type":2},{"text":" and an imaginary ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"L'adresse d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" est la somme d'un ","type":0},{"text":"nombre","type":2},{"text":" réel et d'un ","type":0},{"text":"nombre","type":2},{"text":" imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"How does the real number system compare to the complex number system?","toRaw":"Que représente le nombre complexe par rapport au nombre complexe ?","from":[{"text":"How does the real ","type":0},{"text":"number","type":2},{"text":" system compare to the ","type":0},{"text":"complex number","type":1},{"text":" system?","type":0}],"to":[{"text":"Que représente le ","type":0},{"text":"nombre complexe","type":1},{"text":" par rapport au ","type":0},{"text":"nombre complexe","type":1},{"text":" ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"real number a, we identify a with the complex number (a, 0).","toRaw":"– Pour tout nombre réel a , nous conviendrons d’identifier le nombre complexe (a, 0) avec le réel a .","from":[{"text":"real ","type":0},{"text":"number","type":2},{"text":" a, we identify a with the ","type":0},{"text":"complex number","type":1},{"text":" (a, 0).","type":0}],"to":[{"text":"– Pour tout ","type":0},{"text":"nombre","type":2},{"text":" réel a , nous conviendrons d’identifier le ","type":0},{"text":"nombre complexe","type":1},{"text":" (a, 0) avec le réel a .","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"This makes sense because every real number is also a complex number.","toRaw":"L’implication est vraie, puisque tout nombre entier est aussi un nombre réel.","from":[{"text":"This makes sense because every real ","type":0},{"text":"number","type":2},{"text":" is also a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"L’implication est vraie, puisque tout ","type":0},{"text":"nombre","type":2},{"text":" entier est aussi un ","type":0},{"text":"nombre","type":2},{"text":" réel.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"For example, if we are asked to arithmetically combine an ordinary number with a complex number, we can view the ordinary number as a complex number whose imaginary part is zero.","toRaw":"Par exemple, si on nous demande de combiner arithmétiquement un nombre rationnel avec un nombre complexe, nous pouvons considérer le nombre rationnel comme un nombre complexe dont la partie imaginaire est nulle.","from":[{"text":"For example, if we are asked to arithmetically combine an ordinary ","type":0},{"text":"number","type":2},{"text":" with a ","type":0},{"text":"complex number","type":1},{"text":", we can view the ordinary ","type":0},{"text":"number","type":2},{"text":" as a ","type":0},{"text":"complex number","type":1},{"text":" whose imaginary part is zero.","type":0}],"to":[{"text":"Par exemple, si on nous demande de combiner arithmétiquement un ","type":0},{"text":"nombre","type":2},{"text":" rationnel avec un ","type":0},{"text":"nombre complexe","type":1},{"text":", nous pouvons considérer le ","type":0},{"text":"nombre","type":2},{"text":" rationnel comme un ","type":0},{"text":"nombre complexe","type":1},{"text":" dont la partie imaginaire est nulle.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"For example, if we are asked to arithmetically combine a rational number with a complex number, we can view the rational number as a complex number whose imaginary part is zero.","toRaw":"Par exemple, si on nous demande de combiner arithmétiquement un nombre rationnel avec un nombre complexe, nous pouvons considérer le nombre rationnel comme un nombre complexe dont la partie imaginaire est nulle.","from":[{"text":"For example, if we are asked to arithmetically combine a rational ","type":0},{"text":"number","type":2},{"text":" with a ","type":0},{"text":"complex number","type":1},{"text":", we can view the rational ","type":0},{"text":"number","type":2},{"text":" as a ","type":0},{"text":"complex number","type":1},{"text":" whose imaginary part is zero.","type":0}],"to":[{"text":"Par exemple, si on nous demande de combiner arithmétiquement un ","type":0},{"text":"nombre","type":2},{"text":" rationnel avec un ","type":0},{"text":"nombre complexe","type":1},{"text":", nous pouvons considérer le ","type":0},{"text":"nombre","type":2},{"text":" rationnel comme un ","type":0},{"text":"nombre complexe","type":1},{"text":" dont la partie imaginaire est nulle.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"An illustration of a complex number plotted on the complex plane using Euler's formula","toRaw":"Une illustration d'un nombre complexe placé dans le plan complexe en utilisant la formule d'Euler","from":[{"text":"An illustration of a ","type":0},{"text":"complex number","type":1},{"text":" plotted on the ","type":0},{"text":"complex","type":2},{"text":" plane using Euler's formula","type":0}],"to":[{"text":"Une illustration d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" placé dans le plan ","type":0},{"text":"complexe","type":2},{"text":" en utilisant la formule d'Euler","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Use the COMPLEX function to create a complex number from real and imaginary parts.","toRaw":"Utilisez la fonction COMPLEXE pour convertir des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"Use the ","type":0},{"text":"COMPLEX","type":2},{"text":" function to create a ","type":0},{"text":"complex number","type":1},{"text":" from real and imaginary parts.","type":0}],"to":[{"text":"Utilisez la fonction ","type":0},{"text":"COMPLEXE","type":2},{"text":" pour convertir des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Their Complex Number Computer, completed 8 January 1940, was able to calculate complex numbers.","toRaw":"Leur complexe Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des nombres complexes.","from":[{"text":"Their ","type":0},{"text":"Complex Number","type":1},{"text":" Computer, completed 8 January 1940, was able to calculate ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Leur ","type":0},{"text":"complexe","type":2},{"text":" Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Their Complex Number Calculator, completed January 8, 1940, was able to calculate complex numbers.","toRaw":"Leur complexe Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des nombres complexes.","from":[{"text":"Their ","type":0},{"text":"Complex Number","type":1},{"text":" Calculator, completed January 8, 1940, was able to calculate ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Leur ","type":0},{"text":"complexe","type":2},{"text":" Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Their Complex Number Calculator , completed January 8, 1940, was able to calculate complex numbers.","toRaw":"Leur complexe Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des nombres complexes.","from":[{"text":"Their ","type":0},{"text":"Complex Number","type":1},{"text":" Calculator , completed January 8, 1940, was able to calculate ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Leur ","type":0},{"text":"complexe","type":2},{"text":" Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Their Complex Number Computer, completed January 8, 1940, was able to calculate complex numbers.","toRaw":"Leur complexe Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des nombres complexes.","from":[{"text":"Their ","type":0},{"text":"Complex Number","type":1},{"text":" Computer, completed January 8, 1940, was able to calculate ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Leur ","type":0},{"text":"complexe","type":2},{"text":" Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"For a real number , , so for any complex number, there is a real number such that .","toRaw":"Déterminer des nombres réels et tels que pour tout nombre complexe , on ait .","from":[{"text":"For a real ","type":0},{"text":"number","type":2},{"text":" , , so for any ","type":0},{"text":"complex number","type":1},{"text":", there is a real ","type":0},{"text":"number","type":2},{"text":" such that .","type":0}],"to":[{"text":"Déterminer des ","type":0},{"text":"nombre","type":2},{"text":"s réels et tels que pour tout ","type":0},{"text":"nombre complexe","type":1},{"text":" , on ait .","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"For example, if LabVIEW coerces a complex number to a non-complex number, you might lose the imaginary part of the data.","toRaw":"Par exemple, si LabVIEW contraint un nombre complexe à un nombre non complexe, vous risquez de perdre la partie imaginaire des données.","from":[{"text":"For example, if LabVIEW coerces a ","type":0},{"text":"complex number","type":1},{"text":" to a non-","type":0},{"text":"complex number","type":1},{"text":", you might lose the imaginary part of the data.","type":0}],"to":[{"text":"Par exemple, si LabVIEW contraint un ","type":0},{"text":"nombre complexe","type":1},{"text":" à un ","type":0},{"text":"nombre","type":2},{"text":" non ","type":0},{"text":"complexe","type":2},{"text":", vous risquez de perdre la partie imaginaire des données.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Trigonometric and exponential forms of a complex number.","toRaw":"2 Formes trigonométrique et exponentielle d un nombre complexe.","from":[{"text":"Trigonometric and exponential forms of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"2 Formes trigonométrique et exponentielle d un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Graph complex numbers on the complex plane in rectangular and polar form and explain why the rectangular and polar forms of a given complex number represent the same number.","toRaw":"Représenter les nombres complexes sur le plan complexe de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un nombre complexe représentent le même nombre.","from":[{"text":"Graph ","type":0},{"text":"complex number","type":1},{"text":"s on the ","type":0},{"text":"complex","type":2},{"text":" plane in rectangular and polar form and explain why the rectangular and polar forms of a given ","type":0},{"text":"complex number","type":1},{"text":" represent the same ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Représenter les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sur le plan ","type":0},{"text":"complexe","type":2},{"text":" de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" représentent le même ","type":0},{"text":"nombre","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Trigonometric and exponential forms of a complex number.","toRaw":"Forme trigonométrique et forme exponentielle d’un nombre complexe.","from":[{"text":"Trigonometric and exponential forms of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Forme trigonométrique et forme exponentielle d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Every nonzero complex number has three cube roots.","toRaw":"Un complexe non nul possède trois racines cubiques.","from":[{"text":"Every nonzero ","type":0},{"text":"complex number","type":1},{"text":" has three cube roots.","type":0}],"to":[{"text":"Un ","type":0},{"text":"complexe","type":2},{"text":" non nul possède trois racines cubiques.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"returning the imaginary part of the complex number.","toRaw":"retourne le coefficient de la partie imaginaire du nombre complexe .","from":[{"text":"returning the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"retourne le coefficient de la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":" .","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Absolute value (or modulus) of the complex number.","toRaw":"Renvoie la valeur absolue (module) d'un nombre complexe.","from":[{"text":"Absolute value (or modulus) of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Renvoie la valeur absolue (module) d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Returns the imaginary coefficient of a complex number","toRaw":"Renvoie le coefficient imaginaire d’un nombre complexe.","from":[{"text":"Returns the imaginary coefficient of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Renvoie le coefficient imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The number 31 is another complex number consisting of twins 1 and 3.","toRaw":"Le numéro d'anniversaire 31 est complexe, il est composé des jumeaux 1 et 3.","from":[{"text":"The ","type":0},{"text":"number","type":2},{"text":" 31 is another ","type":0},{"text":"complex number","type":1},{"text":" consisting of twins 1 and 3.","type":0}],"to":[{"text":"Le numéro d'anniversaire 31 est ","type":0},{"text":"complexe","type":2},{"text":", il est composé des jumeaux 1 et 3.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"So we have shown that any real number is also a complex number.","toRaw":"Nous avons donc montré que tout nombre réel est aussi un nombre complexe.","from":[{"text":"So we have shown that any real ","type":0},{"text":"number","type":2},{"text":" is also a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Nous avons donc montré que tout ","type":0},{"text":"nombre","type":2},{"text":" réel est aussi un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"2 is a real number, but it’s a complex number when b = 0.","toRaw":"Cela signifie que tout nombre réel est un nombre complexe lorsque b = 0.","from":[{"text":"2 is a real ","type":0},{"text":"number","type":2},{"text":", but it’s a ","type":0},{"text":"complex number","type":1},{"text":" when b = 0.","type":0}],"to":[{"text":"Cela signifie que tout ","type":0},{"text":"nombre","type":2},{"text":" réel est un ","type":0},{"text":"nombre complexe","type":1},{"text":" lorsque b = 0.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The product of a complex number and its conjugate is a real number.","toRaw":"Le produit d’un complexe et de son conjugué est-il toujours un nombre réel ?","from":[{"text":"The product of a ","type":0},{"text":"complex number","type":1},{"text":" and its conjugate is a real ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Le produit d’un ","type":0},{"text":"complexe","type":2},{"text":" et de son conjugué est-il toujours un ","type":0},{"text":"nombre","type":2},{"text":" réel ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"This whole expression is therefore a complex number","toRaw":"Donc toute cette expression est bien de type complexe,","from":[{"text":"This whole expression is therefore a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Donc toute cette expression est bien de type ","type":0},{"text":"complexe","type":2},{"text":",","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The product of a complex number and its conjugate is a real number.","toRaw":"Le produit d'un nombre complexe et de son conjugué est toujours un nombre réel.","from":[{"text":"The product of a ","type":0},{"text":"complex number","type":1},{"text":" and its conjugate is a real ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Le produit d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" et de son conjugué est toujours un ","type":0},{"text":"nombre","type":2},{"text":" réel.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Absolute value (or modulus) of the complex number.","toRaw":"Renvoie la valeur absolue (module) d’un nombre complexe.","from":[{"text":"Absolute value (or modulus) of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Renvoie la valeur absolue (module) d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":" Calculating the nth roots of a complex number.","toRaw":"calculer les racines carrées d'un nombre complexe.","from":[{"text":" Calculating the nth roots of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"calculer les racines carrées d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Complex numbers are the sum of a real number and an imaginary number.","toRaw":"L'adresse d'un nombre complexe est la somme d'un nombre réel et d'un nombre imaginaire.","from":[{"text":"","type":0},{"text":"Complex number","type":1},{"text":"s are the sum of a real ","type":0},{"text":"number","type":2},{"text":" and an imaginary ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"L'adresse d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" est la somme d'un ","type":0},{"text":"nombre","type":2},{"text":" réel et d'un ","type":0},{"text":"nombre","type":2},{"text":" imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Trigonometric and exponential forms of a complex number.","toRaw":"Formes cartésienne, trigonométrique et exponentielle d'un nombre complexe.","from":[{"text":"Trigonometric and exponential forms of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Formes cartésienne, trigonométrique et exponentielle d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"returning the imaginary part of the complex number.","toRaw":"Retourne la partie imaginaire du nombre complexe.","from":[{"text":"returning the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Retourne la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The Number , the imaginary unit of complex numbers","toRaw":"Le nombre i, l unité imaginaire des nombres complexes.","from":[{"text":"The ","type":0},{"text":"Number","type":2},{"text":" , the imaginary unit of ","type":0},{"text":"complex number","type":1},{"text":"s","type":0}],"to":[{"text":"Le ","type":0},{"text":"nombre","type":2},{"text":" i, l unité imaginaire des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"IMAGINARY The imaginary coefficient of a complex number.","toRaw":"Renvoie le coefficient imaginaire d’un nombre complexe.","from":[{"text":"IMAGINARY The imaginary coefficient of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Renvoie le coefficient imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A transcendental number is a real or complex number that is not algebraic.","toRaw":"Un nombre réel ou complexe est donc transcendant si et seulement si il n'est pas algébrique.","from":[{"text":"A transcendental ","type":0},{"text":"number","type":2},{"text":" is a real or ","type":0},{"text":"complex number","type":1},{"text":" that is not algebraic.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre","type":2},{"text":" réel ou ","type":0},{"text":"complexe","type":2},{"text":" est donc transcendant si et seulement si il n'est pas algébrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Returns the imaginary part of this complex number.","toRaw":"Retourne la partie imaginaire du nombre complexe.","from":[{"text":"Returns the imaginary part of this ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Retourne la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The product of a complex number and its conjugate is a real number,","toRaw":"Le produit d'un nombre complexe et de son conjugué est toujours un nombre réel.","from":[{"text":"The product of a ","type":0},{"text":"complex number","type":1},{"text":" and its conjugate is a real ","type":0},{"text":"number","type":2},{"text":",","type":0}],"to":[{"text":"Le produit d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" et de son conjugué est toujours un ","type":0},{"text":"nombre","type":2},{"text":" réel.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A Transcendental number is a real or complex number that is not algebraic.","toRaw":"Un nombre réel ou complexe est donc transcendant si et seulement si il n'est pas algébrique.","from":[{"text":"A Transcendental ","type":0},{"text":"number","type":2},{"text":" is a real or ","type":0},{"text":"complex number","type":1},{"text":" that is not algebraic.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre","type":2},{"text":" réel ou ","type":0},{"text":"complexe","type":2},{"text":" est donc transcendant si et seulement si il n'est pas algébrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The product of a complex number and its conjugate gives a real number.","toRaw":"Le produit d'un nombre complexe et de son conjugué est toujours un nombre réel.","from":[{"text":"The product of a ","type":0},{"text":"complex number","type":1},{"text":" and its conjugate gives a real ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Le produit d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" et de son conjugué est toujours un ","type":0},{"text":"nombre","type":2},{"text":" réel.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The product of the complex number and its conjugate is a real number!","toRaw":"Le produit d'un nombre complexe et de son conjugué est toujours un nombre réel.","from":[{"text":"The product of the ","type":0},{"text":"complex number","type":1},{"text":" and its conjugate is a real ","type":0},{"text":"number","type":2},{"text":"!","type":0}],"to":[{"text":"Le produit d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" et de son conjugué est toujours un ","type":0},{"text":"nombre","type":2},{"text":" réel.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The real and imaginary parts of the complex number correspond to the projections of the complex number displayed as a vector on the α and βaxes in the complex plane.","toRaw":"Les parties imaginaire et réelle de ce nombre complexe correspondent à la projection de ce nombre complexe, représenté sous la forme de vecteur, sur les axes α et β fixes du plan complexe.","from":[{"text":"The real and imaginary parts of the ","type":0},{"text":"complex number","type":1},{"text":" correspond to the projections of the ","type":0},{"text":"complex number","type":1},{"text":" displayed as a vector on the α and βaxes in the ","type":0},{"text":"complex","type":2},{"text":" plane.","type":0}],"to":[{"text":"Les parties imaginaire et réelle de ce ","type":0},{"text":"nombre complexe","type":1},{"text":" correspondent à la projection de ce ","type":0},{"text":"nombre complexe","type":1},{"text":", représenté sous la forme de vecteur, sur les axes α et β fixes du plan ","type":0},{"text":"complexe","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The real and imaginary parts of the complex number correspond to the projections of the complex number displayed as a vector on the α and βaxes in the complex plane.","toRaw":"Ses parties réelle et imaginaire correspondent aux projections de ce nombre complexe, appelé vecteur de champ et représenté graphiquement comme tel, sur les axes, α et β dans le plan complexe.","from":[{"text":"The real and imaginary parts of the ","type":0},{"text":"complex number","type":1},{"text":" correspond to the projections of the ","type":0},{"text":"complex number","type":1},{"text":" displayed as a vector on the α and βaxes in the ","type":0},{"text":"complex","type":2},{"text":" plane.","type":0}],"to":[{"text":"Ses parties réelle et imaginaire correspondent aux projections de ce ","type":0},{"text":"nombre complexe","type":1},{"text":", appelé vecteur de champ et représenté graphiquement comme tel, sur les axes, α et β dans le plan ","type":0},{"text":"complexe","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is one that expresses the sum between a real number and an imaginary number.","toRaw":"Un nombre complexe est celui qui exprime la somme entre un nombre réel et un nombre imaginaire.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is one that expresses the sum between a real ","type":0},{"text":"number","type":2},{"text":" and an imaginary ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" est celui qui exprime la somme entre un ","type":0},{"text":"nombre","type":2},{"text":" réel et un ","type":0},{"text":"nombre","type":2},{"text":" imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex conjugate of the complex number z = x + yi is given by x - yi.","toRaw":"Le conjugué complexe du nombre complexe z = x + yi est donné par x − yi.","from":[{"text":"The ","type":0},{"text":"complex","type":2},{"text":" conjugate of the ","type":0},{"text":"complex number","type":1},{"text":" z = x + yi is given by x - yi.","type":0}],"to":[{"text":"Le conjugué ","type":0},{"text":"complexe","type":2},{"text":" du ","type":0},{"text":"nombre complexe","type":1},{"text":" z = x + yi est donné par x − yi.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex conjugate of the complex number z = x + yi is given by x − yi.","toRaw":"Le conjugué complexe du nombre complexe z = x + yi est donné par x − yi.","from":[{"text":"The ","type":0},{"text":"complex","type":2},{"text":" conjugate of the ","type":0},{"text":"complex number","type":1},{"text":" z = x + yi is given by x − yi.","type":0}],"to":[{"text":"Le conjugué ","type":0},{"text":"complexe","type":2},{"text":" du ","type":0},{"text":"nombre complexe","type":1},{"text":" z = x + yi est donné par x − yi.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"If a is a complex number, this function converts the real and imaginary parts of the number separately and then returns a complex number that consists of those parts.","toRaw":"Si a est un nombre complexe, cette fonction convertit séparément les parties réelle et imaginaire du nombre puis renvoie un nombre complexe composé de ces parties.","from":[{"text":"If a is a ","type":0},{"text":"complex number","type":1},{"text":", this function converts the real and imaginary parts of the ","type":0},{"text":"number","type":2},{"text":" separately and then returns a ","type":0},{"text":"complex number","type":1},{"text":" that consists of those parts.","type":0}],"to":[{"text":"Si a est un ","type":0},{"text":"nombre complexe","type":1},{"text":", cette fonction convertit séparément les parties réelle et imaginaire du ","type":0},{"text":"nombre","type":2},{"text":" puis renvoie un ","type":0},{"text":"nombre complexe","type":1},{"text":" composé de ces parties.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"• The real number b is called the imaginary part of the complex number.","toRaw":"Le nombre réel b s’appelle la partie imaginaire du nombre complexe z.","from":[{"text":"• The real ","type":0},{"text":"number","type":2},{"text":" b is called the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Le ","type":0},{"text":"nombre","type":2},{"text":" réel b s’appelle la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":" z.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The real number a is called the part of the complex number a + bi.","toRaw":"Le nombre réel a est la partie réelle de a + bi.","from":[{"text":"The real ","type":0},{"text":"number","type":2},{"text":" a is called the part of the ","type":0},{"text":"complex number","type":1},{"text":" a + bi.","type":0}],"to":[{"text":"Le ","type":0},{"text":"nombre","type":2},{"text":" réel a est la partie réelle de a + bi.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is the result of adding a real and an imaginary number.","toRaw":"Rappelez-vous qu’un nombre complexe est le résultat de l’addition d’un nombre réel et d’un autre imaginaire.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is the result of adding a real and an imaginary ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Rappelez-vous qu’un ","type":0},{"text":"nombre complexe","type":1},{"text":" est le résultat de l’addition d’un ","type":0},{"text":"nombre","type":2},{"text":" réel et d’un autre imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"It is important to note that any real number is also a complex number.","toRaw":"Nous avons donc montré que tout nombre réel est aussi un nombre complexe.","from":[{"text":"It is important to note that any real ","type":0},{"text":"number","type":2},{"text":" is also a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Nous avons donc montré que tout ","type":0},{"text":"nombre","type":2},{"text":" réel est aussi un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is the result of adding a real and an imaginary number.","toRaw":"Un nombre complexe est le résultat de l’addition d’un nombre réel et d’un nombre imaginaire.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is the result of adding a real and an imaginary ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" est le résultat de l’addition d’un ","type":0},{"text":"nombre","type":2},{"text":" réel et d’un ","type":0},{"text":"nombre","type":2},{"text":" imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"• The real number b is called the imaginary part of the complex number.","toRaw":"Le nombre réel b s'appelle la partie imaginaire du nombre complexe z.","from":[{"text":"• The real ","type":0},{"text":"number","type":2},{"text":" b is called the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Le ","type":0},{"text":"nombre","type":2},{"text":" réel b s'appelle la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":" z.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Figure \\(\\PageIndex{1}\\): Trigonometric form of a complex number.","toRaw":"[Math 8] - Forme trigonométrique d'un nombre complexe.","from":[{"text":"Figure \\(\\PageIndex{1}\\): Trigonometric form of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"[Math 8] - Forme trigonométrique d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"First of all, as the return type here, we have chosen to return a complex number instead of a reference to a complex number.","toRaw":"Nous avons ici défini comme type de retour une référence sur un nombre complexe,","from":[{"text":"First of all, as the return type here, we have chosen to return a ","type":0},{"text":"complex number","type":1},{"text":" instead of a reference to a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Nous avons ici défini comme type de retour une référence sur un ","type":0},{"text":"nombre complexe","type":1},{"text":",","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"How to calculate the conjugate of a complex number?","toRaw":"Comment calculer le conjugué d'un nombre complexe ?","from":[{"text":"How to calculate the conjugate of a ","type":0},{"text":"complex number","type":1},{"text":"?","type":0}],"to":[{"text":"Comment calculer le conjugué d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"clog, clogf, clogl - natural logarithm of a complex number","toRaw":"clog, clogf, clogl - Logarithmes népériens de nombres complexes","from":[{"text":"clog, clogf, clogl - natural logarithm of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"clog, clogf, clogl - Logarithmes népériens de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Complex Number Calculator Precision 36 is programmed in C#.","toRaw":"La précision 45 de calculatrice de nombre complexe est programmée dans C#.","from":[{"text":"","type":0},{"text":"Complex Number","type":1},{"text":" Calculator Precision 36 is programmed in C#.","type":0}],"to":[{"text":"La précision 45 de calculatrice de ","type":0},{"text":"nombre complexe","type":1},{"text":" est programmée dans C#.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"real: Returns the real component of a complex number.","toRaw":"Fonction : Renvoie la partie réelle d’un nombre complexe.","from":[{"text":"real: Returns the real component of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Fonction : Renvoie la partie réelle d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"clog, clogf, clogl − natural logarithm of a complex number","toRaw":"clog, clogf, clogl - Logarithmes népériens de nombres complexes","from":[{"text":"clog, clogf, clogl − natural logarithm of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"clog, clogf, clogl - Logarithmes népériens de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Converts real and imaginary coefficients into a complex number","toRaw":"Convertit des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Convertit des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"How to calculate the argument of a complex number?","toRaw":"Comment calculer l'argument d'un nombre complexe ?","from":[{"text":"How to calculate the argument of a ","type":0},{"text":"complex number","type":1},{"text":"?","type":0}],"to":[{"text":"Comment calculer l'argument d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"movement represents the imaginary part of the complex number.","toRaw":"imag Retourne la partie imaginaire du nombre complexe.","from":[{"text":"movement represents the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"imag Retourne la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Converts real and imaginary coefficients into a complex number","toRaw":"Ingénierie : convertit des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Ingénierie : convertit des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"real: Returns the real component of a complex number.","toRaw":"real Retourne la partie réelle du nombre complexe.","from":[{"text":"real: Returns the real component of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"real Retourne la partie réelle du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Clearly, the return type is thus a complex number.","toRaw":"Donc le type de retour ici, clairement, est un nombre complexe.","from":[{"text":"Clearly, the return type is thus a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Donc le type de retour ici, clairement, est un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Identify the imaginary part of the following complex number:","toRaw":"Identifier la partie réelle et la partie imaginaire du nombre complexe suivant :","from":[{"text":"Identify the imaginary part of the following ","type":0},{"text":"complex number","type":1},{"text":":","type":0}],"to":[{"text":"Identifier la partie réelle et la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":" suivant :","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Figure \\(\\PageIndex{1}\\): Trigonometric form of a complex number.","toRaw":"» [Math 8] - Forme trigonométrique d'un nombre complexe.","from":[{"text":"Figure \\(\\PageIndex{1}\\): Trigonometric form of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"» [Math 8] - Forme trigonométrique d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Converts real and imaginary coefficients into a complex number.","toRaw":"Convertit des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Convertit des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A quaternion is an extension of the complex number.","toRaw":"Les quaternions sont une généralisation des nombres complexes.","from":[{"text":"A quaternion is an extension of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Les quaternions sont une généralisation des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number for which you want the conjugate.","toRaw":"Un nombre complexe pour lequel vous souhaitez obtenir la cosécante.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" for which you want the conjugate.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" pour lequel vous souhaitez obtenir la cosécante.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The Trigonometric or Polar Form of a Complex Number","toRaw":"Forme polaire ou trigonométrique d'un nombre complexe.","from":[{"text":"The Trigonometric or Polar Form of a ","type":0},{"text":"Complex Number","type":1}],"to":[{"text":"Forme polaire ou trigonométrique d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Every complex number can be written uniquely as a sum of a real number and an imaginary number.","toRaw":"L'ensemble des nombres complexes est l'ensemble des nombres qui peuvent s'écrire sous la forme de la somme d'un nombre réel et un nombre imaginaire.","from":[{"text":"Every ","type":0},{"text":"complex number","type":1},{"text":" can be written uniquely as a sum of a real ","type":0},{"text":"number","type":2},{"text":" and an imaginary ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"L'ensemble des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s est l'ensemble des ","type":0},{"text":"nombre","type":2},{"text":"s qui peuvent s'écrire sous la forme de la somme d'un ","type":0},{"text":"nombre","type":2},{"text":" réel et un ","type":0},{"text":"nombre","type":2},{"text":" imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Converts real and imaginary coefficients into a complex number.","toRaw":"COMPLEXE Ingénierie Convertit des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"","type":0},{"text":"COMPLEXE","type":2},{"text":" Ingénierie Convertit des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Returns the the imaginary part of a complex number.","toRaw":"Calcule la partie imaginaire d’un nombre complexe.","from":[{"text":"Returns the the imaginary part of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Calcule la partie imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"IMAGINARY Returns the imaginary coefficient of a complex number.","toRaw":"COMPLEXE.IMAGINAIRE Ingénierie Renvoie le coefficient imaginaire d'un nombre complexe.","from":[{"text":"IMAGINARY Returns the imaginary coefficient of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"","type":0},{"text":"COMPLEXE","type":2},{"text":".IMAGINAIRE Ingénierie Renvoie le coefficient imaginaire d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Imag: Returns the imaginary part of a complex number.","toRaw":"Fonction : Renvoie la partie imaginaire d’un nombre complexe.","from":[{"text":"Imag: Returns the imaginary part of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Fonction : Renvoie la partie imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Converts real and imaginary coefficients into a complex number","toRaw":"COMPLEXE Ingénierie Convertit des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"","type":0},{"text":"COMPLEXE","type":2},{"text":" Ingénierie Convertit des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"CTANH calculates the hyperbolic tangent of a complex number","toRaw":"Argtanhz : Calcule l'arg-tangente hyperbolique d'un nombre complexe","from":[{"text":"CTANH calculates the hyperbolic tangent of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Argtanhz : Calcule l'arg-tangente hyperbolique d'un ","type":0},{"text":"nombre complexe","type":1}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Converts real and imaginary coefficients into a complex number","toRaw":"Convertit des coefficients réels et imaginaires en un nombre complexe.","from":[{"text":"Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Convertit des coefficients réels et imaginaires en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Each complex number Fk is called a Fourier coefficient.","toRaw":"Le coefficient cn est un nombre complexe appelé coefficient de Fourier.","from":[{"text":"Each ","type":0},{"text":"complex number","type":1},{"text":" Fk is called a Fourier coefficient.","type":0}],"to":[{"text":"Le coefficient cn est un ","type":0},{"text":"nombre complexe","type":1},{"text":" appelé coefficient de Fourier.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number for which you want the conjugate.","toRaw":"Un nombre complexe pour lequel vous souhaitez obtenir la sécante.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" for which you want the conjugate.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" pour lequel vous souhaitez obtenir la sécante.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"IMREAL Returns the real coefficient of a complex number","toRaw":"COMPLEXE.REEL Ingénierie Renvoie le coefficient réel d’un nombre complexe.","from":[{"text":"IMREAL Returns the real coefficient of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"","type":0},{"text":"COMPLEXE","type":2},{"text":".REEL Ingénierie Renvoie le coefficient réel d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Returns the the imaginary part of a complex number.","toRaw":"Renvoie la partie imaginaire d'un nombre complexe.","from":[{"text":"Returns the the imaginary part of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Renvoie la partie imaginaire d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Converts real and imaginary coefficients into a complex number.","toRaw":"Convertit des coefficients réels et imaginaires en un nombre complexe.","from":[{"text":"Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Convertit des coefficients réels et imaginaires en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"IMAGINARY Returns the imaginary coefficient of a complex number.","toRaw":"COMPLEXE.IMAGINAIRE Ingénierie Renvoie le coefficient imaginaire d’un nombre complexe.","from":[{"text":"IMAGINARY Returns the imaginary coefficient of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"","type":0},{"text":"COMPLEXE","type":2},{"text":".IMAGINAIRE Ingénierie Renvoie le coefficient imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"imag: Returns the imaginary component of a complex number.","toRaw":"imag Retourne la partie imaginaire du nombre complexe.","from":[{"text":"imag: Returns the imaginary component of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"imag Retourne la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"IMAGINARY Returns the imaginary coefficient of a complex number.","toRaw":"Ingénierie : renvoie le coefficient imaginaire d’un nombre complexe.","from":[{"text":"IMAGINARY Returns the imaginary coefficient of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Ingénierie : renvoie le coefficient imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Converts real and imaginary coefficients into a complex number.","toRaw":"Ingénierie : convertit des coefficients réel et imaginaire en un nombre complexe.","from":[{"text":"Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Ingénierie : convertit des coefficients réel et imaginaire en un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"IMAGINARY Returns the imaginary coefficient of a complex number.","toRaw":"Renvoie le coefficient imaginaire d'un nombre complexe.","from":[{"text":"IMAGINARY Returns the imaginary coefficient of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Renvoie le coefficient imaginaire d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Any complex number is determined by two real numbers.","toRaw":"Un nombre complexe est uniquement déterminé par deux nombres réels.","from":[{"text":"Any ","type":0},{"text":"complex number","type":1},{"text":" is determined by two real ","type":0},{"text":"number","type":2},{"text":"s.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" est uniquement déterminé par deux ","type":0},{"text":"nombre","type":2},{"text":"s réels.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"In particular, conjugating twice gives the original complex number: .","toRaw":"Conjuguer deux fois donne le nombre complexe original.","from":[{"text":"In particular, conjugating twice gives the original ","type":0},{"text":"complex number","type":1},{"text":": .","type":0}],"to":[{"text":"Conjuguer deux fois donne le ","type":0},{"text":"nombre complexe","type":1},{"text":" original.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The real and imaginary components of a complex number.","toRaw":"Parties réelle et imaginaire d’un nombre complexe.","from":[{"text":"The real and imaginary components of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Parties réelle et imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number for which you want the argument .","toRaw":"Un nombre complexe pour lequel vous souhaitez obtenir la tangente.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" for which you want the argument .","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" pour lequel vous souhaitez obtenir la tangente.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"IMREAL Returns the real coefficient of a complex number","toRaw":"COMPLEXE.REEL Ingénierie Renvoie le coefficient réel d'un nombre complexe.","from":[{"text":"IMREAL Returns the real coefficient of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"","type":0},{"text":"COMPLEXE","type":2},{"text":".REEL Ingénierie Renvoie le coefficient réel d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"imag: Returns the imaginary component of a complex number.","toRaw":"Fonction : Renvoie la partie imaginaire d’un nombre complexe.","from":[{"text":"imag: Returns the imaginary component of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Fonction : Renvoie la partie imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"IMAGINARY Returns the imaginary coefficient of a complex number.","toRaw":"Renvoie le coefficient imaginaire d’un nombre complexe.","from":[{"text":"IMAGINARY Returns the imaginary coefficient of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Renvoie le coefficient imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"CTANH calculates the hyperbolic tangent of a complex number","toRaw":"Tanhz : Calcule la tangente hyperbolique d'un nombre complexe","from":[{"text":"CTANH calculates the hyperbolic tangent of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Tanhz : Calcule la tangente hyperbolique d'un ","type":0},{"text":"nombre complexe","type":1}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Identify the imaginary part of the following complex number:","toRaw":"Calculer la partie réelle et la partie imaginaire du nombre complexe suivant :","from":[{"text":"Identify the imaginary part of the following ","type":0},{"text":"complex number","type":1},{"text":":","type":0}],"to":[{"text":"Calculer la partie réelle et la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":" suivant :","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Graphical representation of a complex number and its conjugate","toRaw":"Les nombres complexes Sommation d'un nombre complexe et de son conjugué","from":[{"text":"Graphical representation of a ","type":0},{"text":"complex number","type":1},{"text":" and its conjugate","type":0}],"to":[{"text":"Les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s Sommation d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" et de son conjugué","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"If z = a + bi is a complex number, then its complex conjugate is: z* = a - bi.","toRaw":"Soit z = a + ib un nombre complexe alors son conjugué est : z = a ib.","from":[{"text":"If z = a + bi is a ","type":0},{"text":"complex number","type":1},{"text":", then its ","type":0},{"text":"complex","type":2},{"text":" conjugate is: z* = a - bi.","type":0}],"to":[{"text":"Soit z = a + ib un ","type":0},{"text":"nombre complexe","type":1},{"text":" alors son conjugué est : z = a ib.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The Complex Number Computer was completed by January 8, 1940, was able to calculate complex numbers.","toRaw":"Leur complexe Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des nombres complexes.","from":[{"text":"The ","type":0},{"text":"Complex Number","type":1},{"text":" Computer was completed by January 8, 1940, was able to calculate ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Leur ","type":0},{"text":"complexe","type":2},{"text":" Number Calculator [50], complété 8 Janvier 1940, a été en mesure de calculer des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The value of the transfer function at a given frequency is the ratio of the complex number representing the output sinusoid to the complex number representing the input sinusoid, using complex division.","toRaw":"La valeur de la fonction de transfert à une fréquence donnée est le rapport entre le nombre complexe représentant la sinusoïde de sortie au nombre complexe représentant la sinusoïde d'entrée, à l'aide de la division des complexes.","from":[{"text":"The value of the transfer function at a given frequency is the ratio of the ","type":0},{"text":"complex number","type":1},{"text":" representing the output sinusoid to the ","type":0},{"text":"complex number","type":1},{"text":" representing the input sinusoid, using ","type":0},{"text":"complex","type":2},{"text":" division.","type":0}],"to":[{"text":"La valeur de la fonction de transfert à une fréquence donnée est le rapport entre le ","type":0},{"text":"nombre complexe","type":1},{"text":" représentant la sinusoïde de sortie au ","type":0},{"text":"nombre complexe","type":1},{"text":" représentant la sinusoïde d'entrée, à l'aide de la division des ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Muses also envisioned a mathematical number concept, Musean hypernumbers, that includes hypercomplex number algebras such as complex numbers and split-complex numbers as primitive types.","toRaw":"Musès a aussi envisagé un système de nombre mathématique, les hypernombres muséens, qui incluent les algèbres de nombres hypercomplexes telles que les nombres complexes et les nombres complexes déployés comme types primitifs.","from":[{"text":"Muses also envisioned a mathematical ","type":0},{"text":"number","type":2},{"text":" concept, Musean hyper","type":0},{"text":"number","type":2},{"text":"s, that includes hyper","type":0},{"text":"complex number","type":1},{"text":" algebras such as ","type":0},{"text":"complex number","type":1},{"text":"s and split-","type":0},{"text":"complex number","type":1},{"text":"s as primitive types.","type":0}],"to":[{"text":"Musès a aussi envisagé un système de ","type":0},{"text":"nombre","type":2},{"text":" mathématique, les hyper","type":0},{"text":"nombre","type":2},{"text":"s muséens, qui incluent les algèbres de ","type":0},{"text":"nombre","type":2},{"text":"s hyper","type":0},{"text":"complexe","type":2},{"text":"s telles que les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s et les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s déployés comme types primitifs.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The amplitude is the modulus of the complex number at the right and its phase is the argument of the complex number at the right.","toRaw":"Son amplitude sera égale au module du nombre complexe de droite et sa phase sera égale à l'argument du nombre complexe de droite.","from":[{"text":"The amplitude is the modulus of the ","type":0},{"text":"complex number","type":1},{"text":" at the right and its phase is the argument of the ","type":0},{"text":"complex number","type":1},{"text":" at the right.","type":0}],"to":[{"text":"Son amplitude sera égale au module du ","type":0},{"text":"nombre complexe","type":1},{"text":" de droite et sa phase sera égale à l'argument du ","type":0},{"text":"nombre complexe","type":1},{"text":" de droite.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The amplitude is the modulus of the complex number at the right and its phase is the argument of the complex number at the right.","toRaw":"L'amplitude du signal est représentée par le module du nombre complexe ; l'argument, quant à lui, représente la phase du signal.","from":[{"text":"The amplitude is the modulus of the ","type":0},{"text":"complex number","type":1},{"text":" at the right and its phase is the argument of the ","type":0},{"text":"complex number","type":1},{"text":" at the right.","type":0}],"to":[{"text":"L'amplitude du signal est représentée par le module du ","type":0},{"text":"nombre complexe","type":1},{"text":" ; l'argument, quant à lui, représente la phase du signal.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The real number a is called the real part of the complex number a + bi.","toRaw":"Le nombre réel a est la partie réelle de a + bi.","from":[{"text":"The real ","type":0},{"text":"number","type":2},{"text":" a is called the real part of the ","type":0},{"text":"complex number","type":1},{"text":" a + bi.","type":0}],"to":[{"text":"Le ","type":0},{"text":"nombre","type":2},{"text":" réel a est la partie réelle de a + bi.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number whose imaginary part is zero can be viewed as a real number.","toRaw":"Un nombre complexe dont la partie imaginaire est nulle est un réel.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" whose imaginary part is zero can be viewed as a real ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" dont la partie imaginaire est nulle est un réel.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Represent complex numbers on the complex plane in rectangular form and polar form (including real and imaginary numbers), and explain why the rectangular form of a given complex number represents the same number.","toRaw":"Représenter les nombres complexes sur le plan complexe de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un nombre complexe représentent le même nombre.","from":[{"text":"Represent ","type":0},{"text":"complex number","type":1},{"text":"s on the ","type":0},{"text":"complex","type":2},{"text":" plane in rectangular form and polar form (including real and imaginary ","type":0},{"text":"number","type":2},{"text":"s), and explain why the rectangular form of a given ","type":0},{"text":"complex number","type":1},{"text":" represents the same ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Représenter les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sur le plan ","type":0},{"text":"complexe","type":2},{"text":" de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" représentent le même ","type":0},{"text":"nombre","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"and so the product of a complex number with its conjugate is a real number.","toRaw":"Le produit d’un nombre complexe par son conjugué est donc toujours un nombre réel positif.","from":[{"text":"and so the product of a ","type":0},{"text":"complex number","type":1},{"text":" with its conjugate is a real ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Le produit d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" par son conjugué est donc toujours un ","type":0},{"text":"nombre","type":2},{"text":" réel positif.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Is the following statement true or false: Any real number is also a complex number?","toRaw":"L'assertion suivante est-elle vraie ou fausse : Tout nombre complexe est aussi un nombre réel ?","from":[{"text":"Is the following statement true or false: Any real ","type":0},{"text":"number","type":2},{"text":" is also a ","type":0},{"text":"complex number","type":1},{"text":"?","type":0}],"to":[{"text":"L'assertion suivante est-elle vraie ou fausse : Tout ","type":0},{"text":"nombre complexe","type":1},{"text":" est aussi un ","type":0},{"text":"nombre","type":2},{"text":" réel ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"complexnumber is text representing a complex number, for example as a+bi or a+bj. number is a number.","toRaw":"nombre_complexe est un texte représentant un nombre complexe, par exemple a+bi ou a+bj. nombre est un nombre.","from":[{"text":"","type":0},{"text":"complexnumber","type":2},{"text":" is text representing a ","type":0},{"text":"complex number","type":1},{"text":", for example as a+bi or a+bj. ","type":0},{"text":"number","type":2},{"text":" is a ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"","type":0},{"text":"nombre","type":2},{"text":"_","type":0},{"text":"complexe","type":2},{"text":" est un texte représentant un ","type":0},{"text":"nombre complexe","type":1},{"text":", par exemple a+bi ou a+bj. ","type":0},{"text":"nombre","type":2},{"text":" est un ","type":0},{"text":"nombre","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.","toRaw":"Trouver le conjugué d'un nombre complexe et utiliser conjugués pour trouver des modules et des quotients de nombres complexes.","from":[{"text":"(+) Find the conjugate of a ","type":0},{"text":"complex number","type":1},{"text":"; use conjugates to find moduli and quotients of ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Trouver le conjugué d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" et utiliser conjugués pour trouver des modules et des quotients de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.","toRaw":"Trouver le conjugué d'un nombre complexe et utiliser conjugués pour trouver des modules et des quotients de nombres complexes.","from":[{"text":"Find the conjugate of a ","type":0},{"text":"complex number","type":1},{"text":"; use conjugates to find moduli and quotients of ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Trouver le conjugué d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" et utiliser conjugués pour trouver des modules et des quotients de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex conjugate of a complex number is given by changing the sign of the imaginary part.","toRaw":"Conjuguer un nombre complexe revient à changer le signe de la partie imaginaire.","from":[{"text":"The ","type":0},{"text":"complex","type":2},{"text":" conjugate of a ","type":0},{"text":"complex number","type":1},{"text":" is given by changing the sign of the imaginary part.","type":0}],"to":[{"text":"Conjuguer un ","type":0},{"text":"nombre complexe","type":1},{"text":" revient à changer le signe de la partie imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The new System.Numerics.Complex structure represents a complex number that supports arithmetic and trigonometric operations with complex numbers.","toRaw":"La nouvelle structure System.Numerics.Complex représente un nombre complexe qui prend en charge des opérations arithmétiques et trigonométriques avec des nombres complexes.","from":[{"text":"The new System.Numerics.","type":0},{"text":"Complex","type":2},{"text":" structure represents a ","type":0},{"text":"complex number","type":1},{"text":" that supports arithmetic and trigonometric operations with ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"La nouvelle structure System.Numerics.Complex représente un ","type":0},{"text":"nombre complexe","type":1},{"text":" qui prend en charge des opérations arithmétiques et trigonométriques avec des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex conjugate of a complex number is obtained by changing the sign of its imaginary part.","toRaw":"Conjuguer un nombre complexe revient à changer le signe de la partie imaginaire.","from":[{"text":"The ","type":0},{"text":"complex","type":2},{"text":" conjugate of a ","type":0},{"text":"complex number","type":1},{"text":" is obtained by changing the sign of its imaginary part.","type":0}],"to":[{"text":"Conjuguer un ","type":0},{"text":"nombre complexe","type":1},{"text":" revient à changer le signe de la partie imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex conjugate of a complex number is obtained by changing the sign of its imaginary part.","toRaw":"Conjuguer » un nombre complexe consiste simplement à changer le signe de sa partie imaginaire.","from":[{"text":"The ","type":0},{"text":"complex","type":2},{"text":" conjugate of a ","type":0},{"text":"complex number","type":1},{"text":" is obtained by changing the sign of its imaginary part.","type":0}],"to":[{"text":"Conjuguer » un ","type":0},{"text":"nombre complexe","type":1},{"text":" consiste simplement à changer le signe de sa partie imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex number calculator is able to calculate complex numbers when they are in their algebraic form.","toRaw":"La calculatrice de nombres complexes est capable de calculer les nombres complexes lorsqu'ils sont sous leur forme algébrique.","from":[{"text":"The ","type":0},{"text":"complex number","type":1},{"text":" calculator is able to calculate ","type":0},{"text":"complex number","type":1},{"text":"s when they are in their algebraic form.","type":0}],"to":[{"text":"La calculatrice de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s est capable de calculer les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s lorsqu'ils sont sous leur forme algébrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex conjugate of a complex number is obtained by changing the sign of its imaginary part.","toRaw":"N’oubliez pas que nous trouvons le conjugué d’un nombre complexe en changeant le signe de sa partie imaginaire.","from":[{"text":"The ","type":0},{"text":"complex","type":2},{"text":" conjugate of a ","type":0},{"text":"complex number","type":1},{"text":" is obtained by changing the sign of its imaginary part.","type":0}],"to":[{"text":"N’oubliez pas que nous trouvons le conjugué d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" en changeant le signe de sa partie imaginaire.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Well, the result is gonna be some complex number on the unit circle in the complex plane.","toRaw":"Le résultat sera un nombre complexe sur le cercle unitaire dans le plan complexe.","from":[{"text":"Well, the result is gonna be some ","type":0},{"text":"complex number","type":1},{"text":" on the unit circle in the ","type":0},{"text":"complex","type":2},{"text":" plane.","type":0}],"to":[{"text":"Le résultat sera un ","type":0},{"text":"nombre complexe","type":1},{"text":" sur le cercle unitaire dans le plan ","type":0},{"text":"complexe","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The complex number calculator is able to calculate complex numbers when they are in their algebraic form.","toRaw":"Le calculateur de nombre complexe permet de manipuler les nombres complexes lorsqu'ils sont sous leur forme algébrique.","from":[{"text":"The ","type":0},{"text":"complex number","type":1},{"text":" calculator is able to calculate ","type":0},{"text":"complex number","type":1},{"text":"s when they are in their algebraic form.","type":0}],"to":[{"text":"Le calculateur de ","type":0},{"text":"nombre complexe","type":1},{"text":" permet de manipuler les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s lorsqu'ils sont sous leur forme algébrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.","toRaw":"Représenter les nombres complexes sur le plan complexe de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un nombre complexe représentent le même nombre.","from":[{"text":"Represent ","type":0},{"text":"complex number","type":1},{"text":"s on the ","type":0},{"text":"complex","type":2},{"text":" plane in rectangular and polar form (including real and imaginary ","type":0},{"text":"number","type":2},{"text":"s), and explain why the rectangular and polar forms of a given ","type":0},{"text":"complex number","type":1},{"text":" represent the same ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Représenter les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sur le plan ","type":0},{"text":"complexe","type":2},{"text":" de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" représentent le même ","type":0},{"text":"nombre","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"When in the standard form \\(a\\) is called the real part of the complex number and \\(b\\) is called the imaginary part of the complex number.","toRaw":"Lorsque dans le formulaire standard (a ) est appelé la partie réelle du nombre complexe et (b ) est appelé la partie imaginaire du nombre complexe.","from":[{"text":"When in the standard form \\(a\\) is called the real part of the ","type":0},{"text":"complex number","type":1},{"text":" and \\(b\\) is called the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Lorsque dans le formulaire standard (a ) est appelé la partie réelle du ","type":0},{"text":"nombre complexe","type":1},{"text":" et (b ) est appelé la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"When in the standard form \\(a\\) is called the real part of the complex number and \\(b\\) is called the imaginary part of the complex number. )","toRaw":"Lorsque dans le formulaire standard (a ) est appelé la partie réelle du nombre complexe et (b ) est appelé la partie imaginaire du nombre complexe.","from":[{"text":"When in the standard form \\(a\\) is called the real part of the ","type":0},{"text":"complex number","type":1},{"text":" and \\(b\\) is called the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":". )","type":0}],"to":[{"text":"Lorsque dans le formulaire standard (a ) est appelé la partie réelle du ","type":0},{"text":"nombre complexe","type":1},{"text":" et (b ) est appelé la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"complexIs an approximation of a complex number (typically two float).","toRaw":"complex est une approximation d'un nombre complexe (typiquement deux floats).","from":[{"text":"","type":0},{"text":"complex","type":2},{"text":"Is an approximation of a ","type":0},{"text":"complex number","type":1},{"text":" (typically two float).","type":0}],"to":[{"text":"complex est une approximation d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" (typiquement deux floats).","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"(+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.","toRaw":"Représenter les nombres complexes sur le plan complexe de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un nombre complexe représentent le même nombre.","from":[{"text":"(+) Represent ","type":0},{"text":"complex number","type":1},{"text":"s on the ","type":0},{"text":"complex","type":2},{"text":" plane in rectangular and polar form (including real and imaginary ","type":0},{"text":"number","type":2},{"text":"s), and explain why the rectangular and polar forms of a given ","type":0},{"text":"complex number","type":1},{"text":" represent the same ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Représenter les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sur le plan ","type":0},{"text":"complexe","type":2},{"text":" de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" représentent le même ","type":0},{"text":"nombre","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"When in the standard form a is called the real part of the complex number and b is called the imaginary part of the complex number.","toRaw":"Lorsque dans le formulaire standard (a ) est appelé la partie réelle du nombre complexe et (b ) est appelé la partie imaginaire du nombre complexe.","from":[{"text":"When in the standard form a is called the real part of the ","type":0},{"text":"complex number","type":1},{"text":" and b is called the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Lorsque dans le formulaire standard (a ) est appelé la partie réelle du ","type":0},{"text":"nombre complexe","type":1},{"text":" et (b ) est appelé la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers) and explain why the rectangular and polar forms of a given complex number represent the same number.","toRaw":"Représenter les nombres complexes sur le plan complexe de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un nombre complexe représentent le même nombre.","from":[{"text":"Represent ","type":0},{"text":"complex number","type":1},{"text":"s on the ","type":0},{"text":"complex","type":2},{"text":" plane in rectangular and polar form (including real and imaginary ","type":0},{"text":"number","type":2},{"text":"s) and explain why the rectangular and polar forms of a given ","type":0},{"text":"complex number","type":1},{"text":" represent the same ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Représenter les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sur le plan ","type":0},{"text":"complexe","type":2},{"text":" de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" représentent le même ","type":0},{"text":"nombre","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The absolute value of a complex number is equal to its magnitude, or the square root of the product of the number and its complex conjugate.","toRaw":"La valeur absolue d'un nombre complexe est égale à son module, ou la racine carrée du produit du nombre et de son complexe conjugué.","from":[{"text":"The absolute value of a ","type":0},{"text":"complex number","type":1},{"text":" is equal to its magnitude, or the square root of the product of the ","type":0},{"text":"number","type":2},{"text":" and its ","type":0},{"text":"complex","type":2},{"text":" conjugate.","type":0}],"to":[{"text":"La valeur absolue d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" est égale à son module, ou la racine carrée du produit du ","type":0},{"text":"nombre","type":2},{"text":" et de son ","type":0},{"text":"complexe","type":2},{"text":" conjugué.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The result of the integration is a single complex number for each bit of the received signal.","toRaw":"Le résultat de l'intégration est un seul nombre complexe pour chaque bit du signal reçu.","from":[{"text":"The result of the integration is a single ","type":0},{"text":"complex number","type":1},{"text":" for each bit of the received signal.","type":0}],"to":[{"text":"Le résultat de l'intégration est un seul ","type":0},{"text":"nombre complexe","type":1},{"text":" pour chaque bit du signal reçu.","type":0}],"d2":"6826","d0":"13","collapsed":false,"source":"wipo.int"},{"fromRaw":"Zero (complex analysis) — In complex analysis, a zero of a holomorphic function f is a complex number a such that f(a) = 0.","toRaw":"Zéro (analyse complexe) — Zéro d une fonction holomorphe En analyse complexe, on appelle zéro d une fonction analytique ou holomorphe f un nombre complexe a tel que f(a) = 0.","from":[{"text":"Zero (","type":0},{"text":"complex","type":2},{"text":" analysis) — In ","type":0},{"text":"complex","type":2},{"text":" analysis, a zero of a holomorphic function f is a ","type":0},{"text":"complex number","type":1},{"text":" a such that f(a) = 0.","type":0}],"to":[{"text":"Zéro (analyse ","type":0},{"text":"complexe","type":2},{"text":") — Zéro d une fonction holomorphe En analyse ","type":0},{"text":"complexe","type":2},{"text":", on appelle zéro d une fonction analytique ou holomorphe f un ","type":0},{"text":"nombre complexe","type":1},{"text":" a tel que f(a) = 0.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"To the complex number we associate the point with coordinates .","toRaw":"À tout nombre complexe avec on associe le point de coordonnées .","from":[{"text":"To the ","type":0},{"text":"complex number","type":1},{"text":" we associate the point with coordinates .","type":0}],"to":[{"text":"À tout ","type":0},{"text":"nombre complexe","type":1},{"text":" avec on associe le point de coordonnées .","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"What is the modulus of the complex number 3−𝑖?","toRaw":"Quel est le module du nombre complexe 3 + 4 𝑖 ?","from":[{"text":"What is the modulus of the ","type":0},{"text":"complex number","type":1},{"text":" 3−𝑖","type":0}],"to":[{"text":"Quel est le module du ","type":0},{"text":"nombre complexe","type":1},{"text":" 3 + 4 𝑖 ","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"ComplexRI constructs a complex number from real and imaginary parts.","toRaw":"ComplexRI construit un nombre complexe à partir de parties réelles et imaginaires.","from":[{"text":"","type":0},{"text":"Complex","type":2},{"text":"RI constructs a ","type":0},{"text":"complex number","type":1},{"text":" from real and imaginary parts.","type":0}],"to":[{"text":"ComplexRI construit un ","type":0},{"text":"nombre complexe","type":1},{"text":" à partir de parties réelles et imaginaires.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Since we have decided to return a complex number here,","toRaw":"puisqu'on a décidé ici de retourner un nombre complexe,","from":[{"text":"Since we have decided to return a ","type":0},{"text":"complex number","type":1},{"text":" here,","type":0}],"to":[{"text":"puisqu'on a décidé ici de retourner un ","type":0},{"text":"nombre complexe","type":1},{"text":",","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Find the multiplicative inverse of the complex number 4 – 3i","toRaw":"Exemple : Obtenir la partie réelle du nombre complexe 3 – 4i","from":[{"text":"Find the multiplicative inverse of the ","type":0},{"text":"complex number","type":1},{"text":" 4 – 3i","type":0}],"to":[{"text":"Exemple : Obtenir la partie réelle du ","type":0},{"text":"nombre complexe","type":1},{"text":" 3 – 4i","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number,Z , can be expressed in several ways:","toRaw":"Les nombres complexes, notés généralement z, peuvent ainsi être présentés de plusieurs manières :","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":",Z , can be expressed in several ways:","type":0}],"to":[{"text":"Les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s, notés généralement z, peuvent ainsi être présentés de plusieurs manières :","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"POTENZA, Returns a complex number raised to an integer power .","toRaw":"Ingénierie : renvoie un nombre complexe élevé à une puissance entière.","from":[{"text":"POTENZA, Returns a ","type":0},{"text":"complex number","type":1},{"text":" raised to an integer power .","type":0}],"to":[{"text":"Ingénierie : renvoie un ","type":0},{"text":"nombre complexe","type":1},{"text":" élevé à une puissance entière.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number can be described by two real coordinates.","toRaw":"Un nombre complexe peut être décrit avec deux types de coordonnées réelles.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" can be described by two real coordinates.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" peut être décrit avec deux types de coordonnées réelles.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Write the complex number z=-2+2i in trigonometric form.","toRaw":"Écrire le complexe z = 1 + i sous forme trigonométrique.","from":[{"text":"Write the ","type":0},{"text":"complex number","type":1},{"text":" z=-2+2i in trigonometric form.","type":0}],"to":[{"text":"Écrire le ","type":0},{"text":"complexe","type":2},{"text":" z = 1 + i sous forme trigonométrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Geometric Construction of the Square Roots of a Complex Number","toRaw":"Construction géométrique des racines carrées d'un complexe","from":[{"text":"Geometric Construction of the Square Roots of a ","type":0},{"text":"Complex Number","type":1}],"to":[{"text":"Construction géométrique des racines carrées d'un ","type":0},{"text":"complexe","type":2}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Write the complex number z=-2+2i in trigonometric form.","toRaw":"Ecrire le nombre complexe z = 1 + i√3 sous forme trigonométrique.","from":[{"text":"Write the ","type":0},{"text":"complex number","type":1},{"text":" z=-2+2i in trigonometric form.","type":0}],"to":[{"text":"Ecrire le ","type":0},{"text":"nombre complexe","type":1},{"text":" z = 1 + i√3 sous forme trigonométrique.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Now we can define the logarithm of a complex number.","toRaw":"Grâce à la forme exponentielle, on peut définir le logarithme d’un nombre complexe.","from":[{"text":"Now we can define the logarithm of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Grâce à la forme exponentielle, on peut définir le logarithme d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Let be a complex number with strictly positive imaginary part.","toRaw":"Soit z un nombre complexe de partie imaginaire strictement positive.","from":[{"text":"Let be a ","type":0},{"text":"complex number","type":1},{"text":" with strictly positive imaginary part.","type":0}],"to":[{"text":"Soit z un ","type":0},{"text":"nombre complexe","type":1},{"text":" de partie imaginaire strictement positive.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The impedance is a complex number, in general noted Z.","toRaw":"L'impédance est un nombre complexe, en général noté Z","from":[{"text":"The impedance is a ","type":0},{"text":"complex number","type":1},{"text":", in general noted Z.","type":0}],"to":[{"text":"L'impédance est un ","type":0},{"text":"nombre complexe","type":1},{"text":", en général noté Z","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Find the multiplicative inverse of the complex number 4 - 3i","toRaw":"Exemple : Obtenir la partie réelle du nombre complexe 3 – 4i","from":[{"text":"Find the multiplicative inverse of the ","type":0},{"text":"complex number","type":1},{"text":" 4 - 3i","type":0}],"to":[{"text":"Exemple : Obtenir la partie réelle du ","type":0},{"text":"nombre complexe","type":1},{"text":" 3 – 4i","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Complex numbers are an extension of the real number system.","toRaw":"Les nombres complexes constituent une extension des nombres réels.","from":[{"text":"","type":0},{"text":"Complex number","type":1},{"text":"s are an extension of the real ","type":0},{"text":"number","type":2},{"text":" system.","type":0}],"to":[{"text":"Les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s constituent une extension des ","type":0},{"text":"nombre","type":2},{"text":"s réels.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"What is the modulus of the complex number 3+7𝑖?","toRaw":"Quel est le module du nombre complexe 3 + 4 𝑖 ?","from":[{"text":"What is the modulus of the ","type":0},{"text":"complex number","type":1},{"text":" 3+7𝑖","type":0}],"to":[{"text":"Quel est le module du ","type":0},{"text":"nombre complexe","type":1},{"text":" 3 + 4 𝑖 ","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Now picture a fixed complex number on the unit circle","toRaw":"» Intégrale du conjugué d'un nombre complexe sur le cercle unité","from":[{"text":"Now picture a fixed ","type":0},{"text":"complex number","type":1},{"text":" on the unit circle","type":0}],"to":[{"text":"» Intégrale du conjugué d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" sur le cercle unité","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Each point on the plane represents a complex number c.","toRaw":"A un point du plan, l'on fait correspondre un nombre complexe c.","from":[{"text":"Each point on the plane represents a ","type":0},{"text":"complex number","type":1},{"text":" c.","type":0}],"to":[{"text":"A un point du plan, l'on fait correspondre un ","type":0},{"text":"nombre complexe","type":1},{"text":" c.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Find the real and imaginary part of a complex number","toRaw":"Déterminer la partie réelle et imaginaire d'un nombre complexe","from":[{"text":"Find the real and imaginary part of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Déterminer la partie réelle et imaginaire d'un ","type":0},{"text":"nombre complexe","type":1}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"This is called the \"rectangular form\" of a complex number.","toRaw":"C'est ce qu'on appelle la forme trigonométrique d'un nombre complexe.","from":[{"text":"This is called the \"rectangular form\" of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"C'est ce qu'on appelle la forme trigonométrique d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Returns the complex number with modulus r and phase phi.","toRaw":"Renvoie le nombre complexe x dont les coordonnées polaires sont r et phi.","from":[{"text":"Returns the ","type":0},{"text":"complex number","type":1},{"text":" with modulus r and phase phi.","type":0}],"to":[{"text":"Renvoie le ","type":0},{"text":"nombre complexe","type":1},{"text":" x dont les coordonnées polaires sont r et phi.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"complexIs an approximation of a complex number (typically two float).","toRaw":"complex est une approximation d'un nombre complexe (typiquement deux float).","from":[{"text":"","type":0},{"text":"complex","type":2},{"text":"Is an approximation of a ","type":0},{"text":"complex number","type":1},{"text":" (typically two float).","type":0}],"to":[{"text":"complex est une approximation d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" (typiquement deux float).","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number for which you want the absolute value.","toRaw":"Représente le nombre réel dont vous voulez obtenir la valeur absolue.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" for which you want the absolute value.","type":0}],"to":[{"text":"Représente le ","type":0},{"text":"nombre","type":2},{"text":" réel dont vous voulez obtenir la valeur absolue.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"➠ Complex Number Conjugate — Transpose of a Matrix — Conjugate Transpose Matrix","toRaw":"➠ Conjugué de Nombre Complexe — Transposée d'une Matrice — Matrice Adjointe","from":[{"text":"➠ ","type":0},{"text":"Complex Number","type":1},{"text":" Conjugate — Transpose of a Matrix — Conjugate Transpose Matrix","type":0}],"to":[{"text":"➠ Conjugué de ","type":0},{"text":"Nombre Complexe","type":1},{"text":" — Transposée d'une Matrice — Matrice Adjointe","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The number i, the imaginary unit of the complex numbers.","toRaw":"Le nombre i, l unité imaginaire des nombres complexes.","from":[{"text":"The ","type":0},{"text":"number","type":2},{"text":" i, the imaginary unit of the ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Le ","type":0},{"text":"nombre","type":2},{"text":" i, l unité imaginaire des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"complexIs an approximation of a complex number (typically two float).","toRaw":"complex est une approximation d'un nombre complexe (typiquement deux floats)","from":[{"text":"","type":0},{"text":"complex","type":2},{"text":"Is an approximation of a ","type":0},{"text":"complex number","type":1},{"text":" (typically two float).","type":0}],"to":[{"text":"complex est une approximation d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" (typiquement deux floats)","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number can be described by two real coordinates.","toRaw":"Un nombre complexe peut être décrit avec deux types de coordonnées","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" can be described by two real coordinates.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" peut être décrit avec deux types de coordonnées","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"We call it the absolute value of the complex number.","toRaw":"En fait, nous appelons cela l’argument du nombre complexe.","from":[{"text":"We call it the absolute value of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"En fait, nous appelons cela l’argument du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Find the real and imaginary parts of the complex number","toRaw":"Déterminer la partie réelle et imaginaire d'un nombre complexe","from":[{"text":"Find the real and imaginary parts of the ","type":0},{"text":"complex number","type":1}],"to":[{"text":"Déterminer la partie réelle et imaginaire d'un ","type":0},{"text":"nombre complexe","type":1}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The argument of a complex number 0 does not exist.","toRaw":"attention le nombre complexe 0 n'a pas d'argument.","from":[{"text":"The argument of a ","type":0},{"text":"complex number","type":1},{"text":" 0 does not exist.","type":0}],"to":[{"text":"attention le ","type":0},{"text":"nombre complexe","type":1},{"text":" 0 n'a pas d'argument.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"where Im denotes the imaginary part of a complex number.","toRaw":"Calcule la partie imaginaire d’un nombre complexe.","from":[{"text":"where Im denotes the imaginary part of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Calcule la partie imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Complex numbers are an extension of the real number system.","toRaw":"Les nombres complexes sont considérés comme une extension du système de nombres réels.","from":[{"text":"","type":0},{"text":"Complex number","type":1},{"text":"s are an extension of the real ","type":0},{"text":"number","type":2},{"text":" system.","type":0}],"to":[{"text":"Les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sont considérés comme une extension du système de ","type":0},{"text":"nombre","type":2},{"text":"s réels.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The imaginary part of our complex number is negative five.","toRaw":"La partie imaginaire de notre nombre complexe est moins cinq.","from":[{"text":"The imaginary part of our ","type":0},{"text":"complex number","type":1},{"text":" is negative five.","type":0}],"to":[{"text":"La partie imaginaire de notre ","type":0},{"text":"nombre complexe","type":1},{"text":" est moins cinq.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"(ii) Re (z) ≤ |z| , where z is a complex number.","toRaw":"conjugue(z), où z représente un nombre complexe.","from":[{"text":"(ii) Re (z) ≤ |z| , where z is a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"conjugue(z), où z représente un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"NAME clog, clogf, clogl - natural logarithm of a complex number","toRaw":"clog, clogf, clogl - Logarithmes népériens de nombres complexes","from":[{"text":"NAME clog, clogf, clogl - natural logarithm of a ","type":0},{"text":"complex number","type":1}],"to":[{"text":"clog, clogf, clogl - Logarithmes népériens de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number for which you want the absolute value.","toRaw":"Mesure pour laquelle vous souhaitez obtenir la valeur absolue.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" for which you want the absolute value.","type":0}],"to":[{"text":"Mesure pour laquelle vous souhaitez obtenir la valeur absolue.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"How to determine the nth roots of a complex number?","toRaw":"Comment déterminer les racines carrées d’un nombre complexe ?","from":[{"text":"How to determine the nth roots of a ","type":0},{"text":"complex number","type":1},{"text":"?","type":0}],"to":[{"text":"Comment déterminer les racines carrées d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"ComplexMA constructs a complex number from a magnitude and angle.","toRaw":"ComplexMA construit un nombre complexe à partir d'un module et d'un angle.","from":[{"text":"","type":0},{"text":"Complex","type":2},{"text":"MA constructs a ","type":0},{"text":"complex number","type":1},{"text":" from a magnitude and angle.","type":0}],"to":[{"text":"ComplexMA construit un ","type":0},{"text":"nombre complexe","type":1},{"text":" à partir d'un module et d'un angle.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The impedance is a complex number, in general noted Z.","toRaw":"L'impédance est un nombre complexe, en général noté Z.","from":[{"text":"The impedance is a ","type":0},{"text":"complex number","type":1},{"text":", in general noted Z.","type":0}],"to":[{"text":"L'impédance est un ","type":0},{"text":"nombre complexe","type":1},{"text":", en général noté Z.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"This is called the \"rectangular form\" of a complex number.","toRaw":"Cette écriture est appelée forme trigonométrique d’un nombre complexe.","from":[{"text":"This is called the \"rectangular form\" of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Cette écriture est appelée forme trigonométrique d’un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"What is the polar (trigonometric) form of a complex number?","toRaw":"Qu’est-ce que la forme trigonométrique d’un nombre complexe ?","from":[{"text":"What is the polar (trigonometric) form of a ","type":0},{"text":"complex number","type":1},{"text":"?","type":0}],"to":[{"text":"Qu’est-ce que la forme trigonométrique d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Where Re denotes the real part of a complex number.","toRaw":"la notation Re désignant la partie réelle d'un nombre complexe.","from":[{"text":"Where Re denotes the real part of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"la notation Re désignant la partie réelle d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"How to determine the nth roots of a complex number?","toRaw":"← Comment déterminer les racines carrées d’un nombre complexe ?","from":[{"text":"How to determine the nth roots of a ","type":0},{"text":"complex number","type":1},{"text":"?","type":0}],"to":[{"text":"← Comment déterminer les racines carrées d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"How to determine the nth roots of a complex number?","toRaw":"Comment déterminer les racines carrées d’un nombre complexe ? →","from":[{"text":"How to determine the nth roots of a ","type":0},{"text":"complex number","type":1},{"text":"?","type":0}],"to":[{"text":"Comment déterminer les racines carrées d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" ? →","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"understand the polar or trigonometrical form of a complex number.","toRaw":"Forme polaire ou trigonométrique d'un nombre complexe.","from":[{"text":"understand the polar or trigonometrical form of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Forme polaire ou trigonométrique d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Hence, find the modulus and argument of the complex number.","toRaw":"Déterminer le module et un argument du nombre complexe .","from":[{"text":"Hence, find the modulus and argument of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Déterminer le module et un argument du ","type":0},{"text":"nombre complexe","type":1},{"text":" .","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"What is the modulus of the complex number 3+7𝑖?","toRaw":"Quel est le module du nombre complexe 3+7𝑖 ?","from":[{"text":"What is the modulus of the ","type":0},{"text":"complex number","type":1},{"text":" 3+7𝑖","type":0}],"to":[{"text":"Quel est le module du ","type":0},{"text":"nombre complexe","type":1},{"text":" 3+7𝑖 ","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"In this case we can see that the complex number is in fact a real number.","toRaw":"Nous en déduisons que le nombre complexe est un nombre réel.","from":[{"text":"In this case we can see that the ","type":0},{"text":"complex number","type":1},{"text":" is in fact a real ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Nous en déduisons que le ","type":0},{"text":"nombre complexe","type":1},{"text":" est un ","type":0},{"text":"nombre","type":2},{"text":" réel.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Therefore, we conclude that the statement “any real number is also a complex number” is true.","toRaw":"On peut dire que l’affirmation « tout nombre réel est aussi un nombre complexe » est vraie.","from":[{"text":"Therefore, we conclude that the statement “any real ","type":0},{"text":"number","type":2},{"text":" is also a ","type":0},{"text":"complex number","type":1},{"text":"” is true.","type":0}],"to":[{"text":"On peut dire que l’affirmation « tout ","type":0},{"text":"nombre","type":2},{"text":" réel est aussi un ","type":0},{"text":"nombre complexe","type":1},{"text":" » est vraie.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"We can say that the statement “Any real number is also a complex number” is true.","toRaw":"On peut dire que l’affirmation « tout nombre réel est aussi un nombre complexe » est vraie.","from":[{"text":"We can say that the statement “Any real ","type":0},{"text":"number","type":2},{"text":" is also a ","type":0},{"text":"complex number","type":1},{"text":"” is true.","type":0}],"to":[{"text":"On peut dire que l’affirmation « tout ","type":0},{"text":"nombre","type":2},{"text":" réel est aussi un ","type":0},{"text":"nombre complexe","type":1},{"text":" » est vraie.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Complex numbers can also be represented in polar form, which associates each complex number with its distance from the origin (its magnitude) and with a particular angle known as the argument of this complex number.","toRaw":"Les nombres complexes peuvent également être représentés sous forme polaire, qui associe chaque nombre complexe avec sa distance de l`origine (sa magnitude) et avec un angle particulier connu comme l`argument de ce nombre complexe.","from":[{"text":"","type":0},{"text":"Complex number","type":1},{"text":"s can also be represented in polar form, which associates each ","type":0},{"text":"complex number","type":1},{"text":" with its distance from the origin (its magnitude) and with a particular angle known as the argument of this ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s peuvent également être représentés sous forme polaire, qui associe chaque ","type":0},{"text":"nombre complexe","type":1},{"text":" avec sa distance de l`origine (sa magnitude) et avec un angle particulier connu comme l`argument de ce ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"3: (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.","toRaw":"Trouver le conjugué d'un nombre complexe et utiliser conjugués pour trouver des modules et des quotients de nombres complexes.","from":[{"text":"3: (+) Find the conjugate of a ","type":0},{"text":"complex number","type":1},{"text":"; use conjugates to find moduli and quotients of ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Trouver le conjugué d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" et utiliser conjugués pour trouver des modules et des quotients de ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A homography is a transformation of the complex plane which associates its image to a complex number z","toRaw":"Une homographie est une transformation du plan complexe qui, à un nombre complexe z associe son image","from":[{"text":"A homography is a transformation of the ","type":0},{"text":"complex","type":2},{"text":" plane which associates its image to a ","type":0},{"text":"complex number","type":1},{"text":" z","type":0}],"to":[{"text":"Une homographie est une transformation du plan ","type":0},{"text":"complexe","type":2},{"text":" qui, à un ","type":0},{"text":"nombre complexe","type":1},{"text":" z associe son image","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"(complex analysis), mathematics, complex number describing the behavior of line integrals of a meromorphic function around a singularity","toRaw":"(Mathématiques) Nombre complexe qui décrit le comportement de l’intégrale curviligne d’une fonction holomorphe aux alentours d’une singularité.","from":[{"text":"(","type":0},{"text":"complex","type":2},{"text":" analysis), mathematics, ","type":0},{"text":"complex number","type":1},{"text":" describing the behavior of line integrals of a meromorphic function around a singularity","type":0}],"to":[{"text":"(Mathématiques) ","type":0},{"text":"Nombre complexe","type":1},{"text":" qui décrit le comportement de l’intégrale curviligne d’une fonction holomorphe aux alentours d’une singularité.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"COMPLEX Converts real and imaginary coefficients into a complex number of the form x + yi or x + yj.","toRaw":"Convertit des coefficients réels et imaginaires en un nombre complexe de la forme x + yi ou x + yj.","from":[{"text":"","type":0},{"text":"COMPLEX","type":2},{"text":" Converts real and imaginary coefficients into a ","type":0},{"text":"complex number","type":1},{"text":" of the form x + yi or x + yj.","type":0}],"to":[{"text":"Convertit des coefficients réels et imaginaires en un ","type":0},{"text":"nombre complexe","type":1},{"text":" de la forme x + yi ou x + yj.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"(complex analysis), mathematics, complex number describing the behavior of line integrals of a meromorphic function around a singularity","toRaw":"(Analyse) Nombre complexe qui décrit le comportement de l’intégrale curviligne d’une fonction holomorphe aux alentours d’une singularité.","from":[{"text":"(","type":0},{"text":"complex","type":2},{"text":" analysis), mathematics, ","type":0},{"text":"complex number","type":1},{"text":" describing the behavior of line integrals of a meromorphic function around a singularity","type":0}],"to":[{"text":"(Analyse) ","type":0},{"text":"Nombre complexe","type":1},{"text":" qui décrit le comportement de l’intégrale curviligne d’une fonction holomorphe aux alentours d’une singularité.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Complex Number Primer - This is a brief introduction to some of the basic ideas involved with Complex Numbers.","toRaw":"Complex Number Number – Ceci est une brève introduction à quelques-unes des idées de base impliquées dans les nombres complexes.","from":[{"text":"","type":0},{"text":"Complex Number","type":1},{"text":" Primer - This is a brief introduction to some of the basic ideas involved with ","type":0},{"text":"Complex Number","type":1},{"text":"s.","type":0}],"to":[{"text":"Complex Number Number – Ceci est une brève introduction à quelques-unes des idées de base impliquées dans les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"We can see from the above that a single complex number is a point in the complex plane.","toRaw":"Nous pouvons voir à partir de ce qui précède qu'un nombre complexe unique est un point dans le plan complexe.","from":[{"text":"We can see from the above that a single ","type":0},{"text":"complex number","type":1},{"text":" is a point in the ","type":0},{"text":"complex","type":2},{"text":" plane.","type":0}],"to":[{"text":"Nous pouvons voir à partir de ce qui précède qu'un ","type":0},{"text":"nombre complexe","type":1},{"text":" unique est un point dans le plan ","type":0},{"text":"complexe","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Understand and use complex numbers, including real and imaginary numbers, on the complex plane in rectangular and polar form, and explain why the rectangular and polar forms of a given complex number represent the same number.","toRaw":"Représenter les nombres complexes sur le plan complexe de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un nombre complexe représentent le même nombre.","from":[{"text":"Understand and use ","type":0},{"text":"complex number","type":1},{"text":"s, including real and imaginary ","type":0},{"text":"number","type":2},{"text":"s, on the ","type":0},{"text":"complex","type":2},{"text":" plane in rectangular and polar form, and explain why the rectangular and polar forms of a given ","type":0},{"text":"complex number","type":1},{"text":" represent the same ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Représenter les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sur le plan ","type":0},{"text":"complexe","type":2},{"text":" de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" représentent le même ","type":0},{"text":"nombre","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"N.CN.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.","toRaw":"Représenter les nombres complexes sur le plan complexe de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un nombre complexe représentent le même nombre.","from":[{"text":"N.CN.4 Represent ","type":0},{"text":"complex number","type":1},{"text":"s on the ","type":0},{"text":"complex","type":2},{"text":" plane in rectangular and polar form (including real and imaginary ","type":0},{"text":"number","type":2},{"text":"s), and explain why the rectangular and polar forms of a given ","type":0},{"text":"complex number","type":1},{"text":" represent the same ","type":0},{"text":"number","type":2},{"text":".","type":0}],"to":[{"text":"Représenter les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sur le plan ","type":0},{"text":"complexe","type":2},{"text":" de forme rectangulaire et polaire et expliquer pourquoi les formes rectangulaires et polaires d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" représentent le même ","type":0},{"text":"nombre","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The number a is called the real part and the number b is called the imaginary part of the complex number.","toRaw":"Dans cette écriture, le nombre a s’appelle la partie réelle et le nombre b s’appelle la partie imaginaire du nombre complexe.","from":[{"text":"The ","type":0},{"text":"number","type":2},{"text":" a is called the real part and the ","type":0},{"text":"number","type":2},{"text":" b is called the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Dans cette écriture, le ","type":0},{"text":"nombre","type":2},{"text":" a s’appelle la partie réelle et le ","type":0},{"text":"nombre","type":2},{"text":" b s’appelle la partie imaginaire du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"To begin with, we’ll simply define what we mean by an imaginary number and a complex number.","toRaw":"Pour commencer, nous allons simplement définir ce que signifie un nombre imaginaire et un nombre complexe.","from":[{"text":"To begin with, we’ll simply define what we mean by an imaginary ","type":0},{"text":"number","type":2},{"text":" and a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Pour commencer, nous allons simplement définir ce que signifie un ","type":0},{"text":"nombre","type":2},{"text":" imaginaire et un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The absolute value of a complex number z= a + bi is a nonnegative real number defined by:","toRaw":"Le quotient d'un nombre complexe z = a + b i par un réel k non nul est le nombre complexe défini par :","from":[{"text":"The absolute value of a ","type":0},{"text":"complex number","type":1},{"text":" z= a + bi is a nonnegative real ","type":0},{"text":"number","type":2},{"text":" defined by:","type":0}],"to":[{"text":"Le quotient d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" z = a + b i par un réel k non nul est le ","type":0},{"text":"nombre complexe","type":1},{"text":" défini par :","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is an imaginary number if and only if its imaginary part is non-zero.","toRaw":"Un nombre complexe est donc réel si et seulement si sa partie imaginaire est nulle.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is an imaginary ","type":0},{"text":"number","type":2},{"text":" if and only if its imaginary part is non-zero.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" est donc réel si et seulement si sa partie imaginaire est nulle.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is an imaginary number if and only if its imaginary part is non-zero.","toRaw":"En particulier un nombre complexe est réel si et seulement si sa partie imaginaire est nulle.","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is an imaginary ","type":0},{"text":"number","type":2},{"text":" if and only if its imaginary part is non-zero.","type":0}],"to":[{"text":"En particulier un ","type":0},{"text":"nombre complexe","type":1},{"text":" est réel si et seulement si sa partie imaginaire est nulle.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The absolute value of a complex number z= a + bi is a nonnegative real number defined by:","toRaw":"Le quotient d'un nombre complexe z=a+bi par un réel k non nul est le nombre complexe défini par :","from":[{"text":"The absolute value of a ","type":0},{"text":"complex number","type":1},{"text":" z= a + bi is a nonnegative real ","type":0},{"text":"number","type":2},{"text":" defined by:","type":0}],"to":[{"text":"Le quotient d'un ","type":0},{"text":"nombre complexe","type":1},{"text":" z=a+bi par un réel k non nul est le ","type":0},{"text":"nombre complexe","type":1},{"text":" défini par :","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"For a complex number \" z1 \" we would write it like this :","toRaw":"par exemple pour un complexe « z1 », qu'on pourrait écrire comme ça","from":[{"text":"For a ","type":0},{"text":"complex number","type":1},{"text":" \" z1 \" we would write it like this :","type":0}],"to":[{"text":"par exemple pour un ","type":0},{"text":"complexe","type":2},{"text":" « z1 », qu'on pourrait écrire comme ça","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"A complex number is a number consisting of a real part (Physical)and an imaginary part (spiritual).","toRaw":"Un nombre complexe a toujours une partie réelle (vert) et une partie imaginaire (cyan).","from":[{"text":"A ","type":0},{"text":"complex number","type":1},{"text":" is a ","type":0},{"text":"number","type":2},{"text":" consisting of a real part (Physical)and an imaginary part (spiritual).","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" a toujours une partie réelle (vert) et une partie imaginaire (cyan).","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The quaternions are a number system that extends the complex numbers.","toRaw":"Les quaternions sont un type de nombres hypercomplexes, constituant une extension des nombres complexes.","from":[{"text":"The quaternions are a ","type":0},{"text":"number","type":2},{"text":" system that extends the ","type":0},{"text":"complex number","type":1},{"text":"s.","type":0}],"to":[{"text":"Les quaternions sont un type de ","type":0},{"text":"nombre","type":2},{"text":"s hyper","type":0},{"text":"complexe","type":2},{"text":"s, constituant une extension des ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"In fact, we call this the argument of the complex number.","toRaw":"En fait, nous appelons cela l’argument du nombre complexe.","from":[{"text":"In fact, we call this the argument of the ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"En fait, nous appelons cela l’argument du ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"How are complex numbers an extension of the real number system?","toRaw":"Les nombres complexes sont considérés comme une extension du système de nombres réels.","from":[{"text":"How are ","type":0},{"text":"complex number","type":1},{"text":"s an extension of the real ","type":0},{"text":"number","type":2},{"text":" system?","type":0}],"to":[{"text":"Les ","type":0},{"text":"nombre","type":2},{"text":"s ","type":0},{"text":"complexe","type":2},{"text":"s sont considérés comme une extension du système de ","type":0},{"text":"nombre","type":2},{"text":"s réels.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Alternately, this can be regarded (using complex number notation) as simple amplitude modulation of a complex-valued carrier wave by a single complex-valued signal.","toRaw":"C'est à dire, cela peut être reconnu (utilisant une notation en nombre complexe) comme une simple modulation d'amplitude d'une onde, exprimée en complexe, par un signal, exprimé en complexe.","from":[{"text":"Alternately, this can be regarded (using ","type":0},{"text":"complex number","type":1},{"text":" notation) as simple amplitude modulation of a ","type":0},{"text":"complex","type":2},{"text":"-valued carrier wave by a single ","type":0},{"text":"complex","type":2},{"text":"-valued signal.","type":0}],"to":[{"text":"C'est à dire, cela peut être reconnu (utilisant une notation en ","type":0},{"text":"nombre complexe","type":1},{"text":") comme une simple modulation d'amplitude d'une onde, exprimée en ","type":0},{"text":"complexe","type":2},{"text":", par un signal, exprimé en ","type":0},{"text":"complexe","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"What are the real and imaginary parts of the complex number ?","toRaw":"Que sont les parties réelle et imaginaire d’un nombre complexe ?","from":[{"text":"What are the real and imaginary parts of the ","type":0},{"text":"complex number","type":1},{"text":" ?","type":0}],"to":[{"text":"Que sont les parties réelle et imaginaire d’un ","type":0},{"text":"nombre complexe","type":1},{"text":" ?","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"DESCRIPTION A complex number can be described by two real coordinates.","toRaw":"Un nombre complexe peut être décrit avec deux types de coordonnées réelles.","from":[{"text":"DESCRIPTION A ","type":0},{"text":"complex number","type":1},{"text":" can be described by two real coordinates.","type":0}],"to":[{"text":"Un ","type":0},{"text":"nombre complexe","type":1},{"text":" peut être décrit avec deux types de coordonnées réelles.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"The next step in the discussion is the complex number system.","toRaw":"La étape suivante dans la discussion celle du système de nombre complexes.","from":[{"text":"The next step in the discussion is the ","type":0},{"text":"complex number","type":1},{"text":" system.","type":0}],"to":[{"text":"La étape suivante dans la discussion celle du système de ","type":0},{"text":"nombre complexe","type":1},{"text":"s.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"These return the real and imaginary parts of a complex number.","toRaw":"Ces outils identifient tout simplement la partie réelle ou imaginaire d'un nombre complexe.","from":[{"text":"These return the real and imaginary parts of a ","type":0},{"text":"complex number","type":1},{"text":".","type":0}],"to":[{"text":"Ces outils identifient tout simplement la partie réelle ou imaginaire d'un ","type":0},{"text":"nombre complexe","type":1},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"a is called the real part of the complex number z.","toRaw":"Le nombre réel a s’appelle la partie réelle du nombre complexe z.","from":[{"text":"a is called the real part of the ","type":0},{"text":"complex number","type":1},{"text":" z.","type":0}],"to":[{"text":"Le ","type":0},{"text":"nombre","type":2},{"text":" réel a s’appelle la partie réelle du ","type":0},{"text":"nombre complexe","type":1},{"text":" z.","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"7 Find the imaginary part of the complex number −3 − 5i","toRaw":"Exemple : Obtenir la partie réelle du nombre complexe 3 – 4i","from":[{"text":"7 Find the imaginary part of the ","type":0},{"text":"complex number","type":1},{"text":" −3 − 5i","type":0}],"to":[{"text":"Exemple : Obtenir la partie réelle du ","type":0},{"text":"nombre complexe","type":1},{"text":" 3 – 4i","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"Alternately, this can be regarded (using complex number notation) as simple amplitude modulation of a complex-valued carrier wave by a single complex-valued signal.","toRaw":"Autrement dit, cela peut être considéré (utilisant une notation en nombre complexe) comme une simple modulation d'amplitude d'une onde, exprimée en complexe, par un signal, exprimé en complexe.","from":[{"text":"Alternately, this can be regarded (using ","type":0},{"text":"complex number","type":1},{"text":" notation) as simple amplitude modulation of a ","type":0},{"text":"complex","type":2},{"text":"-valued carrier wave by a single ","type":0},{"text":"complex","type":2},{"text":"-valued signal.","type":0}],"to":[{"text":"Autrement dit, cela peut être considéré (utilisant une notation en ","type":0},{"text":"nombre complexe","type":1},{"text":") comme une simple modulation d'amplitude d'une onde, exprimée en ","type":0},{"text":"complexe","type":2},{"text":", par un signal, exprimé en ","type":0},{"text":"complexe","type":2},{"text":".","type":0}],"d2":"0","d0":"11","collapsed":false,"source":"CCMatrix (Wikipedia + CommonCrawl)"},{"fromRaw":"complex number 2 - work 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