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rdf:type: owl:Thing
rdfs:label: Komplexa tal Número complejo Bilangan kompleks Número complexo Zenbaki konplexu 複素数 عدد مركب Komplexní číslo Complex getal Nombre complex 복소수 Комплексне число Complex number 复数 (数学) Komplexe Zahl Numero complesso Kompleksa nombro Μιγαδικός αριθμός Uimhir choimpléascach Nombre complexe Liczby zespolone Комплексное число
rdfs:comment: En matemàtiques, un nombre complex és un nombre, , que es pot expressar en la forma , on i són nombres reals, i és la unitat imaginària, que satisfà la propietat fonamental . En l'expressió donada, s'anomea la part real del nombre complex, , i la part imaginària, Komplexní čísla (z latinského complexus, složený) vznikají rozšířením oboru reálných čísel tak, aby v něm každá algebraická rovnice měla příslušný počet řešení podle základní věty algebry. Například kvadratická rovnice x2 + 1 = 0 nemá v oboru reálných čísel řešení, protože její diskriminant (−4) je záporný a jeho odmocnina zde není definována. Komplexní číslo má dvě složky, reálnou a imaginární, a zapisuje se nejčastěji jako a + bi, přičemž i znamená imaginární jednotku, definovanou vztahem i2 = −1. Zmíněná rovnice pak má dvě řešení, ± i. Pro operace s komplexními čísly platí pravidla pro počítání s dvojčleny. Množinu všech komplexních čísel obvykle značíme ℂ. Liczby zespolone – liczby będące elementami rozszerzenia ciała liczb rzeczywistych o jednostkę urojoną to znaczy pierwiastek wielomianu Liczby zespolone rozszerzają koncepcję jednowymiarowej osi liczbowej do dwuwymiarowej płaszczyzny zespolonej, przy zastosowaniu osi poziomej do oznaczenia liczb rzeczywistych, a pionowej do oznaczenia liczb urojonych. Liczba zespolona postaci może być określona za pomocą współrzędnych na płaszczyźnie zespolonej. En mathématiques, l'ensemble des nombres complexes a été progressivement introduit au XVIe siècle par l’école mathématique italienne (Jérôme Cardan, Raphaël Bombelli, Tartaglia) afin d'exprimer les solutions des équations du troisième degré en toute généralité par les formules de Cardan, en utilisant notamment des « nombres » de carré négatif. Cet ensemble est actuellement défini comme une extension de l'ensemble des nombres réels, contenant en particulier un nombre imaginaire noté i tel que i2 = −1. Le carré de (−i) est aussi égal à −1 : (−i)2 = −1. In de wiskunde zijn complexe getallen een uitbreiding van de reële getallen. Zoals de reële getallen overeenkomen met punten op een rechte lijn, correspondeert elk complex getal met een punt uit een vlak. Een complex getal is zodoende een paar reële getallen en , dat gewoonlijk weergegeven wordt als . Hierin is (soms wordt ook gebruikt) een bijzonder complex getal, de imaginaire eenheid, met als eigenschap . Met complexe getallen in de vorm kan gewoon gerekend worden, met de extra rekenregel dat overal vervangen wordt door −1. Zenbaki konplexuak zenbaki erreal pare batez osatutako zenbakiak dira, hurrengo eran idatz daitezkeenak: , non i unitate irudikaria den propietatea betetzen duena. z zenbaki konplexuaren adierazpen binomikoa da. eta bi zenbaki erreal dira, a z-ren zati erreala eta b z-ren zati irudikaria direla diogu eta , idatzi ohi da. Adibidez, hau zenbaki konplexua da: non parte erreala den eta parte irudikaria. Zenbaki errealen mutzoa zenbaki konplexuen parte dira. Zenbaki konplexuen multzoa ikurarren budez adieraziko dugu eta honela definitu: 수학에서 복소수(複素數, 영어: complex number)는 (는 실수) 꼴의 수이다. 여기서 는 허수 단위라고 불리는 수이며, 을 만족시킨다. 는 실수부(real part), 는 허수부(imaginary part)라고 한다. 전자•전기를 다루는 계통에서는 전류 기호와의 혼동을 피하기 위해 대신 기호 를 쓰기도 한다. 복소수의 집합은 체를 이루며, 대수학의 기본 정리가 성립한다. 그러나 실수와 달리 표준적인 전순서를 줄 수 없다. 기하학적 관점에서, 복소수의 공간은 2차원 복소평면과 같으며, 이는 실수 공간을 나타내는 1차원 실수선을 확장하여 얻을 수 있다. 추상대수학적 관점에서, 복소수의 집합은 유일한 2차원 노름 이다. 복소수 집합의 기호는 또는 이다. Στα μαθηματικά, οι μιγαδικοί αριθμοί είναι μία επέκταση του συνόλου των πραγματικών αριθμών με την προσθήκη του στοιχείου , που λέγεται φανταστική μονάδα, και έχει την ιδιότητα: Κάθε μιγαδικός αριθμός μπορεί να γραφτεί με τη μορφή , όπου τα και είναι πραγματικοί αριθμοί και λέγονται πραγματικό μέρος και φανταστικό μέρος του μιγαδικού αριθμού, αντίστοιχα. Για παράδειγμα, ο είναι ένας μιγαδικός, με πραγματικό μέρος και φανταστικό μέρος . Un numero complesso è definito come un numero della forma , con e numeri reali e una soluzione dell'equazione detta unità immaginaria. I numeri complessi sono usati in tutti i campi della matematica, in molti campi della fisica (notoriamente in meccanica quantistica), nonché in ingegneria (specialmente in elettronica, telecomunicazioni ed elettrotecnica) per la loro utilità nel rappresentare onde elettromagnetiche e correnti elettriche ad andamento temporale sinusoidale. Ко́мпле́ксные чи́сла (от лат. complexus — связь, сочетание; о двойном ударении см. примечание) — числа вида где — вещественные числа, — мнимая единица, то есть число, для которого выполняется равенство: Множество комплексных чисел обычно обозначается символом Вещественные числа можно рассматривать как частный случай комплексных, они имеют вид Главное свойство — в нём выполняется основная теорема алгебры, то есть любой многочлен -й степени имеет корней. Доказано, что система комплексных чисел логически непротиворечива. Los números complejos, designados con la notación , son una extensión de los números reales y forman un cuerpo algebraicamente cerrado. Entre ambos conjuntos de números se cumple que , es decir: está estrictamente contenido en . Los números complejos incluyen todas las raíces de los polinomios, a diferencia de los reales. Todo número complejo puede representarse como la suma de un número real y un número imaginario (que es un múltiplo real de la unidad imaginaria, que se indica con la letra i, o en ). Em matemática, um número complexo é um elemento de um sistema numérico que contém os números reais e um elemento específico denotado i, chamado de unidade imaginária, e que satisfaz a equação i2 = −1. O fato de um número negativo não ter raiz quadrada parece ter sido claro para os matemáticos que se depararam com esta questão, até a concepção do modelo dos números complexos. Um número complexo é um número que pode ser escrito na forma , sendo e números reais e denota a unidade imaginária. Esta tem a propriedade sendo que e são chamados respectivamente parte real e parte imaginária de . Kompleksa nombro estas nombro, kiu havas aspekton z=a+bi, kie a kaj b estas reelaj nombroj, kaj i² egalas al la nombro -1. La signo i estas por imaginara unuo, a = Re z nomiĝas reela parto de kompleksa nombro kaj b = Im z - imaginara parto. Reelaj nombroj estas aparta kazo de kompleksaj nombroj, kie b=0. Operacioj de adicio kaj multipliko por kompleksaj nombroj estas difinitaj nature laŭ la koncernaj reguloj sur plurtermoj kaj kun kondiĉo , t.e. 数学における複素数（ふくそすう、（英: complex number）とは、2つの実数 a, b と虚数単位 i = √−1 を用いて z = a + bi と表すことのできる数のことである。1, i は実数体上線型独立であり、複素数は、係数体を実数とする、1, i の線型結合である。実数体 R 上の二次拡大環の元であるため、二元数の一つである。複素数全体からなる集合を、太字の C あるいは黒板太字で ℂ と表す。C は可換体である。体論の観点からは、複素数体 C は、実数体 R に √−1 を添加して得られる拡大体である。代数学の基本定理により、複素数体は代数的閉体である。複素数体はケーリー＝ディクソン代数（四元数、八元数、十六元数など）の基点となる体系であり、またさまざまな超複素数系の中で最もよく知られた例である。複素数の概念は、一次元の実数直線を二次元の複素数平面に拡張する。複素数全体に通常の大小関係を入れることはできない。つまり、複素数体 C は順序体でない。数学での分野、概念や構成において、考えている体構造が複素数体であるとき、それを、それらの概念等の名称に、多くは接頭辞「複素-」を付けることで反映させる。例えば、複素解析、複素行列、複素（係数）多項式、複素リー代数など。複數，為實數的延伸，它使任一多項式方程都有根。複數當中有個「虛數單位」，它是的一个平方根，即。任一複數都可表達為，其中及皆為實數，分別稱為複數之「實部」和「虛部」。複數的發現源於三次方程的根的表達式。數學上，「複」字表明所討論的數體為複數，如複矩陣、複變函數等。形式上，複數系統可以定義為普通實數的虛數i的代數擴展。這意味著複數可以作為變量i中的多項式進行加，減和乘，並施加規則。此外，複數也可以除以非零複數。總體而言，複數系統是一個域。在幾何上，複數通過將水平軸用於實部，將垂直軸用於虛部，將一維數線的概念擴展到二維複平面。這些數字的點位於複平面的垂直軸上。虛部為零的複數可以看作是實數。但是，複數允許使用更豐富的代數結構，其中包括在向量空間中不一定可用的附加運算。例如，兩個複數的乘積總是再次產生一個複數，並且不應將其誤認為是涉及向量的常規“乘積”。 Sa mhatamaitic, is éard is uimhir choimpléascach ann ná ball de chóras uimhreach a leathnaíonn na réaduimhreacha le huimhir speisialta darb ainm i, ar a dtugtar an t-aonad samhailteach agus a chomhlíonann an chothromóid ; is féidir gach uimhir choimpléascach a scríobh san fhoirm a + bi; is réaduimhreacha iad a agus b anseo. Toisc nach gcomhlíonann aon réaduimhir an chothromóid thuas, thug René Descartes uimhir shamhailteach ar i. San uimhir choimpléascach a+bi, tugtar an chuid réadach ar a agus tugtar an chuid shamhailteach ar b. Úsáidtear an tsiombail ar thacar na n-uimhreacha coimpléascacha. In ainneoin na hainmníochta stairiúla “samhailteach”, meastar go bhfuil uimhreacha coimpléascacha díreach chomh “réadach” leis na réaduimhreacha sa mhatamaitic, agus imríonn siad ról bunúsach sa chu Bilangan kompleks dalam matematika, adalah bilangan yang dinotasikan oleh , di mana a dan b adalah bilangan riil, dan i adalah suatu bilangan imajiner di mana i 2 = −1. Bilangan riil a disebut juga bagian riil dari bilangan kompleks, dan bilangan real b disebut bagian imajiner. Jika pada suatu bilangan kompleks, nilai b adalah 0, maka bilangan kompleks tersebut menjadi sama dengan bilangan real a. Sebagai contoh, 3 + 2i adalah bilangan kompleks dengan bagian riil 3 dan bagian imajiner 2i. In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Ко́мпле́ксні чи́сла — розширення поля дійсних чисел, зазвичай позначається .Будь-яке комплексне число може бути представлене як формальна сума , де і — дійсні числа, — уявна одиниця. Комплексні числа утворюють алгебрично замкнуте поле — це означає, що многочлен степеня n із комплексними коефіцієнтами має рівно n комплексних коренів (основна теорема алгебри). Це головна причина широкого застосування комплексних чисел у математиці. Крім того, застосування комплексних чисел дозволяє зручно і компактно формулювати багато математичних моделей у фізиці. * * Die komplexen Zahlen stellen eine Erweiterung der reellen Zahlen dar.Ziel der Erweiterung ist es, algebraische Gleichungen wie bzw. lösbar zu machen. Im Gegensatz zu den Erweiterungen reicht es hier nicht mehr aus, die Zahlen „linksseitig“ zu erweitern (ganze Zahlen) oder „dichter zu stopfen“ (rationale und reelle Zahlen), sondern man wechselt von einer Zahlengeraden zu einer Zahlenebene. De komplexa talen kan ses som en utvidgning av de reella talen. Ett komplext tal kan skrivas som där det reella talet a är realdelen, det reella talet b är imaginärdelen och i är den imaginära enheten med egenskapen Om b ≠ 0 så är z ett icke reellt komplext tal (till exempel 2 + 4i), och om a = 0 kallas talet rent imaginärt (till exempel 4i). Mängden av komplexa tal betecknas med C eller ℂ, och utgör en kropp. العدد المركب أو العدد العقدي (بالإنجليزية: Complex number)‏ هو أي عدد يكتب على الصورة حيث و عددان حقيقيان و عدد خيالي مربعه يساوي 1- (أي أن ) ويسمى وحدة تخيلية. ويسمى العدد الحقيقي بالجزء الحقيقي، والعدد الحقيقي بالجزء التخيلي. فمثلا، 3 + 2i هو عدد مركب، فيه 3 هو الجزء الحقيقي و 2 هو الجزء التخيلي. و عندما يكون "" (أي الجزء التخيلي) مساويا ل 0، فإن قيمة العدد المركب تساوي قيمة الجزء الحقيقي "" فقط، ويسمي العدد عددا حقيقيـا صرفا. وعندما يكون "" (أي الجزء الحقيقي) مساويا ل 0، يكون العدد تخيليـا صرفـا.
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dbp:cs1Dates: l
dbp:date: June 2020
dbp:first: E.D.
dbp:id: c/c024140
dbp:last: Solomentsev
dbp:ref: none
dbp:title: Complex number
dbp:year: 2001
dbo:abstract: Em matemática, um número complexo é um elemento de um sistema numérico que contém os números reais e um elemento específico denotado i, chamado de unidade imaginária, e que satisfaz a equação i2 = −1. O fato de um número negativo não ter raiz quadrada parece ter sido claro para os matemáticos que se depararam com esta questão, até a concepção do modelo dos números complexos. Um número complexo é um número que pode ser escrito na forma , sendo e números reais e denota a unidade imaginária. Esta tem a propriedade sendo que e são chamados respectivamente parte real e parte imaginária de . O conjunto dos números complexos, denotado por , contém o conjunto dos números reais. Munido de operações de adição e multiplicação obtidas por extensão das operações de mesma denominação nos números reais, adquire uma estrutura algébrica denominada corpo algebricamente fechado, sendo que esse fechamento consiste na propriedade que tem o conjunto de possuir todas as soluções de qualquer equação polinomial com coeficientes naquele mesmo conjunto (no caso, o conjunto dos complexos). O conjunto dos números complexos também pode ser entendido por seu isomorfismo com um espaço vetorial sobre , o conjunto dos reais. Além disso, a cada número complexo podemos atribuir um número real positivo chamado módulo, dado por: Liczby zespolone – liczby będące elementami rozszerzenia ciała liczb rzeczywistych o jednostkę urojoną to znaczy pierwiastek wielomianu Liczby zespolone rozszerzają koncepcję jednowymiarowej osi liczbowej do dwuwymiarowej płaszczyzny zespolonej, przy zastosowaniu osi poziomej do oznaczenia liczb rzeczywistych, a pionowej do oznaczenia liczb urojonych. Liczba zespolona postaci może być określona za pomocą współrzędnych na płaszczyźnie zespolonej. Liczby zespolone pozbawione części rzeczywistej, a zatem leżące bezpośrednio na osi pionowej płaszczyzny zespolonej, nazywane są liczbami urojonymi, zaś liczby pozbawione części urojonej, a więc leżące bezpośrednio na osi poziomej, to liczby rzeczywiste. Zbiór liczb zespolonych zawiera zatem w sobie zbiór liczb rzeczywistych, rozszerzony w celu umożliwienia rozwiązywania takich problemów, które nie posiadają rozwiązania w zbiorze liczb rzeczywistych. Poza matematyką liczby zespolone znajdują zastosowanie także w innych dziedzinach nauki, jak fizyka, chemia, biologia, ekonomia, elektrotechnika i statystyka. In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equationhas no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions and . Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule combined with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with {1, i} as a standard basis. This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm. In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two. Ко́мпле́ксні чи́сла — розширення поля дійсних чисел, зазвичай позначається .Будь-яке комплексне число може бути представлене як формальна сума , де і — дійсні числа, — уявна одиниця. Комплексні числа утворюють алгебрично замкнуте поле — це означає, що многочлен степеня n із комплексними коефіцієнтами має рівно n комплексних коренів (основна теорема алгебри). Це головна причина широкого застосування комплексних чисел у математиці. Крім того, застосування комплексних чисел дозволяє зручно і компактно формулювати багато математичних моделей у фізиці. Поле комплексних чисел можна розглядати як розширення поля дійсних чисел, в якому многочлен має корінь. Наступна модель показує можливість побудови такої системи чисел. Усі змісти комплексних чисел є ізоморфними розширеннями поля дійсних чисел , як і будь-які інші конструкції поля розкладу многочлена . Комплексне число можна визначити як упорядковану пару дійсних чисел . Введемо операції додавання і множення таких пар наступним чином: * * Дійсні числа є в цій моделі підмножиною множини комплексних чисел і представлені парами виду , причому операції з такими парами узгоджені зі звичайними додаванням і множенням дійсних чисел. Нуль зображується парою , одиниця — , а уявна одиниця — . На множині комплексних чисел нуль і одиниця мають ті ж властивості, що і на множині дійсних, а квадрат уявної одиниці, як легко перевірити, дорівнює , тобто . Нескладно показати, що визначені вище операції мають ті ж властивості, що й аналогічні операції з числами. Винятком є тільки властивості, пов'язані з відношенням порядку (більше-менше), тому що розширити порядок дійсних чисел, включивши в нього всі комплексні числа і при цьому зберігши звичайні властивості порядку, неможливо. Відомо також кілька узагальнень комплексних чисел, таких як кватерніони. De komplexa talen kan ses som en utvidgning av de reella talen. Ett komplext tal kan skrivas som där det reella talet a är realdelen, det reella talet b är imaginärdelen och i är den imaginära enheten med egenskapen Om b ≠ 0 så är z ett icke reellt komplext tal (till exempel 2 + 4i), och om a = 0 kallas talet rent imaginärt (till exempel 4i). Mängden av komplexa tal betecknas med C eller ℂ, och utgör en kropp. 수학에서 복소수(複素數, 영어: complex number)는 (는 실수) 꼴의 수이다. 여기서 는 허수 단위라고 불리는 수이며, 을 만족시킨다. 는 실수부(real part), 는 허수부(imaginary part)라고 한다. 전자•전기를 다루는 계통에서는 전류 기호와의 혼동을 피하기 위해 대신 기호 를 쓰기도 한다. 복소수의 집합은 체를 이루며, 대수학의 기본 정리가 성립한다. 그러나 실수와 달리 표준적인 전순서를 줄 수 없다. 기하학적 관점에서, 복소수의 공간은 2차원 복소평면과 같으며, 이는 실수 공간을 나타내는 1차원 실수선을 확장하여 얻을 수 있다. 추상대수학적 관점에서, 복소수의 집합은 유일한 2차원 노름 이다. 복소수 집합의 기호는 또는 이다. Komplexní čísla (z latinského complexus, složený) vznikají rozšířením oboru reálných čísel tak, aby v něm každá algebraická rovnice měla příslušný počet řešení podle základní věty algebry. Například kvadratická rovnice x2 + 1 = 0 nemá v oboru reálných čísel řešení, protože její diskriminant (−4) je záporný a jeho odmocnina zde není definována. Komplexní číslo má dvě složky, reálnou a imaginární, a zapisuje se nejčastěji jako a + bi, přičemž i znamená imaginární jednotku, definovanou vztahem i2 = −1. Zmíněná rovnice pak má dvě řešení, ± i. Pro operace s komplexními čísly platí pravidla pro počítání s dvojčleny. Množinu všech komplexních čísel obvykle značíme ℂ. Komplexní čísla lze interpretovat geometricky. Zde je příklad v kartézských pravoúhlých souřadnicích. Jako se reálná čísla zobrazují na reálné ose Re, budou imaginární čísla zobrazena na kolmé imaginární ose Im a každé komplexní číslo se zobrazí jako bod v rovině se souřadnicemi [x, y]. Číslo tvaru [x, 0] je reálné, číslo tvaru [0, y] je ryze imaginární. Absolutní hodnota komplexního čísla je pak vzdálenost bodu [x, y] od počátku souřadnic a číslo komplexně sdružené (tj. číslo [x, −y]) je zrcadlovým obrazem bodu [x, y] podle reálné osy x, tedy Re. Komplexní čísla jsou významná nejen v matematice, ale také ve fyzice, především v elektrotechnice, optice a hydrodynamice. En matemàtiques, un nombre complex és un nombre, , que es pot expressar en la forma , on i són nombres reals, i és la unitat imaginària, que satisfà la propietat fonamental . En l'expressió donada, s'anomea la part real del nombre complex, , i la part imaginària, El conjunt dels nombres complexos es representa per C, o per . Com que cada nombre complex ve determinat per les seves parts real i imaginària, geomètricament es pot identificar C amb els punts d'un pla, el pla complex, o pla d'Argand.En aquest pla, l'eix d'abscisses correspon als complexos amb part imaginària nul·la, que es poden identificar amb els nombres reals; així, el pla complex conté la recta numèrica. En el mateix pla, l'eix d'ordenades correspon als complexos amb part real nul·la, anomenats imaginaris purs; són els complexos de la forma , amb real. A banda de la seva importància en àlgebra, els nombres complexos són una eina fonamental en pràcticament totes les branques de les matemàtiques. Igual com amb funcions de variable real, es pot fer anàlisi matemàtica amb funcions de variable complexa; la teoria corresponent s'anomena anàlisi complexa, i té característiques que la fan molt diferent de l'anàlisi real. Més enllà de les matemàtiques, els nombres complexos tenen aplicacions en la major part de les ciències i la tecnologia. Moltes d'aquestes aplicacions són simplement una conseqüència dels avantatges de treballar amb nombres complexos en lloc de reals, però en alguns camps específics, com ara la mecànica quàntica o la teoria quàntica de camps, l'ús de la variable complexa en la descripció de les entitats físiques és essencial. Zenbaki konplexuak zenbaki erreal pare batez osatutako zenbakiak dira, hurrengo eran idatz daitezkeenak: , non i unitate irudikaria den propietatea betetzen duena. z zenbaki konplexuaren adierazpen binomikoa da. eta bi zenbaki erreal dira, a z-ren zati erreala eta b z-ren zati irudikaria direla diogu eta , idatzi ohi da. Adibidez, hau zenbaki konplexua da: non parte erreala den eta parte irudikaria. Zenbaki errealen mutzoa zenbaki konplexuen parte dira. Zenbaki konplexuen multzoa ikurarren budez adieraziko dugu eta honela definitu: Zenbaki errealen hedapen bezala, hauen eragiketak betetzen dituzte, eta beste propietate garrantzitsu batzuk ere betetzen dituzte. Adibidez, zenbaki errealetan ez bezala, polinomio orok ebazpena dauka zenbaki konplexuen multzoan. 数学における複素数（ふくそすう、（英: complex number）とは、2つの実数 a, b と虚数単位 i = √−1 を用いて z = a + bi と表すことのできる数のことである。1, i は実数体上線型独立であり、複素数は、係数体を実数とする、1, i の線型結合である。実数体 R 上の二次拡大環の元であるため、二元数の一つである。複素数全体からなる集合を、太字の C あるいは黒板太字で ℂ と表す。C は可換体である。体論の観点からは、複素数体 C は、実数体 R に √−1 を添加して得られる拡大体である。代数学の基本定理により、複素数体は代数的閉体である。複素数体はケーリー＝ディクソン代数（四元数、八元数、十六元数など）の基点となる体系であり、またさまざまな超複素数系の中で最もよく知られた例である。複素数の概念は、一次元の実数直線を二次元の複素数平面に拡張する。複素数全体に通常の大小関係を入れることはできない。つまり、複素数体 C は順序体でない。数学での分野、概念や構成において、考えている体構造が複素数体であるとき、それを、それらの概念等の名称に、多くは接頭辞「複素-」を付けることで反映させる。例えば、複素解析、複素行列、複素（係数）多項式、複素リー代数など。 Kompleksa nombro estas nombro, kiu havas aspekton z=a+bi, kie a kaj b estas reelaj nombroj, kaj i² egalas al la nombro -1. La signo i estas por imaginara unuo, a = Re z nomiĝas reela parto de kompleksa nombro kaj b = Im z - imaginara parto. Reelaj nombroj estas aparta kazo de kompleksaj nombroj, kie b=0. Operacioj de adicio kaj multipliko por kompleksaj nombroj estas difinitaj nature laŭ la koncernaj reguloj sur plurtermoj kaj kun kondiĉo , t.e. Multobligado kaj adiciado estas komutecaj kaj asociecaj kaj estas ligitaj kun rilato de distribueco. Por ili ekzistas ankaŭ inversaj operacioj, t.e. subtraho kaj divido (escepte de divido je 0). Tiamaniere, kompleksaj nombroj faras kampon kaj estas signataj per . Tial unu prezento de kompleksaj nombroj estas per vektoroj, tiel formantaj la kompleksan ebenon: (anstataŭ a ofte uzatas x, kaj anstataŭ b uzatas y) Sa mhatamaitic, is éard is uimhir choimpléascach ann ná ball de chóras uimhreach a leathnaíonn na réaduimhreacha le huimhir speisialta darb ainm i, ar a dtugtar an t-aonad samhailteach agus a chomhlíonann an chothromóid ; is féidir gach uimhir choimpléascach a scríobh san fhoirm a + bi; is réaduimhreacha iad a agus b anseo. Toisc nach gcomhlíonann aon réaduimhir an chothromóid thuas, thug René Descartes uimhir shamhailteach ar i. San uimhir choimpléascach a+bi, tugtar an chuid réadach ar a agus tugtar an chuid shamhailteach ar b. Úsáidtear an tsiombail ar thacar na n-uimhreacha coimpléascacha. In ainneoin na hainmníochta stairiúla “samhailteach”, meastar go bhfuil uimhreacha coimpléascacha díreach chomh “réadach” leis na réaduimhreacha sa mhatamaitic, agus imríonn siad ról bunúsach sa chur síos eolaíochta ar an domhan nádúrtha. Ba é ón Iodáil a rinne cur síos ar na rialacha ailgéabracha do na huimhreacha coimpléascacha. Rinne an tÉireannach William Rowan Hamilton forbairt air seo le teoiric na gceathairníon. Los números complejos, designados con la notación , son una extensión de los números reales y forman un cuerpo algebraicamente cerrado. Entre ambos conjuntos de números se cumple que , es decir: está estrictamente contenido en . Los números complejos incluyen todas las raíces de los polinomios, a diferencia de los reales. Todo número complejo puede representarse como la suma de un número real y un número imaginario (que es un múltiplo real de la unidad imaginaria, que se indica con la letra i, o en ). Los números complejos son la herramienta de trabajo del álgebra, análisis, así como de ramas de las matemáticas puras y aplicadas como variable compleja, ecuaciones diferenciales, facilita el cálculo de integrales, en aerodinámica, hidrodinámica y electromagnetismo entre otras de gran importancia. Además, los números complejos se utilizan por doquier en matemáticas, en muchos campos de la física (notoriamente en la mecánica cuántica) y en ingeniería, especialmente en la electrónica y las telecomunicaciones, por su utilidad para representar las ondas electromagnéticas y la corriente eléctrica. En matemáticas, estos números constituyen un cuerpo y, en general, se consideran como puntos del plano: el plano complejo. Este cuerpo contiene a los números reales y los imaginarios puros. Bilangan kompleks dalam matematika, adalah bilangan yang dinotasikan oleh , di mana a dan b adalah bilangan riil, dan i adalah suatu bilangan imajiner di mana i 2 = −1. Bilangan riil a disebut juga bagian riil dari bilangan kompleks, dan bilangan real b disebut bagian imajiner. Jika pada suatu bilangan kompleks, nilai b adalah 0, maka bilangan kompleks tersebut menjadi sama dengan bilangan real a. Sebagai contoh, 3 + 2i adalah bilangan kompleks dengan bagian riil 3 dan bagian imajiner 2i. Bilangan kompleks dapat ditambah, dikurang, dikali, dan dibagi seperti bilangan riil; namun bilangan kompleks juga mempunyai sifat-sifat tambahan yang menarik. Misalnya, setiap persamaan aljabar polinomial mempunyai solusi bilangan kompleks, tidak seperti bilangan riil yang hanya memiliki sebagian. Dalam bidang-bidang tertentu (seperti teknik elektro, di mana i digunakan sebagai simbol untuk arus listrik), bilangan kompleks ditulis a + bj. Στα μαθηματικά, οι μιγαδικοί αριθμοί είναι μία επέκταση του συνόλου των πραγματικών αριθμών με την προσθήκη του στοιχείου , που λέγεται φανταστική μονάδα, και έχει την ιδιότητα: Κάθε μιγαδικός αριθμός μπορεί να γραφτεί με τη μορφή , όπου τα και είναι πραγματικοί αριθμοί και λέγονται πραγματικό μέρος και φανταστικό μέρος του μιγαδικού αριθμού, αντίστοιχα. Για παράδειγμα, ο είναι ένας μιγαδικός, με πραγματικό μέρος και φανταστικό μέρος . Για τους μιγαδικούς αριθμούς ορίζονται οι πράξεις της πρόσθεσης, της αφαίρεσης, του πολλαπλασιασμού και της διαίρεσης, όπως και στους πραγματικούς αριθμούς. Στην ορολογία των μαθηματικών, αυτό σημαίνει ότι το σύνολο των μιγαδικών είναι σώμα. Η βασική διαφορά των μιγαδικών αριθμών με τους πραγματικούς είναι η ύπαρξη του στοιχείου i και των πολλαπλασίων του, που όταν υψωθούν στο τετράγωνο δίνουν αρνητικούς πραγματικούς αριθμούς. Επιπλέον, στους μιγαδικούς δεν ορίζεται η διάταξη, δηλαδή δεν έχει έννοια να συγκρίνουμε δύο μιγαδικούς ώστε να πούμε ότι ένας μιγαδικός αριθμός είναι μεγαλύτερος ή μικρότερος από κάποιον άλλον μιγαδικό αριθμό Οι μιγαδικοί αριθμοί έχουν, μεταξύ άλλων, σημαντικές εφαρμογές στη λύση διαφορικών εξισώσεων αλλά και στη μελέτη διάφορων φυσικών προβλημάτων οπτικής, κυματικής, κβαντομηχανικής και ηλεκτρονικής. Ко́мпле́ксные чи́сла (от лат. complexus — связь, сочетание; о двойном ударении см. примечание) — числа вида где — вещественные числа, — мнимая единица, то есть число, для которого выполняется равенство: Множество комплексных чисел обычно обозначается символом Вещественные числа можно рассматривать как частный случай комплексных, они имеют вид Главное свойство — в нём выполняется основная теорема алгебры, то есть любой многочлен -й степени имеет корней. Доказано, что система комплексных чисел логически непротиворечива. Так же как и для вещественных чисел, для комплексных чисел определены операции сложения, вычитания, умножения и деления. Однако многие свойства комплексных чисел отличаются от свойств вещественных чисел; например, нельзя указать, какое из двух комплексных чисел больше или меньше. Удобно представлять комплексные числа точками на комплексной плоскости; например, для изображения сопряжённых чисел используется операция отражения относительно горизонтальной оси. Альтернативное представление комплексного числа в тригонометрической записи оказалось полезным для вычисления степеней и корней. Функции комплексного аргумента изучаются в комплексном анализе. Первоначально идея о необходимости использования комплексных чисел возникла в результате формального решения кубических уравнений, при котором в формуле Кардано под знаком квадратного корня получалось отрицательное число. Большой вклад в исследование комплексных чисел внесли такие математики, как Эйлер, который ввёл общепризнанное обозначение для мнимой единицы, Декарт, Гаусс. Сам термин «комплексное число» ввёл в науку Гаусс в 1831 году. Уникальные свойства комплексных чисел и функций нашли широкое применение для решения многих практических задач в различных областях математики, физики и техники: в обработке сигналов, теории управления, электромагнетизме, теории колебаний, теории упругости и многих других. Преобразования комплексной плоскости оказались полезны в картографии и гидродинамике. Современная физика полагается на описание мира с помощью квантовой механики, которая опирается на систему комплексных чисел. Известно также несколько обобщений комплексных чисел — например, кватернионы. En mathématiques, l'ensemble des nombres complexes a été progressivement introduit au XVIe siècle par l’école mathématique italienne (Jérôme Cardan, Raphaël Bombelli, Tartaglia) afin d'exprimer les solutions des équations du troisième degré en toute généralité par les formules de Cardan, en utilisant notamment des « nombres » de carré négatif. Cet ensemble est actuellement défini comme une extension de l'ensemble des nombres réels, contenant en particulier un nombre imaginaire noté i tel que i2 = −1. Le carré de (−i) est aussi égal à −1 : (−i)2 = −1. Tout nombre complexe peut s'écrire sous la forme a + i b où a et b sont des nombres réels. On peut munir l'ensemble des nombres complexes d'une addition et d'une multiplication qui en font un corps commutatif contenant le corps des nombres réels. Il est appelé corps des nombres complexes et se note ℂ. La notion de valeur absolue définie sur l'ensemble des nombres réels peut être étendue à l'ensemble des nombres complexes et prend alors le nom de module. Mais on ne peut pas munir l'ensemble des nombres complexes d'une relation d'ordre qui en ferait un corps totalement ordonné, c'est-à-dire qu'il n'est pas possible de comparer deux complexes en respectant les règles opératoires valables pour les nombres réels. Ce n'est qu'à partir du XIXe siècle, sous l'impulsion de l'abbé Buée et de Jean-Robert Argand (plan d'Argand), puis avec les travaux de Gauss et de Cauchy, que se développe l'aspect géométrique des nombres complexes. On les associe à des vecteurs ou des points du plan. Les transformations du plan s'expriment alors sous forme de transformations complexes. En algèbre, le théorème de d'Alembert-Gauss énonce qu'un polynôme complexe non constant possède toujours au moins une racine complexe. Le corps des nombres complexes est dit algébriquement clos. On peut ainsi identifier le degré d'un polynôme complexe non nul au nombre de ses racines comptées avec leur ordre de multiplicité. En analyse, l'exponentielle complexe permet de simplifier l'étude des séries de Fourier, puis de définir la transformée de Fourier. La branche de l'analyse complexe concerne l'étude des fonctions dérivables au sens complexe, appelées fonctions holomorphes. En physique, les nombres complexes sont utilisés pour décrire le comportement d'oscillateurs électriques ou les phénomènes ondulatoires en électromagnétisme (Re(eiωt) représentant une sinusoïde). Dans le domaine de l'électricité et notamment de l'électrocinétique, on note souvent j l'unité imaginaire, la notation usuelle pouvant prêter à confusion avec le symbole d'une intensité électrique. Ils sont aussi essentiels dans la formulation mathématique de la mécanique quantique. Die komplexen Zahlen stellen eine Erweiterung der reellen Zahlen dar.Ziel der Erweiterung ist es, algebraische Gleichungen wie bzw. lösbar zu machen. Im Gegensatz zu den Erweiterungen reicht es hier nicht mehr aus, die Zahlen „linksseitig“ zu erweitern (ganze Zahlen) oder „dichter zu stopfen“ (rationale und reelle Zahlen), sondern man wechselt von einer Zahlengeraden zu einer Zahlenebene. Da die Quadrate aller reellen Zahlen größer oder gleich 0 sind, kann die Lösung der Gleichung keine reelle Zahl sein. Man braucht eine ganz neue Zahl, die man üblicherweise nennt, mit der Eigenschaft . Diese Zahl wird als imaginäre Einheit bezeichnet. Komplexe Zahlen werden nun als Summe definiert, wobei und reelle Zahlen sind und die oben definierte imaginäre Einheit ist.Auf die so definierten komplexen Zahlen lassen sich die üblichen Rechenregeln für reelle Zahlen anwenden, wobei wie eine Konstante verwendet wird und durch ersetzt werden kann und umgekehrt. Für die Menge der komplexen Zahlen wird das Symbol (ℂ als Unicode-Zeichen U+2102, siehe Buchstaben mit Doppelstrich) verwendet. Der so konstruierte Zahlenbereich der komplexen Zahlen bildet einen Erweiterungskörper der reellen Zahlen und hat eine Reihe vorteilhafter Eigenschaften, die sich in vielen Bereichen der Natur- und Ingenieurwissenschaften als äußerst nützlich erwiesen haben. Einer der Gründe für diese positiven Eigenschaften ist die algebraische Abgeschlossenheit der komplexen Zahlen. Dies bedeutet, dass jede algebraische Gleichung positiven Grades über den komplexen Zahlen eine Lösung besitzt, was für reelle Zahlen nicht gilt. Diese Eigenschaft ist der Inhalt des Fundamentalsatzes der Algebra. Ein weiterer Grund ist ein Zusammenhang zwischen trigonometrischen Funktionen und der Exponentialfunktion (Eulerformel), der über die komplexen Zahlen hergestellt werden kann. Ferner ist jede auf einer offenen Menge einmal komplex differenzierbare Funktion dort auch beliebig oft differenzierbar – anders als in der Analysis der reellen Zahlen. Die Eigenschaften von Funktionen mit komplexen Argumenten sind Gegenstand der Funktionentheorie, auch komplexe Analysis genannt. In der Elektrotechnik wird stattdessen der Buchstabe verwendet, um einer Verwechslung mit einer (durch oder bezeichneten) von der Zeit abhängigen Stromstärke vorzubeugen, allerdings erhöht dies die Verwechslungsgefahr mit der Stromdichte in der Elektrodynamik. In de wiskunde zijn complexe getallen een uitbreiding van de reële getallen. Zoals de reële getallen overeenkomen met punten op een rechte lijn, correspondeert elk complex getal met een punt uit een vlak. Een complex getal is zodoende een paar reële getallen en , dat gewoonlijk weergegeven wordt als . Hierin is (soms wordt ook gebruikt) een bijzonder complex getal, de imaginaire eenheid, met als eigenschap . Met complexe getallen in de vorm kan gewoon gerekend worden, met de extra rekenregel dat overal vervangen wordt door −1. De schrijfwijze laat zien dat een complex getal de som is van een reëel getal en een imaginair getal, of anders gezegd, een lineaire combinatie met reële coëfficiënten van 1 en . De extra mogelijkheden die het rekenen met complexe getallen biedt, hebben geleid tot allerlei nuttige toepassingen in vooral alles wat met trillingen en golven te maken heeft, zoals het grootste deel van de natuurkunde, de elektrotechniek, de meet- en regeltechniek en vele andere technische disciplines. Uit de definitie van complexe getallen blijkt dat reële getallen ook complexe getallen zijn. Binnen een context waarin nog niet is gespecificeerd of alleen reële of alle complexe getallen worden beschouwd betekent "complex getal" meestal "niet noodzakelijk reëel complex getal". Afhankelijk van de context wordt met een complex getal ook weleens bedoeld een niet-reëel complex getal. Un numero complesso è definito come un numero della forma , con e numeri reali e una soluzione dell'equazione detta unità immaginaria. I numeri complessi sono usati in tutti i campi della matematica, in molti campi della fisica (notoriamente in meccanica quantistica), nonché in ingegneria (specialmente in elettronica, telecomunicazioni ed elettrotecnica) per la loro utilità nel rappresentare onde elettromagnetiche e correnti elettriche ad andamento temporale sinusoidale. In matematica i numeri complessi formano un campo (nonché un'algebra reale bidimensionale) e sono generalmente visualizzati come punti di un piano, detto piano complesso. La proprietà più importante dei numeri complessi è basata sul teorema fondamentale dell'algebra, secondo il quale qualunque equazione polinomiale di grado ha soluzioni complesse, non necessariamente distinte. العدد المركب أو العدد العقدي (بالإنجليزية: Complex number)‏ هو أي عدد يكتب على الصورة حيث و عددان حقيقيان و عدد خيالي مربعه يساوي 1- (أي أن ) ويسمى وحدة تخيلية. ويسمى العدد الحقيقي بالجزء الحقيقي، والعدد الحقيقي بالجزء التخيلي. فمثلا، 3 + 2i هو عدد مركب، فيه 3 هو الجزء الحقيقي و 2 هو الجزء التخيلي. و عندما يكون "" (أي الجزء التخيلي) مساويا ل 0، فإن قيمة العدد المركب تساوي قيمة الجزء الحقيقي "" فقط، ويسمي العدد عددا حقيقيـا صرفا. وعندما يكون "" (أي الجزء الحقيقي) مساويا ل 0، يكون العدد تخيليـا صرفـا. من الممكن إجراء العمليات الحسابية العادية على الأعداد المركبة كالجمع والطرح والضرب والقسمة بطريقة تماثل الأعداد الحقيقية مع بعض الاختلافات خاصة في عملية القسمة. ولكنها أيضـا تتمتع بخصائص أخرى تمكنها من حل كافة المعادلات الجبرية العادية التي يصعب حلها باستخدام الأعداد الحقيقية فقط. عندما وجد الرياضيون أن المعادلة مستحيلة الحل في مجموعة الأعداد الحقيقية كان لا بد من وضع حل لها. لذلك تمّ إيجاد عدد جديد هو العدد التخيلي i. وتعريف العدد i هو الجذر التربيعي للعدد 1-.وهنا يكمن التعقيد. فمن المعلوم أنه ليس للعدد 1- جذر تربيعي، ولكن هذا في الأعداد الحقيقية. فكما أنه لا وجود للعدد 5- في الأعداد الطبيعية ولكنه موجود في الأعداد الصحيحة (والحال نفسه بالنسبة للعدد ) فالرياضيات هي علم وضعه البشر ولهم الحق في تطويره وتجديده وفق قواعد واضحة تخضع للمنطق الرياضي ولا تنافي المبادئ الرياضية والموضوعات والبديهيات في علم الرياضيات. 複數，為實數的延伸，它使任一多項式方程都有根。複數當中有個「虛數單位」，它是的一个平方根，即。任一複數都可表達為，其中及皆為實數，分別稱為複數之「實部」和「虛部」。複數的發現源於三次方程的根的表達式。數學上，「複」字表明所討論的數體為複數，如複矩陣、複變函數等。形式上，複數系統可以定義為普通實數的虛數i的代數擴展。這意味著複數可以作為變量i中的多項式進行加，減和乘，並施加規則。此外，複數也可以除以非零複數。總體而言，複數系統是一個域。在幾何上，複數通過將水平軸用於實部，將垂直軸用於虛部，將一維數線的概念擴展到二維複平面。這些數字的點位於複平面的垂直軸上。虛部為零的複數可以看作是實數。但是，複數允許使用更豐富的代數結構，其中包括在向量空間中不一定可用的附加運算。例如，兩個複數的乘積總是再次產生一個複數，並且不應將其誤認為是涉及向量的常規“乘積”。
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