# Complex number

A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i satisfies i2 = −1.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a solution of the equation x2 = −1. Because no real number satisfies this equation, i is called an imaginary number. For the complex number a + bi, a is called the real part, and b is called the imaginary part. 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.[note 1][1]

Complex numbers allow solutions to certain equations that have no solutions in real numbers. For example, the equation

${\displaystyle (x+1)^{2}=-9}$

has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with an indeterminate i (sometimes called the imaginary unit) that is taken to satisfy the relation i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1:

${\displaystyle ((-1+3i)+1)^{2}=(3i)^{2}=\left(3^{2}\right)\left(i^{2}\right)=9(-1)=-9,}$
${\displaystyle ((-1-3i)+1)^{2}=(-3i)^{2}=(-3)^{2}\left(i^{2}\right)=9(-1)=-9.}$

According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. In contrast, some polynomial equations with real coefficients have no solution in real numbers. The 16th century Italian mathematician Gerolamo Cardano is credited with introducing complex numbers in his attempts to find solutions to cubic equations.[2]

Formally, the complex number system can be defined as the algebraic extension of the ordinary real numbers by an imaginary number i.[3] This means that complex numbers can be added, subtracted, and multiplied, as polynomials in the variable i, with the rule i2 = −1 imposed. Furthermore, complex numbers can also be divided by nonzero complex numbers. Overall, the complex number system is a field.

Geometrically, complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary; the points for these numbers lie on the vertical axis of the complex plane. A complex number whose imaginary part is zero can be viewed as a real number; its point lies on the horizontal axis of the complex plane. 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.

The geometric identification of the complex numbers with the complex plane, which is a Euclidean plane (${\displaystyle \mathbb {R} ^{2}}$), makes their structure as a real 2-dimensional vector space evident. Real and imaginary parts of a complex number may be taken as components of a vector with respect to the canonical standard basis. The addition of complex numbers is thus immediately depicted as the usual component-wise addition of vectors. However, the complex numbers allow for a richer algebraic structure, comprising additional operations, that are not necessarily available in a vector space; for example, the multiplication of two complex numbers always yields again a complex number, and should not be mistaken for the usual "products" involving vectors, like the scalar multiplication, the scalar product or other (sesqui)linear forms, available in many vector spaces; and the broadly exploited vector product exists only in an orientation-dependent form in three dimensions.

## Definition

An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.

Based on the concept of real numbers, a complex number is a number of the form a + bi, where a and b are real numbers and i is an indeterminate satisfying i2 = −1. For example, 2 + 3i is a complex number.[4]

This way, a complex number is defined as a polynomial with real coefficients in the single indeterminate i, for which the relation i2 + 1 = 0 is imposed. Based on this definition, complex numbers can be added and multiplied, using the addition and multiplication for polynomials. The relation i2 + 1 = 0 induces the equalities i4k = 1, i4k+1 = i, i4k+2 = −1, and i4k+3 = −i, which hold for all integers k; these allow the reduction of any polynomial that results from the addition and multiplication of complex numbers to a linear polynomial in i, again of the form a + bi with real coefficients a, b.

The real number a is called the real part of the complex number a + bi; the real number b is called its imaginary part. To emphasize, the imaginary part does not include a factor i; that is, the imaginary part is b, not bi.[5][6]

Formally, the complex numbers are defined as the quotient ring of the polynomial ring in the indeterminate i, by the ideal generated by the polynomial i2 + 1 (see below).[7]

Other Languages
Afrikaans: Komplekse getal
Alemannisch: Komplexe Zahl
العربية: عدد مركب
aragonés: Numero complexo
অসমীয়া: জটিল সংখ্যা
azərbaycanca: Kompleks ədədlər
Bân-lâm-gú: Ho̍k-cha̍p-sò͘
башҡортса: Комплекслы һан
беларуская: Камплексны лік
беларуская (тарашкевіца)‎: Камплексны лік
bosanski: Kompleksan broj
Cymraeg: Rhif cymhlyg
Deutsch: Komplexe Zahl
emiliàn e rumagnòl: Nómmer cumplês
Esperanto: Kompleksa nombro
فارسی: عدد مختلط
føroyskt: Komplekst tal
français: Nombre complexe

한국어: 복소수
հայերեն: Կոմպլեքս թիվ
hrvatski: Kompleksni broj
Bahasa Indonesia: Bilangan kompleks
interlingua: Numero complexe
íslenska: Tvinntölur
қазақша: Кешен сан
Kiswahili: Namba changamano
Lingua Franca Nova: Numero complesal
la .lojban.: relcimdyna'u
lumbaart: Numer compless
македонски: Комплексен број
Malagasy: Isa haro
മലയാളം: മിശ്രസംഖ്യ
Bahasa Melayu: Nombor kompleks
မြန်မာဘာသာ: ကွန်ပလက်စ်ကိန်း
Nederlands: Complex getal

Nordfriisk: Kompleks taal
norsk nynorsk: Komplekse tal
oʻzbekcha/ўзбекча: Kompleks sonlar
ភាសាខ្មែរ: ចំនួនកុំផ្លិច
Piemontèis: Nùmer compless
português: Número complexo
română: Număr complex
русиньскый: Комплексне чісло
Simple English: Complex number
slovenčina: Komplexné číslo
slovenščina: Kompleksno število
Soomaaliga: Tiro kakan
српски / srpski: Комплексан број
srpskohrvatski / српскохрватски: Kompleksan broj
svenska: Komplexa tal
татарча/tatarça: Комплекс сан
українська: Комплексне число
Tiếng Việt: Số phức

West-Vlams: Complexe getalln