mathematics, a complex number is a number that can be expressed

7 min readMay 29, 2021

In mathematics, 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 symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no “real” number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, 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 {\displaystyle \mathbb {C} }\mathbb {C} 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.[1][2][3][4][a]

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 polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation {\displaystyle (x+1)^{2}=-9}{\displaystyle (x+1)^{2}=-9} has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions −1 + 3i and −1 − 3i.

Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i2 = −1 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 form also 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.

Contents
1 Definition
2 Notation
3 Visualization
3.1 Cartesian complex plane
3.2 Polar complex plane
3.3 Complex graphs
4 History
5 Relations and operations
5.1 Equality
5.2 Ordering
5.3 Conjugate
5.4 Addition and subtraction
5.5 Multiplication
5.6 Reciprocal and division
5.7 Multiplication and division in polar form
5.8 Square root
5.9 Exponential function
5.10 Complex logarithm
5.11 Exponentiation
6 Properties
6.1 Field structure
6.2 Solutions of polynomial equations
6.3 Algebraic characterization
6.4 Characterization as a topological field
7 Formal construction
7.1 Construction as ordered pairs
7.2 Construction as a quotient field
7.3 Matrix representation of complex numbers
8 Complex analysis
8.1 Complex exponential and related functions
8.2 Holomorphic functions
9 Applications
9.1 Geometry
9.2 Algebraic number theory
9.3 Analytic number theory
9.4 Improper integrals
9.5 Dynamic equations
9.6 In applied mathematics
9.7 In physics
10 Generalizations and related notions
11 See also
12 Notes
13 References
13.1 Works cited
14 Further reading
14.1 Mathematical
14.2 Historical
Definition

An illustration of the complex number z = x + iy on the complex plane. The real part is x, and its imaginary part is y.
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.[6][3]

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.[7][8][3]

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).[9]

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mathematics, a complex number is a number that can be expressed

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