Complex number ( $C$ )

A complex number has the form

x + i y

, with conjugate

x - i y

x^{2} + y^{2}

.

Polar form of a complex number ( $z = r e^{i θ}$ )

A nonzero complex number has the form

z = r e^{i θ}

with

r = ∣ z ∣ > 0

and an argument determined modulo

2 π

. Multiplication multiplies moduli and adds arguments. This makes powers and roots of a complex number particularly transparent.

Root of a complex number ( $w^{n} = z$ )

For

z = r e^{i θ} \neq = 0

and a positive integer

n

, all solutions of

w^{n} = z

are

r^{1/ n} e^{i (θ + 2 πk) / n}

for

k = 0, \dots, n - 1

. They differ by multiplication by roots of unity and are distinct. If

z = 0

, the sole root is zero.

Complex modulus ( $∣ z ∣$ )

The complex modulus of

z = x + i y

is

∣ z ∣ = x^{2} + y^{2} = z \overline{z}

. It is the Euclidean norm of

(x, y)

in the complex plane, and satisfies

∣ z w ∣ = ∣ z ∣∣ w ∣

and the triangle inequality.

Imaginary unit ( $i$ )

The imaginary unit is the complex number

i

satisfying

i^{2} = - 1

. Its integer powers repeat with period four.

Real part ( $Re z$ )

For a complex number

z = x + i y

, its real part is

Re z = x

.

Imaginary part ( $Im z$ )

For a complex number

z = x + i y

, its imaginary part is

Im z = y

.

Euler's formula

Euler's formula states that

e^{i θ} = cos θ + i sin θ

.

Modulus ( $∣ z ∣$ )

The modulus of a complex number

z = x + i y

is

∣ z ∣ = x^{2} + y^{2}

.

Complex conjugate ( $\overline{z}$ )

The complex conjugate of

z = x + i y

is

\overline{z} = x - i y

. It satisfies

z \overline{z} = ∣ z ∣^{2}

.

Complex conjugation ( $z \mapsto \overline{z}$ )

Complex conjugation sends a complex number

z = a + ib

to its complex conjugate

\overline{z} = a - ib

. It is an involution and a field automorphism of

C

, with fixed field

R

. Applied componentwise to a complex vector, it produces the conjugated vector appearing in a Choi state for a rank-one Kraus operator.

Complex argument ( $ar g z$ )

An argument of a nonzero complex number

z

is any angle

θ

for which

z = ∣ z ∣ e^{i θ}

. Arguments differ by integer multiples of

2 π

; choosing one representative defines a branch of the argument.

Complex plane

The complex plane identifies the complex number

x + i y

with the point

(x, y)

in the real plane.

Imaginary axis ( $i R$ )

The imaginary axis is the set of complex numbers

i y

with

y \in R

. It is the vertical coordinate axis in the complex plane.

Complex unit circle ( $S^{1}$ )

The complex unit circle is the subgroup

{z \in C : ∣ z ∣ = 1}

under multiplication.

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Complex number by

Wikipedia Bot  1

A complex number is a number that can be expressed in the form \( a + bi \), where: - \( a \) and \( b \) are real numbers, - \( i \) is the imaginary unit, defined as \( i = \sqrt{-1} \). In this representation: - \( a \) is called the **real part** of the complex number, - \( b \) is called the **imaginary part** of the complex number.

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Complex number by

Ciro Santilli  40 Updated 2025-07-16

An ordered pair of two real numbers with the complex addition and multiplication defined.

Forms both a:

division algebra if thought of $R^{2}$ with complex multiplication as the bilinear map of the algebra
field

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