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Complex Numbers

Operations, polar form, powers, and roots on the complex plane.

Inputs

Two complex numbers z₁ and z₂.

z₁ = a + bi
z₂ = c + di
Add, multiply, divide complex numbers in rectangular and polar form.

About this topic

Arithmetic Operations Add, subtract, multiply, and divide complex numbers.

Complex numbers behave like 2D vectors under addition, but multiplication and division use special rules that involve the imaginary unit i (where i² = −1).

Key formulas & rules

  • •Addition: (a+bi) + (c+di) = (a+c) + (b+d)i
  • •Subtraction: (a+bi) − (c+di) = (a−c) + (b−d)i
  • •Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i
  • •Division: multiply numerator and denominator by the conjugate
  • •Addition is commutative: z₁ + z₂ = z₂ + z₁
  • •Multiplication is commutative: z₁·z₂ = z₂·z₁

Examples

Input Output Note
(3+4i) + (1+2i) 4+6i Add matching parts
(3+4i) − (1+2i) 2+2i Subtract matching parts
(3+4i)(1+2i) −5+10i Use FOIL, i² = −1
(3+4i) ÷ (1−2i) −1+2i Multiply by conjugate

Real-world uses

Electrical impedance Signal processing Quantum mechanics
Polar & Rectangular Form Convert between (a + bi) and (r, θ) representations.

A complex number can be written either by its Cartesian coordinates (a, b) or by its distance from the origin r and angle θ from the positive real axis.

Key formulas & rules

  • •Rectangular: z = a + bi
  • •Polar: z = r(cos θ + i sin θ)
  • •Magnitude: r = √(a² + b²)
  • •Argument: θ = atan2(b, a)
  • •Rect from polar: a = r·cos θ, b = r·sin θ
  • •θ is measured counterclockwise from the positive real axis

Examples

Input Output Note
z = 3 + 4i r = 5, θ = 53.13° Pythagorean triple
z = 1 + i r = √2, θ = 45° Classic 45° case
r=2, θ=60° z = 1 + 1.732i Polar → Rect
r=1, θ=180° z = −1 On negative real axis

Real-world uses

Phasor diagrams AC circuit analysis Rotations in graphics
Powers & Roots (De Moivre) Raise to a power or extract all n-th roots.

De Moivre's theorem connects complex powers and roots to trigonometric identities. Every non-zero complex number has exactly n distinct n-th roots, evenly spaced around a circle.

Key formulas & rules

  • •De Moivre: zⁿ = rⁿ(cos nθ + i sin nθ)
  • •Roots: z^(1/n) has exactly n distinct values
  • •Root formula: r^(1/n)·e^(i(θ + 2πk)/n), k = 0, 1, …, n−1
  • •n-th roots are spaced 360°/n apart on a circle of radius r^(1/n)
  • •Euler form: z = r·e^(iθ) simplifies powers
  • •Multiplying z by z rotates by θ and scales by r

Examples

Input Output Note
(1 + i)² 2i Square rotates 90°
(1 + i)⁴ −4 Four rotations
√(−1) i, −i Two square roots
∛(1) 1, ω, ω² Cube roots of unity

Real-world uses

Roots of unity in Fourier analysis Rotations in 2D graphics Signal modulation
Properties & Conjugate Magnitude, argument, conjugate, and Euler form of a single number.

Every complex number has several derived quantities that describe its size, direction, and relationship to the real axis.

Key formulas & rules

  • •Magnitude: |z| = √(a² + b²)
  • •Argument: θ = atan2(b, a)
  • •Conjugate: z̄ = a − bi (mirror over real axis)
  • •z · z̄ = |z|² = a² + b² (always real, non-negative)
  • •Real part: Re(z) = a, Imaginary part: Im(z) = b
  • •Euler form: z = |z|·e^(iθ)

Examples

Input Output Note
z = 3 + 4i |z| = 5, z̄ = 3 − 4i Conjugate mirrors
z = 5 |z| = 5, θ = 0° Real numbers
z = 3i |z| = 3, θ = 90° Pure imaginary
z · z̄ (3+4i) 25 Always |z|²

Real-world uses

Distance calculations Absolute error magnitude Signal power
Imaginary Unit i The square root of −1 that unlocks complex math.

The imaginary unit i is defined so that i² = −1. It extends the real number line into a plane and makes equations like x² + 1 = 0 solvable.

Key formulas & rules

  • •i² = −1
  • •i¹ = i
  • •i² = −1
  • •i³ = −i
  • •i⁴ = 1 (cycle repeats every 4 powers)
  • •i^n follows the cycle: i, −1, −i, 1, ...

Examples

Input Output Note
i² −1 Definition
i³ −i Cycle continues
i⁴ 1 Back to start
√(−9) 3i Real × i

Real-world uses

Solving x² = negative numbers Rotations by 90° Foundation of complex analysis

Modes & formulas

1

Operations

• Add, subtract, multiply, and divide complex numbers z₁ and z₂. • Enter the real and imaginary parts of both numbers. • The calculator returns the result in rectangular form and plots it on an Argand diagram.
(a+bi)(c+di) = (ac−bd) + (ad+bc)i

For multiplication, FOIL and substitute i² = −1. For division, multiply numerator and denominator by the conjugate of the denominator.

2

Polar ⇄ Rect

• Convert z₁ = a + bi to polar form (r, θ). • Or convert from polar (r, θ) to rectangular (a + bi). • The calculator shows the magnitude r and argument θ.
r = √(a² + b²) | θ = atan2(b, a)

The magnitude is the distance from the origin, and the argument θ is the angle from the positive real axis.

3

Powers & Roots

• Raise z₁ to a power n, or take the n-th root. • De Moivre's theorem computes z^n = r^n(cos nθ + i sin nθ). • Roots are evenly spaced around a circle of radius r^(1/n).
z^(1/n) = r^(1/n) · e^(i(θ + 2πk)/n)

Every non-zero complex number has exactly n distinct n-th roots, spaced 360°/n apart.

4

Properties

• View magnitude, argument, conjugate, and Euler form of a single complex number. • The conjugate z̄ = a − bi mirrors the point across the real axis. • z · z̄ = |z|² is always real and non-negative.
|z| = √(a² + b²) | z̄ = a − bi | z · z̄ = |z|²

The conjugate is essential in division and in computing real-valued magnitudes.

How to use it

Select a mode (Operations, Polar, Powers, or Properties). Enter your values. The result updates instantly, and an Argand diagram visualizes the geometry.

Frequently asked questions

What is i²?
i² = −1 by definition. This is the foundational rule that makes complex arithmetic possible.
Why does division by 0 + 0i fail?
Division by zero is undefined in any number system. The tool detects and rejects division by the zero complex number.
What is the imaginary unit i?
The imaginary unit i is defined as the square root of -1. It extends the real number system into a 2D plane, allowing us to solve equations like x² + 1 = 0.
What is an Argand diagram?
It is a 2D graph where the x-axis represents the real part and the y-axis represents the imaginary part of a complex number.
How do you add complex numbers?
Add the real parts together and the imaginary parts together separately. (a+bi) + (c+di) = (a+c) + (b+d)i.
How do you multiply complex numbers?
Use the FOIL method (First, Outer, Inner, Last) and remember that i² = -1. For example, (a+bi)(c+di) = (ac-bd) + (ad+bc)i.

Related calculators

Vector Math Dot product, cross product, magnitude, and angle between vectors. Matrix Calculator Calculate determinant, transpose, addition, and multiplication. Laplace Transform Reference table of common Laplace transform pairs.
Read the full Complex Numbers guide — examples & 18 FAQs