What are complex numbers?

A complex number has the form a + bi, where a is the real part, b is the imaginary part, and i = √(−1). Complex numbers extend the real number line to the complex plane and arise naturally in electrical engineering, signal processing, quantum mechanics, and many areas of mathematics.

Key operations

Addition: (a + bi) + (c + di) = (a+c) + (b+d)i
Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i
Modulus: |a + bi| = √(a² + b²)
Conjugate: conj(a + bi) = a − bi
Division: multiply numerator and denominator by the conjugate of the denominator.
De Moivre: (r·e^(iθ))^n = rⁿ·e^(inθ)
Euler's formula: e^(iθ) = cos θ + i·sin θ

Input format

Enter numbers in a+bi form: 3+4i, -2-5i, 7, 3i, -i. Spaces around + and − are allowed.

The complex plane (Argand diagram)

Every complex number z = a + bi can be visualized as a point in a 2D plane where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This is the Argand diagram (or complex plane). The number 3 + 4i sits at coordinates (3, 4). The distance from the origin to this point is the modulus |z| = √(3² + 4²) = 5, and the angle from the positive real axis is the argument θ = arctan(4/3) ≈ 53.13°.

Polar and exponential form

Any complex number can be written in rectangular form (a + bi), polar form (r·(cosθ + i·sinθ)), or exponential form (r·e^iθ). Conversion formulas:

r (modulus) = √(a² + b²)
θ (argument) = atan2(b, a) - use atan2 not atan to handle all four quadrants
a = r·cosθ
b = r·sinθ

Multiplication and division are much simpler in polar form: multiply the moduli and add the arguments.

Applications

AC circuit analysis: impedance in AC circuits is a complex number (resistance + j·reactance). Complex arithmetic handles the magnitude and phase of voltages and currents simultaneously.
Fourier transform: decomposes a signal into its frequency components using Euler's formula. The transform output is complex-valued, with the modulus representing amplitude and the argument representing phase.
Control systems: stability analysis (pole-zero plots) is done entirely in the complex plane. Poles in the left half-plane indicate a stable system.
Fluid dynamics: conformal mapping using complex functions transforms complicated flow problems into simpler ones.

Complex roots of unity

The n-th roots of unity are the n solutions to zⁿ = 1. They are evenly spaced around the unit circle at angles 2πk/n (for k = 0, 1, …, n−1). For example, the cube roots of unity (n=3) are 1, e^2πi/3, and e^4πi/3, equally spaced at 120° apart. Roots of unity appear in Fourier analysis, crystallography, and polynomial factoring.