Which complex power and logarithm value does the calculator return?

For general complex powers and ln, the calculator uses the principal branch: arg(z) is atan2(imaginary part, real part) in (-π, π]. Complex powers can have multiple values, so this returns the principal value.

Which zero operations are undefined?

Division by zero, the inverse of zero, and ln(0) are undefined. The calculator also rejects 0 raised to zero, a negative real exponent, or a non-real exponent, instead of displaying NaN or Infinity.

Complex Number Calculator - Add, Multiply, Modulus & More

Coordinate plane data
Element	x₂	y₂
a	3	4
b	1	-2
z	4	2

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.

Domains and principal values

The calculator rejects operations that do not have a finite complex result: division by 0 + 0i, the inverse of zero, and ln(0). It also accepts zero as a power base only when its exponent is a positive real number. For a general complex power it evaluates the principal value exp(b × Log(a)), using the principal argument atan2(imaginary, real) between −π and π. Complex logarithms and powers can have multiple mathematical values, so this convention makes the displayed answer reproducible. Calculations that overflow the finite numeric range show an error instead of NaN or Infinity.

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.