Perform complex number arithmetic — add, subtract, multiply, divide, raise to powers via De Moivre, find modulus, argument, polar form, and conjugate.
Complex numbers extend the real number line into a two-dimensional plane, allowing solutions to equations like x² + 1 = 0 that have no real answer. Every complex number takes the form z = a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit satisfying i² = −1. Despite seeming abstract, complex numbers are indispensable in electrical engineering, quantum mechanics, signal processing, and fluid dynamics.
This calculator handles all fundamental complex-number operations: addition, subtraction, multiplication, and division of two complex numbers, plus raising a complex number to an integer power via De Moivre's theorem. It also computes the modulus (distance from the origin), argument (angle from the positive real axis), polar form (r∠θ), and complex conjugate. Every result comes with step-by-step working, a complex-plane visualization, and a reference table of key identities.
Whether you are a student working through an algebra or engineering course, a hobbyist exploring the beauty of the complex plane, or a professional double-checking a quick calculation, this tool gives you instant answers with full mathematical context.
Complex numbers appear throughout mathematics, engineering, and physics, yet mental arithmetic with them is error-prone. This calculator handles the tedious algebra — especially multiplication, division, and powers — and shows every step so you can learn from the process rather than just getting an answer.
The complex-plane visualization and polar-form output give geometric insight that formulas alone cannot convey. Students gain intuition about rotation and scaling, while professionals can quickly verify intermediate results.
Addition: (a+bi)+(c+di) = (a+c)+(b+d)i. Multiplication: (a+bi)(c+di) = (ac−bd)+(ad+bc)i. Division: multiply by conjugate of denominator. Power (De Moivre): (r(cosθ+i sinθ))ⁿ = rⁿ(cos nθ + i sin nθ). Modulus: |z| = √(a²+b²). Argument: arg(z) = atan2(b, a).
Result: −1 + 5i
(2+3i)(1+i) = 2·1 − 3·1 + (2·1 + 3·1)i = −1 + 5i. Modulus = √(1+25) ≈ 5.099.
Complex numbers were first encountered in the 16th century when Italian mathematicians Cardano and Bombelli needed square roots of negative numbers to solve cubic equations. Initially viewed with suspicion — Descartes coined the term "imaginary" as a pejorative — complex numbers earned full acceptance after Euler, Gauss, and Riemann demonstrated their deep connections to algebra, geometry, and analysis.
Every complex number a + bi corresponds to a point (a, b) in the plane. The horizontal axis is the real axis and the vertical axis is the imaginary axis. Addition corresponds to vector addition; multiplication by a complex number of modulus r and argument θ scales by r and rotates by θ. This geometric interpretation makes complex multiplication a rotation-and-scaling operation, a fact exploited throughout physics and engineering.
In electrical engineering, impedance Z = R + jX (using j instead of i) combines resistance and reactance into one number. In quantum mechanics, the wave function ψ is complex-valued, and probabilities are |ψ|². In fluid dynamics, conformal mappings — functions of a complex variable — transform complicated flow geometries into simpler ones. Even in number theory, the Riemann zeta function ζ(s) for complex s encodes the distribution of prime numbers.
i is defined as the square root of −1: i² = −1. It is the foundation of complex numbers.
It states that (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ), making it easy to raise complex numbers to integer powers using their polar form. Use this as a practical reminder before finalizing the result.
The conjugate of z = a + bi is z̄ = a − bi. Multiplying z by z̄ gives |z|² = a² + b², always a non-negative real number.
Multiply numerator and denominator by the conjugate of the denominator: (a+bi)/(c+di) = (a+bi)(c−di) / (c²+d²). Keep this note short and outcome-focused for reuse.
z = r(cos θ + i sin θ) = r·e^(iθ), where r = |z| is the modulus and θ = arg(z) is the argument (angle). Apply this check where your workflow is most sensitive.
They simplify AC circuit analysis (impedance), signal processing (Fourier transforms), and control theory (transfer functions) by encoding both magnitude and phase in one number. Use this checkpoint when values look unexpected.