Complex number calculator

You supply a = a_real + a_imag·i and b = b_real + b_imag·i, choose add, subtract, multiply or divide, and receive the resulting complex number in both rectangular form and a readable display string. Division is computed with the exact conjugate formula and correctly rejects division by zero. The same deterministic code runs free in your browser and through the paid API, so what you test on this page is exactly what your integration receives.

What a complex number operation actually computes

A complex number is a pair of real numbers written as a + bi, where i is the imaginary unit satisfying i² = −1. Addition and subtraction work component by component: the real parts combine and the imaginary parts combine independently, so (3 + 2i) + (1 − 4i) = 4 − 2i. Multiplication is the interesting case because it uses the defining property of i. Expanding (a + bi)(c + di) with the distributive law gives ac + adi + bci + bdi², and since i² = −1 the result is (ac − bd) + (ad + bc)i. The real part of a product is not the product of the real parts, which is the single most common mistake people make by hand. This complex number calculator applies these closed-form formulas directly, with no iteration and no approximation beyond the final rounding, so the answer is deterministic: the same inputs always produce byte-identical outputs, whether the computation happens in your browser tab or on our edge servers through the API.

Division and the conjugate trick

Dividing complex numbers is the operation that most often needs a calculator, because the formula is not obvious. To compute (a + bi) / (c + di) you multiply numerator and denominator by the conjugate of the denominator, (c − di). The denominator then becomes c² + d², a plain real number, and the numerator expands with the multiplication rule. The final result is ((ac + bd) / (c² + d²)) + ((bc − ad) / (c² + d²))i. The one case where this breaks down is when c² + d² = 0, which happens only when c and d are both zero — the zero complex number. Division by zero is undefined in the complex field just as it is for real numbers, and this endpoint reports it as an invalid input error instead of returning an infinite or NaN result that would silently corrupt a downstream pipeline. Results are rounded to ten decimal places so that floating-point noise never appears in the output, and a negative zero is normalized to zero for clean, comparable responses.

Where complex arithmetic shows up in practice

Complex numbers are not an academic curiosity; they are the working notation of several engineering disciplines. Electrical engineers represent impedances and phasors as complex numbers, so combining two circuit elements in series is an addition and computing transfer functions is a division. Signal processing expresses frequency components as complex values from a Fourier transform, and filtering multiplies them by complex coefficients. Control theory places poles and zeros on the complex plane, and quantum mechanics is written in complex amplitudes from the ground up. In all of these fields the arithmetic itself is mechanical — the four operations on this page cover it completely — and the value of an API is that it can be embedded in a validation step, a teaching tool or a code generator without reimplementing the formulas. Because the capability is stateless and deterministic, it is safe to cache, safe to retry and safe to run in parallel across as many pairs of operands as your workload requires.