Engineers could not analyze your wall outlet's AC power without them, yet complex numbers started as a 16th-century workaround for 'impossible' square roots. A complex number a + bi pairs a real part with an imaginary part built on i = √(−1). Adding is component-wise — (2 + 3i) + (1 − i) = 3 + 2i — while multiplying uses the FOIL pattern plus the rule i² = −1, so (1 + i)(1 − i) = 2. Division clears the denominator with the conjugate, and the modulus √(a² + b²) measures size.

The payoff is real: alternating-current circuits, where voltage and current have both magnitude and phase, are modeled as complex impedance, letting engineers solve AC problems with plain algebra instead of differential equations. The same numbers describe quantum states and digital signal filters. This calculator handles the arithmetic: enter the real and imaginary parts of each operand and it returns the sum, difference, product, or quotient with the work shown.

With the pre-filled defaults — Variable A 10, Variable B 5 — the Complex Number Calculator returns Result 50.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.