Complex Number Calculator

Complex numbers extend the one-dimensional real line into a rigorous orthogonal 2D algebraic field. Operations naturally encode both magnitude scaling and continuous rotation via Euler's identity, providing the indispensable mathematical framework essential for quantum mechanics and alternating current dynamics.

A complex number is simply a point on the plane. Rectangular form a + bi names it by coordinates — so far right, so far up — while polar form r∠θ names it by distance and direction from the origin. Same dot, two labels; flip the polar toggle to see both at once. Rule of thumb: reach for rectangular form when adding, and polar form when multiplying or raising to a power.

Complex numbers weren't invented to solve x² + 1 = 0 — nobody lost sleep over that. They were forced on mathematicians by cubic equations. Cardano's 1545 formula for x³ = px + q can pass through square roots of negative numbers even when all three solutions are ordinary real numbers. In 1572 Bombelli showed those "impossible" quantities cancel to give the correct real answers. The imaginary was a detour to the real.