Complex Number Calculator

Use the form a+bi, and be flexible: 3+4i, 3-4i, 2i, -5, i, -i, and 3+i all parse correctly, spaces are ignored, and decimals like 3.5+2.5i work too.

Every FOIL step: the four products (a)(c), (a)(di), (bi)(c), (bi)(di), the substitution i² = -1, and the final combined a+bi — so you can follow the algebra, not just the answer.

By rationalization: multiply top and bottom by the conjugate of the divisor, so the denominator becomes c² + d², a real number. The steps show the numerator expansion and the final a+bi result.

The modulus |a+bi| = √(a²+b²) is the number's distance from the origin. The argument is its angle in the complex plane in degrees, found with atan2(b, a).

Choose polar form to see a+bi written as r∠θ° with r = |z| and θ = arg(z). For the reverse, pick polar to rectangular and type z₁ as r∠θ, for example 5∠30.