Complex Number Calculator
Add, subtract, multiply, divide and raise complex numbers to a power, with the answer in a + bi and polar form and plotted on an Argand diagram.
About this tool
A complex number a + bi has a real part a and an imaginary part b, where i² = −1. Adding and subtracting work part by part; multiplying expands the brackets using i² = −1; and dividing multiplies the top and bottom by the conjugate of the bottom (a − bi) to make the denominator real. Powers use De Moivre's theorem: raise the modulus to the power and multiply the angle by it, which also handles fractional powers such as ½ for a square root.
Every answer is also shown in polar form, r ∠ θ, where r is the modulus (distance from 0) and θ the argument (angle from the positive real axis, in degrees from −180° to 180°), along with its conjugate. The Argand diagram plots both inputs and the result, which makes it easy to see that multiplying rotates and stretches.