How to write a complex number in polar form
Because a+bi is a point on the plane, it can also be described by how far it sits from the origin and in which direction. The distance is the modulus r=√(a²+b²) and the direction is the argument θ. Written as r(cos θ + i sin θ) that is polar form, and Euler's formula turns the same thing into r·e^(iθ), the exponential form. This converter shows both and lets you read the argument in degrees or radians.
Finding the argument from the arctangent of b divided by a loses the quadrant, since 1/1 and −1/−1 are the same ratio and 45° cannot be told apart from −135°. This tool therefore uses atan2, which reads both signs and returns an argument above −180° and up to 180°. When the modulus is zero there is no direction at all, so the argument is undefined and a notice appears instead of a result.
Polar form pays off in multiplication: moduli multiply while arguments add. A power therefore reduces to raising r to the nth power and multiplying the argument by n, which is De Moivre's theorem. Going the other way, an nth root adds whole multiples of 360° to the argument before dividing by n, which produces exactly n distinct values that all share the same modulus, the nth root of r. Orders from 2 to 12 are supported and irrational values are shown to six decimal places.
Frequently Asked Questions
Taking the arctangent of y divided by x throws the quadrant away, because 1 over 1 and −1 over −1 give the same ratio. atan2 reads both signs and returns the correct argument above −180 degrees and up to 180 degrees.
It is undefined, because a point with modulus zero has no direction. When the real and imaginary parts are both zero the tool shows a notice instead of a result.
Adding 360 degrees to the argument gives the same complex number, so (θ+360k)/n produces n different arguments for k from 0 to n−1. All of them share the same modulus, the nth root of r.