Complex Number Polar Form Calculator
Polar form turns a complex number a + bi into a magnitude and an angle, which makes multiplication, division, and powers simple through De Moivre's theorem. Enter the real and imaginary parts to get the modulus r and the argument theta in both radians and degrees, computed with atan2 so the quadrant is always correct.
Polar conversion formula
r = sqrt(a^2 + b^2)
theta = atan2(b, a) (radians, range -pi to pi)
theta in degrees = theta * 180 / pi
Polar form: a + bi = r (cos theta + i sin theta)
The modulus is the distance from the origin and the argument is the angle from the positive real axis. Using atan2 gives the correct quadrant automatically.
Worked example
For 3 + 4i: r = square root of (3 squared plus 4 squared) = square root of 25 = 5.00. The argument is atan2(4, 3) = 0.9273 radians = 53.13 degrees. So 3 + 4i = 5 (cos 53.13 degrees + i sin 53.13 degrees).
Complex polar form: frequently asked questions
What is the polar form of a complex number?
It expresses a + bi as r times (cos theta plus i sin theta), where r is the modulus, the distance from the origin, and theta is the argument, the angle from the positive real axis. The polar form makes multiplication, division, and powers straightforward.
How are the modulus and argument found?
The modulus is the square root of a squared plus b squared. The argument is atan2(b, a), the two-argument arctangent, which places the angle in the correct quadrant rather than the principal range of plain arctangent.
Why use atan2 instead of arctan(b/a)?
Plain arctan(b/a) loses the sign information of a and b, so it cannot tell the second quadrant from the fourth. The atan2 function takes both parts and returns the true angle in the range from minus pi to pi, the standard principal argument.
What is the argument of zero?
The complex number zero has modulus zero and an undefined argument, since it has no direction. The calculator reports a modulus of zero and treats the argument as zero by convention when both parts are zero.
Sources
- NIST Digital Library of Mathematical Functions: Complex numbers, modulus and argument.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.