Complex to Polar Form Converter
Polar form expresses a complex number by its distance from the origin and its angle, which is often more convenient than rectangular form for multiplication, division and powers. This converter takes a complex number written as a real part plus an imaginary part and returns the modulus r and the argument theta, reporting the angle in both radians and degrees. The argument is computed with the two-argument arctangent so the quadrant is always correct. The result is the principal argument between minus 180 and 180 degrees.
Rectangular to polar formula
r = sqrt(a^2 + b^2)
theta (radians) = atan2(b, a)
theta (degrees) = theta * 180 / pi
Polar form = r * (cos theta + i sin theta) = r e^(i theta)
The argument is the principal value in the range minus pi to pi radians (minus 180 to 180 degrees).
Polar form context
- The modulus r is the distance from the origin in the complex plane.
- The argument theta is measured from the positive real axis, counterclockwise positive.
- The two-argument arctangent keeps the correct quadrant for all signs of a and b.
- Polar form simplifies multiplication, division and powers via De Moivre's theorem.
- At the origin the modulus is zero and the angle is undefined; it is shown as zero by convention.
Complex to polar: frequently asked questions
What is the polar form of a complex number?
Polar form writes a complex number as r times (cos theta plus i sin theta), or r times e to the i theta. Here r is the modulus, the distance from the origin, and theta is the argument, the angle from the positive real axis. The calculator returns r and theta in both radians and degrees.
How is the modulus calculated?
The modulus r of a plus bi is the square root of (a squared plus b squared). It is always non-negative and equals the length of the vector from the origin to the point (a, b) in the complex plane.
How is the argument calculated?
The argument uses the two-argument arctangent, atan2(b, a), which returns the correct angle in all four quadrants between minus pi and pi radians. Using a plain arctangent of b over a would lose the quadrant information, so atan2 is used instead.
What is the range of the angle?
This converter returns the principal argument in the range minus 180 to 180 degrees, equivalently minus pi to pi radians. You can add or subtract a full turn (360 degrees or 2 pi radians) to express the same direction with a different angle if your application needs that.
What happens at the origin?
When both the real and imaginary parts are zero, the modulus is zero and the argument is undefined because there is no direction. The calculator reports a modulus of zero and shows the angle as zero by convention.
Official sources
- NIST Digital Library of Mathematical Functions: Complex Numbers, polar representation.
- U.S. National Institute of Standards and Technology: NIST home, mathematical references.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.