Complex Number to Polar Form Calculator

Convert a rectangular complex number a + bi into polar form r(cos θ + i sin θ). Enter the real and imaginary parts to get the modulus, argument, and quadrant.

Quick Facts

Formula
r = √(a² + b²), θ = atan2(b, a)
atan2 uses the sign of a and b to place θ in the correct quadrant automatically.
Polar notation
z = r(cos θ + i sin θ) = r·e^(iθ)
Also written r∠θ or r cis θ in engineering and physics.

Your Results

Calculated
Modulus (r)
-
Distance from origin, |z|
Argument (θ)
-
Angle from positive real axis
Polar form
-
z = r(cos θ + i sin θ)
Quadrant
-
Location in the complex plane

Ready

Enter the real and imaginary parts, then press Calculate.

About the Complex Number to Polar Form Calculator

Every complex number z = a + bi can be plotted as a point (a, b) in the complex plane, where the horizontal axis is the real part and the vertical axis is the imaginary part. Polar form describes that same point using a distance from the origin and an angle instead of horizontal/vertical coordinates — the same idea as converting Cartesian (x, y) coordinates to polar (r, θ) coordinates in ordinary geometry.

The conversion formulas

  • Modulus (r): r = √(a² + b²) — the distance from the origin to the point, always non-negative.
  • Argument (θ): θ = atan2(b, a) — the angle from the positive real axis, using the signs of a and b to select the correct quadrant.
  • Polar form: z = r(cos θ + i sin θ), often abbreviated r cis θ, or written in exponential form as z = r·e^(iθ) using Euler's formula.

Why atan2 instead of arctan

A plain arctan(b/a) only returns angles between −90° and 90°, so it cannot tell Quadrant I from Quadrant III or Quadrant II from Quadrant IV — both give the same ratio. The two-argument atan2(b, a) function looks at the sign of a and b separately, so it returns the correct angle across the full −180° to 180° range (or −π to π radians) without extra sign-checking logic.

Going back to rectangular form

The conversion is reversible: given r and θ, the rectangular components are a = r cos θ and b = r sin θ. This round trip underlies multiplying and dividing complex numbers in polar form, where moduli multiply and arguments add — a shortcut that is much harder to see in rectangular a + bi form.

Frequently Asked Questions

What is the formula for converting a complex number to polar form?
To convert z = a + bi to polar form, compute the modulus r = √(a² + b²) and the argument θ = atan2(b, a). The polar form is then z = r(cos θ + i sin θ), also written r∠θ or r·e^(iθ).
What is the difference between modulus and argument?
The modulus r is the distance from the origin to the point (a, b) in the complex plane and is always zero or positive. The argument θ is the angle, measured counterclockwise from the positive real axis, that locates the point around that distance — typically reported between −180° and 180° (or −π and π radians).
Why use atan2 instead of a simple arctangent?
Plain arctan(b/a) only returns values between −90° and 90°, so it can't distinguish Quadrant I from III or II from IV — both give the same ratio. atan2(b, a) uses the sign of each part separately to place θ in the correct quadrant automatically.
Can I convert the polar form back to rectangular form?
Yes. Given r and θ, recover the rectangular parts with a = r cos θ and b = r sin θ — the reverse of the modulus and argument formulas used here.

Related calculators