Complex Number to Polar Form Calculator

This calculator converts a complex number written as a + bi into polar form. Enter the real part a and the imaginary part b; you get the modulus r (the distance from the origin) and the argument θ (the angle from the positive real axis) in degrees and in radians.

Formula

r = √(a² + b²) and θ = atan2(b, a), so a + bi = r(cos θ + i sin θ) = r·e^iθ.

• a: the real part (horizontal coordinate on the complex plane)
• b: the imaginary part (vertical coordinate)
• r: the modulus, always zero or positive
• θ: the argument, between −180° and 180° (−π to π radians)

atan2 looks at the signs of both a and b, so it puts the angle in the right quadrant. A plain arctan(b ÷ a) gives the wrong answer when a is negative.

Worked Example

Take z = 3 + 4i.

1. r = √(3² + 4²) = √25 = 5.0000
2. θ = atan2(4, 3) = 53.1301°
3. In radians: 53.1301 × π ÷ 180 = 0.9273 rad
4. So 3 + 4i = 5(cos 53.1301° + i sin 53.1301°)

How to read the result

The modulus is the length of the arrow from 0 to the point (a, b) and the argument is the direction of that arrow. Polar form makes multiplication easy: multiply the moduli and add the arguments. It is also the starting point for powers and roots with De Moivre’s theorem.

A negative angle means the point is below the real axis. Add 360° (or 2π) if you need an angle between 0° and 360°.

Limits and common mistakes

• For z = 0 the modulus is 0 and the argument is not defined; the calculator shows 0°, which is a convention only.
• Do not mix degrees and radians in the same working.
• Rounding θ to whole degrees and converting back will not give exactly the same a and b.

Frequently Asked Questions

What is the polar form of 1 + i?

r = √2 ≈ 1.4142 and θ = 45°, so 1 + i = √2(cos 45° + i sin 45°).

Why use atan2 instead of arctan?

arctan(b ÷ a) only returns angles between −90° and 90°. atan2 uses both signs and returns the true angle in any quadrant.

Can the modulus be negative?

No. It is a distance, so it is zero or positive.

How do I go back to a + bi?

a = r cos θ and b = r sin θ. The polar to rectangular calculator does this.

in a | Real part a | - | 3 | -
in b | Imaginary part b | - | 4 | -
out r | Modulus r | - | sqrt(a^2 + b^2) | 4
out th | Argument θ | ° | atan2(b, a) * 180 / pi | 4
out thr | Argument θ | rad | atan2(b, a) | 4