Polar Form Of Complex Number Calculator

Work out polar form of complex number instantly with clear inputs, formula shown and shareable results.

About Polar Form Of Complex Number Calculator

Polar form writes z = r(cos θ + i sin θ) = re^(iθ). Powers become trivial in this form: the modulus is raised to the power and the argument is multiplied by it, which is De Moivre's theorem.

Formulas

Polar form

z = r(cos θ + i sin θ) = r·e^(iθ)

De Moivre

zⁿ = rⁿ(cos nθ + i sin nθ)

Worked example

With the default values below, Polar Form Of Complex Number Calculator returns Polar form of 4.242641(cos 45° + i sin 45°). Change any input above and every figure here is recalculated in your browser.

Inputs used

Real part: 3
Imaginary part: 3
Also raise to this power: 4

Results

Polar form: 4.242641(cos 45° + i sin 45°)
Exponential form: 4.242641e^(0.785398i)
Modulus after raising to the power: 324.000000
Argument after raising to the power: 180.000000 °

Step by step

1Values used
2Polar form
3De Moivre
4Polar form= 4.242641(cos 45° + i sin 45°)
5Exponential form= 4.242641e^(0.785398i)
6Modulus after raising to the power= 324.000000
7Argument after raising to the power= 180.000000 °

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For 3 + 3i?

The modulus is 4.2426407 and the argument 45°, so the fourth power has modulus 324 and argument 180° — that is −324.

Why is exponential form useful?

It turns multiplication into addition of exponents, which is why it dominates electrical engineering and wave physics.