Complex Number Polar Form Calculator
A complex number in both forms, with its modulus, argument, conjugate, powers by De Moivre and all its n-th roots.
Results
What this tool does
A complex number can be written two ways: as a pair of coordinates, or as a length and an angle. The first is natural for adding, the second for multiplying — in polar form, multiplying means multiplying the lengths and adding the angles, and raising to a power becomes almost trivial. This page converts either way, gives the modulus and argument in both degrees and radians, applies De Moivre's theorem for any power up to twenty, and lists all n distinct roots, which sit at equal angles around a circle.
Formula
r = √(a² + b²) · θ = atan2(b, a) · (r∠θ)ⁿ = rⁿ∠nθ · ⁿ√(r∠θ) = ⁿ√r ∠ (θ + 360°k) ÷ n
Variables
| Symbol | Meaning | Unit |
|---|---|---|
mode | Direction | — |
real | Real part a | — |
imag | Imaginary part b | — |
modulus | Length r | — |
argument | Angle | — |
unit | The angle is in | — |
power | Raise it to the power | — |
OU | The other form | — |
RC | As a + bi | — |
PL | As length and angle | — |
MO | Modulus, its distance from zero | — |
AD | Argument in degrees | — |
AR | Argument in radians | — |
PR | Real part after the power | — |
PI | Imaginary part after the power | — |
PM | Modulus after the power | — |
RT | Root number · real part · imaginary part · angle | — |
CJ | The conjugate | — |
Worked example
- Directionto_polar
- Real part a3
- Imaginary part b4
- Length r5
- Angle53.1301
- The angle is indeg
- Raise it to the power2
- The other form5(cos 53.1301023542 + i sin 53.1301023542)
- As a + bi3 + 4i
- As length and angle5(cos 53.1301023542 + i sin 53.1301023542)
- Modulus, its distance from zero5.0000000000
- Argument in degrees53.13010235
- Argument in radians0.9272952180
- Real part after the power-7.0000000000
- Imaginary part after the power24.0000000000
- Modulus after the power25.0000000000
- Root number · real part · imaginary part · angle0 2 1 26.565051 1 -2 -1 206.565051
- The conjugate3 - 4i
Frequently asked questions
Why does polar form exist at all?
Because it makes multiplication trivial. In rectangular form, multiplying two complex numbers means four products and careful bookkeeping of the signs; in polar form you multiply the two lengths and add the two angles, and that is the whole operation. Raising to a power becomes raising one number to that power and multiplying one angle by it, which is De Moivre's theorem. Addition is the opposite way round — easy in rectangular, awkward in polar — which is why both forms stay in use and why converting between them is a routine step rather than a curiosity.
What are the roots in the table?
The n-th roots of your number, all of them. A real positive number has one real square root you usually think about; a complex number has exactly n distinct n-th roots, and they sit at equal angles around a circle like the points of a star. The cube roots of 1, for instance, are 1 and two others at 120 and 240 degrees — which is why a cubic always has three roots even when only one of them is real. The table gives each root's rectangular coordinates and its angle.