De Moivre's Theorem Tool

Math Algebra • Complex Numbers

Written by STEM Calculators Team Published December 20, 2025 Updated February 24, 2026

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4. De Moivre's Theorem Tool

Compute integer powers \(z^n\) and \(n\)-th roots (all branches) using De Moivre’s theorem, with a clean Argand plot.

1) Choose task

Exponent \(n\)

Integer (can be negative)

Root order \(n\)

Positive integer

2) Enter \(z\)

Input form

Angle unit

Wrap angles

Rectangular

Re(a) Im(b)

Polar

\(r\ge 0\) \(\theta\)

If you enter \(r<0\), the tool normalizes it (\(r\to -r,\ \theta\to\theta+\pi\)).

Preview

3) Display options

Show step-by-step Show \(e^{i\theta}\) form Show full roots table (roots task)

Roots: highlight a branch

Branch \(k\)

Animate highlight (cycles \(k\))

Roots: \(w_k = r^{1/n}\,\mathrm{cis}\!\left(\frac{\theta+2\pi k}{n}\right)\), for \(k=0,1,\dots,n-1\).

Plot options

Equal axis units Label points Show unit circle

Ready

Drag to pan • wheel/pinch to zoom • Power task plots \(z\) and \(z^n\). Roots task plots all \(n\)-th roots and highlights \(w_k\).

Complex plane

Re horizontal, Im vertical. (Auto fit if points go off-screen.)

Re: 0, Im: 0 sx: 70, sy: 70

Choose a task and click Compute.

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Theory

Polar form and cis notation

Any nonzero complex number \(z=a+bi\) can be written in polar form:

\[ z = r(\cos\theta+i\sin\theta) = r\,\mathrm{cis}(\theta) = r e^{i\theta}, \]

where \(r=|z|=\sqrt{a^2+b^2}\) and \(\theta=\operatorname{atan2}(b,a)\). Angles are not unique: \(\theta\) and \(\theta+2k\pi\) represent the same direction. Many calculators use the principal argument \(\operatorname{Arg}(z)\in(-\pi,\pi]\).

De Moivre’s theorem for integer powers

If \(z=r(\cos\theta+i\sin\theta)\) and \(n\in\mathbb{Z}\), then:

\[ z^n = \big[r(\cos\theta+i\sin\theta)\big]^n = r^n\big(\cos(n\theta)+i\sin(n\theta)\big). \]

This is De Moivre’s theorem. It makes powers easy: raise the modulus to \(r^n\) and multiply the angle by \(n\). If you want an angle in a standard range, you can wrap \(n\theta\) back into \((- \pi,\pi]\) (or another interval).

\(n\)-th roots and multiple branches

Solving \(w^n=z\) produces multiple solutions (branches). If \(z=r\,\mathrm{cis}(\theta)\) and \(n\ge 1\), then all \(n\)-th roots are:

\[ w_k = r^{1/n}\,\mathrm{cis}\!\left(\frac{\theta+2\pi k}{n}\right), \qquad k=0,1,\dots,n-1. \]

These roots are equally spaced in angle by \(2\pi/n\) and lie on a circle of radius \(r^{1/n}\). When \(r=1\), they are the \(n\)-th roots of unity.

Solved example: \((1+i)^4\)

First write \(1+i\) in polar form. Its modulus is \(r=\sqrt{2}\) and its angle is \(\theta=\pi/4\):

\[ 1+i = \sqrt{2}\,\mathrm{cis}\!\left(\frac{\pi}{4}\right) = \sqrt{2}\,e^{i\pi/4}. \]

Apply De Moivre with \(n=4\):

\[ (1+i)^4 = (\sqrt{2})^4\,\mathrm{cis}\!\left(4\cdot\frac{\pi}{4}\right) = 4\,\mathrm{cis}(\pi) = 4(\cos\pi+i\sin\pi) = 4(-1+0i) = -4. \]

Common notes

If \(z=0\), then \(z^n=0\) for \(n>0\), while \(0^n\) is undefined for \(n\le 0\).
If you enter polar data with negative \(r\), you can convert it to standard form by using \(r\to -r\) and \(\theta\to\theta+\pi\).
Fractional powers \(z^{p/q}\) are essentially “take a \(q\)-th root, then raise to \(p\)” and are multi-valued. This tool focuses on integer powers and explicit \(n\)-th roots (branches).

Frequently Asked Questions

What does De Moivre's theorem calculate for complex numbers?

It provides a direct way to compute integer powers of a complex number written in polar form. If z = r(cos(theta) + i sin(theta)), then z^n = r^n(cos(n theta) + i sin(n theta)).

How do you find the n-th roots of a complex number using De Moivre's theorem?

Write z in polar form z = r cis(theta), then the n-th roots are w_k = r^(1/n) cis((theta + 2 pi k)/n) for k = 0, 1, ..., n-1. Each k gives a different root (branch) spaced evenly around a circle.

Why are there multiple n-th roots of the same complex number?

Angles in polar form are not unique because theta and theta + 2 pi m represent the same direction. Dividing the angle by n creates n distinct solutions, producing n equally spaced roots on the Argand plane.

When should I use principal argument wrapping versus raw angles?

Use principal wrapping when you want angles reported in a standard interval such as (-pi, pi]. Use raw angles when you want to keep unwrapped angles (for example, to track rotations without reducing modulo 2 pi).