Roots of Unity

Principle

The \(n\)-th roots of unity are the complex solutions of \(z^n=1\). They lie on the unit circle and are equally spaced in argument.

For a positive integer \(n\), this equation has exactly \(n\) distinct complex solutions.

Notation

• \(n\) is a positive integer.
• \(z=re^{i\theta}\) is the polar form of the unknown complex number.
• \(r=|z|\) is the modulus of \(z\).
• \(\theta\) is an argument of \(z\).
• \(k\) is an integer that records the possible argument shifts by multiples of \(2\pi\).

Method

Write the unknown in polar form:

Raise to the \(n\)-th power:

Write \(1\) with all possible arguments:

Match moduli and arguments:

Because \(r\ge0\), the modulus equation gives \(r=1\). The argument equation gives:

Only \(n\) independent values are needed, because increasing \(k\) by \(n\) adds \(2\pi\) to the argument and gives the same complex number.

Rules

• Each root has modulus \(1\).
• Adjacent roots differ in argument by \(2\pi/n\).
• The points form a regular \(n\)-sided polygon in the complex plane.
• One root is always \(z_0=1\).

Examples

Solve:

Use the roots of unity formula with \(n=5\):

So the five solutions are:

The next value, \(k=5\), gives \(e^{2\pi i}=1\), which repeats the first solution.

Checks

• Do not list every integer \(k\); choose \(k=0,1,\ldots,n-1\) for the independent solutions.
• Remember that arguments are only determined up to multiples of \(2\pi\).
• Check that every root has modulus \(1\).
• Check that substituting a root gives \((e^{2\pi i k/n})^n=e^{2\pi i k}=1\).