Adding the Roots of Unity

The n roots of unity are the solutions to the equation

The square roots of unity are the solutions toThese are 1 and -1.

These add to give zero.

The cube roots of unity are the solutions toThese are 1,and

These add to give zero.

The fourth roots of unity are the solutions toThese areand

These add to give zero.

In fact all the nth roots of unity add to give zero whatever the value ofThis is a consequence of the factorisation of

If the solutions ofarethen we can write

Multiplying out the brackets gives

The coefficient ofinis zero, soand the sum of theroots is zero.

Notice also that for each value ofthe roots are distributed in a regular way on the unit circle about the origin. In fact, the real axis is a mirror line for the roots, so that ifis a root, then so isThis means that all the roots add to give a real number. Rotate each root byEach root will be rotates onto the adjacent root so the sum of the roots will not change, but the sum will be multiplied byThe only way to maintain consistency is for the sum of the roots to be zero.