## Adding the Roots of Unity

The n roots of unity are the solutions to the equation The square roots of unity are the solutions to These are 1 and -1.

The cube roots of unity are the solutions to These are 1, and The fourth roots of unity are the solutions to These are and In fact all the nth roots of unity add to give zero whatever the value of This is a consequence of the factorisation of If the solutions of are then we can write Multiplying out the brackets gives The coefficient of in is zero, so and the sum of the roots is zero.

Notice also that for each value of the 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 if is a root, then so is This means that all the roots add to give a real number. Rotate each root by Each root will be rotates onto the adjacent root so the sum of the roots will not change, but the sum will be multiplied by The only way to maintain consistency is for the sum of the roots to be zero. 