A root of unity is a complex number that when raised to some positive integer will return 1.

It is any complex number #z# which satisfies the following equation:

#z^n = 1#

where #n in NN#, which is to say that n is a natural number. A natural number is any positive integer: (n = 1, 2, 3, ...). This is sometimes referred to as a counting number and the notation for it is #NN#.

For any #n#, there may be multiple #z# values that satisfy that equation, and those values comprise the roots of unity for that n.

When #n = 1#

Roots of unity: #1#

When #n = 2#

Roots of unity: #-1, 1#

When #n = 3#

Roots of unity = #1, (1 + sqrt(3)i)/2, (1 - sqrt(3)i)/2#

When #n = 4#

Roots of unity = #-1, i, 1, -i#