14.8 The roots of unity
The nth roots of unity are the nth roots of one. How can there be more than one () nth root of one? Well, some of them are complex, of course!
Note that 88 8 In this document the natural numbers include zero!
This is because of Euler’s formula.
Therefore, if we take the th roots of both sides, we get that
Which we can use to compute the th roots of unity. Note that by the fundamental theorem of algebra for the roots of unity, there are different values.
Note that we often use the letter (the Greek “omega”) to denote .
The th roots of unity sum to .
Proof: I don’t think you need to know this for A Level Mathematics, but an easy-ish proof is to consider
and note that if we multiply through by we actually get the same value back! That is
The only complex number for which is and therefore the sum of the roots of unity is .
By considering the ninth roots of unity, show that 99 9 I believe this question comes from the textbook ”Further Pure Mathematics”
Using Euler’s formula 1010 10 , see above for more we can write the sum of as
This is actually the sum of eight of the nine roots of unity in disguise! Note that if we add to anything in the form , this has no effect (again, Euler’s formula and the fact that radians is a full rotation). Therefore, the previous expression is the same as
which can be re-ordered as
which are all the ninth roots of unity (except ).
Thus we can write that
which means that