Since for every integer , we can write
These are the th roots of unity. The special case is called a primitive th root of unity, since integer powers of it give all of the others:
The th roots of unity are so frequently used that they are often given a special notation in the signal processing literature:
where denotes a primitive th root of unity.^{3.7} We may also call a generator of the mathematical group consisting of the th roots of unity and their products.
We will learn later that the th roots of unity are used to generate all the sinusoids used by the length- DFT and its inverse. The th complex sinusoid used in a DFT of length is given by
where , , and is the sampling interval in seconds.