Have you noticed: that "Eleven plus two" is an anagram of "Twelve plus one"?

This week's topic: Roots of Unity

Another way to look at this is that the n-th roots of unity are the n distinct roots of the polynomial xⁿ - 1. Using elementary algebra, the first few cases of the n-th roots of unity can be easily found.

n = 1, x - 1 = 0 gives the x = 1 as the only first root of unity.

n = 2, x² - 1 = (x - 1)(x + 1) = 0 gives 1 and -1 as the 2 second roots of unity.

n = 3, x³ - 1 = (x - 1)(x² + x + 1) = 0 gives 1, sqrt(2)/2 + i sqrt(2)/2, and - sqrt(2)/2 - i sqrt(2)/2 as the 3 third roots of unity (you can use the quadratic formula to find the last two roots).

n = 4, x⁴ - 1 = (x² - 1)(x² + 1) = (x - 1)(x + 1)(x² + 1) = 0 gives 1, -1, i, and -i as the 4 fourth roots of unity.

However, once n goes beyond 4, it will be a non-trivial task to find the roots of unity using only elementary algebra. (A side note: the term x - 1 is always a factor of xⁿ - 1, for xⁿ - 1 = (x - 1)(x^n-1 + x^n-2 + ... + x + 1) regardless the specific value of n; therefore confirms what we all know to be true: that 1 is always an n-th root of itself.)

Fortunately, as seen last time, with some slightly more advanced mathematical knowledge we have derived a simple formula to find all the n-th roots of unity, for any n. The formula we came up with last time is:

When k = 0, the corresponding root is of course 1 + 0i = 1, which is always an n-th root of itself for any value of n, as we already knew.

From last time we also know that the modulus (i.e. the distance, on the complex plane, between the point representing the number and the origin) of any root of unity is exactly 1. This means that all the roots of unity are points on the unit circle of the complex plane. However, the converse of this fact is false. It is not true that every point on the unit circle of the complex plane is also a root of unity. That is, take an arbitrary point, representing a complex number z, from the unit circle, there is no guarantee that you can find a positive integer n such that zⁿ = 1. In fact, even though there are infinitely many roots of unity, all of them located on the unit circle, a far greater number of points on the unit circle represent numbers, with modulus 1, who are not roots of unity. (The preceding sentence might not make too much sense to you right now, but it is indeed true. Perhaps someday I will write a column about the concept of infinity that would clarify this.)

If z is an n-th root of unity, then it is also a kn-th root of unity for any multiple of n. This is easily verified by the fact that if zⁿ = 1, then z^kn = (zⁿ)^k = 1^k = 1. Hence, if z is an n-th root of unity then it is also an m-th root of unity for any number m for which n is a factor of. So, for example, since 1 and -1 are the two second roots of unity, they are also two of the fourth-, sixth-, eighth-, etc., roots of unity. If a number z is an n-th root of unity, but is not a k-th root of unity for any k less than n, then z is called a primitive n-th root of unity. For example, 1 is the primitive first root of unity, -1 is the primitive second root of unity, sqrt(2)/2 + i sqrt(2)/2 and - sqrt(2)/2 - i sqrt(2)/2 are primitive third roots of unity, while i and -i are primitive fourth roots of unity. It follows that, for any given n, the number of primitive n-th roots of unity is equal to the number of positive integers less than n that are relatively prime to n (i.e., not sharing any common factor greater than 1 with n). Using results from elementary number theory, we can say a few things about primitive roots of unity: That there is always at least one primitive n-th root of unity for any number n. That there are exactly n-1 such roots if n is a prime number. And that if m and n are relatively prime, and there are j primitive m-th and k primitive n-th roots of unity, then there are exactly jk different primitive mn-th roots of unity.

Lastly, I will point out one important relation between n-th roots of unity and the n-th roots of any positive real number r. As we know by now that each non-zero number r has n distinct n-th roots. If r is a positive real number then those n different roots are given by r^1/nz_k, where r^1/n is the (one and only) positive real n-th root of r, and z_k are the n different n-th roots of unity.

Related topics: roots of a complex number, cyclotomic polynomials

For next time, we will move away from the subject of complex numbers (for now) and explore something that we are more familiar with, the Pythagorean triples.

 
Next time: Pythagorean Triples

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