calculating the nth roots of a complex number
First of all, if , it is clear that all its nth roots will be zero as well.
Suppose is not zero. Then we can write , for some positive real number and some real number (this is de Moivre’s theorem). In fact, we have a choice of values for : for every integer . Usually, we choose so that .
What are the possible values for ? Write in polar form also, as . Then . We are looking for values of and so that . Since every nonzero complex number can be written in polar form in a unique way with and , we can assume that this is true for . So for to equal , we must have and for some integer . The first of these conditions is that be the usual (positive) nth root of the real number . The second, rewritten, says that for some integer . There will be exactly possibilities for which yield : .
Of course, the restriction on the values of is designed to ensure that none of the values obtained for different are actually equal; we could have chosen a different range of values for : in books, you most often see , which still ensures that the values are all distinct but does not ensure that they are between and .
We can write the nth roots of a complex number in another way. First, apply the above expression to compute the nth roots of :
Then observe that if , then . So if is any nth root of , the nth roots of can also be written as
This last way of writing the nth roots of a complex number shows that somehow the nth roots of already capture the unusual behaviour of the nth roots of any number. So in fact, one often wants to look at the roots of unity in any field, whether it is the integers modulo a prime, rational functions, or some more exotic field.
|Title||calculating the nth roots of a complex number|
|Date of creation||2013-03-22 14:13:42|
|Last modified on||2013-03-22 14:13:42|
|Last modified by||archibal (4430)|