calculating the nth roots of a complex number

Fix a complex numberMathworldPlanetmathPlanetmath z. We wish to compute all the nth roots of z. By definition, w is an nth root of z if wn=z.

First of all, if z=0, it is clear that all its nth roots will be zero as well.

Suppose z is not zero. Then we can write z=reiθ, for some positive real number r and some real number θ (this is de Moivre’s theorem). In fact, we have a choice of values for θ: rei(θ+2kπ)=reiθ for every integer k. Usually, we choose θ so that -π<θπ.

What are the possible values for w? Write w in polar form also, as w=ρeiϕ. Then wn=ρneinϕ. We are looking for values of ρ and ϕ so that ρneinϕ=reiθ. Since every nonzero complex number can be written in polar form in a unique way with ρ>0 and -π<ϕπ, we can assume that this is true for w. So for wn to equal z, we must have ρn=r and nϕ=θ+2kπ for some integer k. The first of these conditions is that ρ be the usual (positivePlanetmathPlanetmath) nth root of the real number r. The second, rewritten, says that ϕ=θ/n+2kπ/n for some integer k. There will be exactly n possibilities for k which yield -π<ϕπ: -n/2<kn/2.

Summarizing, if

z=reiθ for -π<θπ,




w=rnei(θn+2knπ) for some k such that -n2<kn2.

Of course, the restriction on the values of k is designed to ensure that none of the values obtained for different k are actually equal; we could have chosen a different range of values for k: in books, you most often see 0k<n, which still ensures that the values are all distinct but does not ensure that they are between -π and π.

Thinking about what this means in polar coordinates, this means that the angles between the nth roots are exactly 1/n of a complete circle, so that they form the vertices of a regular polygonMathworldPlanetmath.

We can write the nth roots of a complex number in another way. First, apply the above expression to compute the nth roots of 1:


Then observe that if wn=z, then (ωkw)n=ωknwn=wn=z. So if w is any nth root of z, the nth roots of z can also be written as

ωkw for 0k<n,


ω0kw for 0k<n.

This last way of writing the nth roots of a complex number shows that somehow the nth roots of 1 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
Canonical name CalculatingTheNthRootsOfAComplexNumber
Date of creation 2013-03-22 14:13:42
Last modified on 2013-03-22 14:13:42
Owner archibal (4430)
Last modified by archibal (4430)
Numerical id 10
Author archibal (4430)
Entry type Example
Classification msc 12D99
Classification msc 30-00
Related topic TopicEntryOnComplexAnalysis
Related topic BinomialEquation