To solve zn=w, where w=r eitheta 0 and n is a positive integer, include every angle theta+2kpi before taking the root. z_k=r 1n(cos theta+2kpin+isin theta+2kpin), k=0,1,,n-1. There are exactly n distinct roots. They have common modulus r 1n, and neighbouring arguments differ by 2pin. For n 3, they form a regular polygon; for n=2, a diametrically opposite pair. If w=0, the only distinct root is zero. Example The four roots of z4=16 have modulus 2 and arguments 0, 2,pi, 3pi2: 2,2i,-2,-2i. Roots of unity set r=1,theta=0. They are e(2kpi i)/(n), k=0,,n-1. For n 2, their sum is zero. For a rational power w pq, w 0, with coprime integers p,q, q>0, use the equation zq=wp when all values are required. Do not assume a single chosen root represents the full solution set.

To solve \(z^n=w\), where \(w=r\mathrm e^{i\theta}\ne0\) and \(n\) is a positive integer, include every angle \(\theta+2k\pi\) before taking the root.

\(z_k=r^{\frac1n}\left(\cos\dfrac{\theta+2k\pi}{n}+i\sin\dfrac{\theta+2k\pi}{n}\right),\quad k=0,1,\ldots,n-1\).

There are exactly \(n\) distinct roots. They have common modulus \(r^{\frac1n}\), and neighbouring arguments differ by \(\dfrac{2\pi}{n}\). For \(n\ge3\), they form a regular polygon; for \(n=2\), a diametrically opposite pair. If \(w=0\), the only distinct root is zero.

The four roots of \(z^4=16\) have modulus \(2\) and arguments \(0,\dfrac\pi2,\pi,\dfrac{3\pi}{2}\): \(2,2i,-2,-2i\).

Roots of unity set \(r=1,\theta=0\). They are \(\mathrm e^{\frac{2k\pi i}{n}}\), \(k=0,\ldots,n-1\). For \(n\ge2\), their sum is zero.

For a rational power \(w^{\frac pq}\), \(w\ne0\), with coprime integers \(p,q\), \(q>0\), use the equation \(z^q=w^p\) when all values are required. Do not assume a single chosen root represents the full solution set.

Complex Numbers

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Complex roots and roots of unity

Complex roots and roots of unity