Complex numbers · Definition

Complex nth root

complex analysiscomplex numbers

Before you start you need to know

It solves equations of the form \(z^n=w\) over the complex numbers, where every nonzero number has exactly \(n\) solutions.

Definition

An \(n\)-th root of a complex number \(w\) is a complex number \(z\) whose \(n\)-th power, with \(n\) a positive integer, is \(w\). If \(w\) has modulus \(r\) and argument \(\theta\), its roots are given below.

\[z_k=\sqrt[n]{r}\left(\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\right),\quad k=0,\dots,n-1\]

To solve \(z^n=w\) write \(z=\rho(\cos\varphi+i\sin\varphi)\) and \(w=r(\cos\theta+i\sin\theta)\). By De Moivre, \(\rho^n=r\) and \(n\varphi=\theta+2\pi k\) with \(k\) an integer. Hence \(\rho=\sqrt[n]{r}\) and \(\varphi=\frac{\theta+2\pi k}{n}\), which is the formula above.

Only \(k=0,1,\dots,n-1\) give distinct numbers: for \(k=n\) the angle grows by \(2\pi\) and the first root repeats. So every \(w\neq0\) has exactly \(n\) \(n\)-th roots, all with the same modulus \(\sqrt[n]{r}\) and spaced \(\frac{2\pi}{n}\) apart: they are the vertices of a regular \(n\)-gon centred at the origin.

Example: the cube roots of \(-8\) have modulus 2 and angles \(\frac\pi3,\pi,\frac{5\pi}3\), i.e. \(1+i\sqrt3,\ -2,\ 1-i\sqrt3\). The roots of \(w=1\) are called roots of unity; their sum is zero for \(n\ge2\), and the roots of any \(w\) come from multiplying one of them by the roots of unity.

Over the reals \(\sqrt{x}\) is one quantity, but over the complex numbers there is no natural choice of sign for \(\sqrt{w}\): that is why the symbol \(\sqrt[n]{w}\) is avoided or defined as the principal root (\(k=0\)). A typical mistake is to give only the principal root and forget the other \(n-1\).

Think about it

How many sixth roots does \(64\) have, and what is their modulus?

Six roots, all of modulus \(\sqrt[6]{64}=2\), forming a regular hexagon.

If \(z\) is a fourth root of \(w\), how do you get the other three?

By multiplying \(z\) by \(i\), \(-1\) and \(-i\), the fourth roots of unity.

Exercises 10

  1. How many distinct cube roots does \(8\) have in \(\mathbb{C}\)?
  2. Find the square roots of \(-4\).
  3. Which of these is NOT a fourth root of \(1\)?
  4. What is the modulus of each fifth root of \(32\)?
  5. The cube root of \(-8\) with \(k=0\) in the formula is:
  6. The sum of the \(n\) \(n\)-th roots of \(1\), with \(n\ge2\), is:
  7. Find the square roots of \(i\).
  8. The angle between two consecutive sixth roots of any nonzero number is:
  9. Which of these is a cube root of \(-27i\)?
  10. The fourth roots of \(-4\) are \(\pm1\pm i\). If \(z\) is one of them, which statement is true?

References

  • Elias M. Stein, Rami Shakarchi, Complex Analysis, 1st edition, Princeton University Press, 2003. Cap. 1, §1.1
  • James Ward Brown, Ruel V. Churchill, Complex Variables and Applications, 9th edition, McGraw-Hill, 2014. §9
  • Lars V. Ahlfors, Complex Analysis, 3rd edition, McGraw-Hill, 1979. Cap. 1, §2.2