## 47. Roots of complex numbers.

We have not, up to the present, attributed any meaning to symbols such as \(\sqrt[n]{a}\), \(a^{m/n}\), when \(a\) is a complex number, and \(m\) and \(n\) integers. It is, however, natural to adopt the definitions which are given in elementary algebra for real values of \(a\). Thus we define \(\sqrt[n]{a}\) or \(a^{1/n}\), where \(n\) is a positive integer, as a number \(z\) which satisfies the equation \(z^{n} = a\); and \(a^{m/n}\), where \(m\) is an integer, as \((a^{1/n})^{m}\). These definitions do not prejudge the question as to whether there are or are not more than one (or any) roots of the equation.

## 48. Solution of the equation \(z^{n} = a\).

Let \[a = \rho(\cos\phi + i\sin\phi),\] where \(\rho\) is positive and \(\phi\) is an angle such that \(-\pi < \phi \leq \pi\). If we put \(z = r(\cos\theta + i\sin\theta)\), the equation takes the form \[r^{n}(\cos n\theta + i\sin n\theta) = \rho(\cos\phi + i \sin\phi);\] so that \[\begin{equation*} r^{n} = \rho,\quad \cos n\theta = \cos\phi,\quad \sin n\theta = \sin\phi. \tag{1} \end{equation*}\]

The only possible value of \(r\) is \(\sqrt[n]{\rho}\), the ordinary arithmetical \(n\)th root of \(\rho\); and in order that the last two equations should be satisfied it is necessary and sufficient that \(n\theta = \phi + 2k\pi\), where \(k\) is an integer, or \[\theta = (\phi + 2k\pi)/n.\] If \(k = pn + q\), where \(p\) and \(q\) are integers, and \(0 \leq q < n\), the value of \(\theta\) is \(2p\pi + (\phi + 2q\pi)/n\), and in this the value of \(p\) is a matter of indifference. Hence *the equation \[z^{n} = a = \rho(\cos\phi + i\sin\phi)\] has \(n\) roots and \(n\) only, given by \(z = r(\cos\theta + i\sin\theta)\), where* \[r = \sqrt[n]{\rho},\quad \theta = (\phi + 2q\pi)/n,\quad (q = 0,\ 1,\ 2,\ \dots, n – 1).\]

That these \(n\) roots are in reality all distinct is easily seen by plotting them on Argand’s diagram. The particular root \[\sqrt[n]{\rho}\{\cos(\phi/n) + i\sin(\phi/n)\}\] is called the *principal value* of \(\sqrt[n]{a}\).

The case in which \(a = 1\), \(\rho = 1\), \(\phi = 0\) is of particular interest. The \(n\) roots of the equation \(x^{n} = 1\) are \[\cos(2q\pi/n) + i\sin(2q\pi/n),\quad (q = 0,\ 1,\ \dots, n – 1).\] These numbers are called the \(n\)th roots of unity; the principal value is unity itself. If we write \(\omega_{n}\) for \(\cos(2\pi/n) + i\sin(2\pi/n)\), we see that the \(n\)th roots of unity are \[1,\quad \omega_{n},\quad \omega_{n}^{2},\ \dots,\quad \omega_{n}^{n-1}.\]

## 49. The general form of De Moivre’s Theorem.

It follows from the results of the last section that if \(q\) is a positive integer then one of the values of \((\cos\theta + i\sin\theta)^{1/q}\) is \[\cos(\theta/q) + i\sin(\theta/q).\] Raising each of these expressions to the power \(p\) (where \(p\) is any integer positive or negative), we obtain the theorem that one of the values of \((\cos\theta + i\sin\theta)^{p/q}\) is \(\cos(p\theta/q) + i\sin(p\theta/q)\), or that *if \(\alpha\) is any rational number then one of the values of \((\cos\theta + i\sin\theta)^{\alpha}\) is* \[\cos\alpha\theta + i\sin\alpha\theta.\] This is a generalised form of De Moivre’s Theorem (SecNo 45).

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