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 \[1,\quad \omega_{n},\quad \omega_{n}^{2},\ \dots,\quad \omega_{n}^{n-1}.\]

Example XXII

1. The two square roots of \(1\) are \(1\), \(-1\); the three cube roots are \(1\), \(\frac{1}{2}(-1 + i\sqrt{3})\), \(\frac{1}{2}(-1 – i\sqrt{3})\); the four fourth roots are \(1\), \(i\), \(-1\), \(-i\); and the five fifth roots are \[\begin{aligned} {4} 1,\quad &\tfrac{1}{4} \Bigl[ &&\sqrt{5} – 1 + i\sqrt{10 + 2\sqrt{5}}\Bigr],\quad && \tfrac{1}{4} \Bigl[-&&\sqrt{5} – 1 + i\sqrt{10 – 2\sqrt{5}}\Bigr],\\ &\tfrac{1}{4} \Bigl[-&&\sqrt{5} – 1 – i\sqrt{10 – 2\sqrt{5}}\Bigr],\quad && \tfrac{1}{4} \Bigl[ &&\sqrt{5} – 1 – i\sqrt{10 + 2\sqrt{5}}\Bigr].\end{aligned}\]

2. Prove that \[1 + \omega_{n} + \omega_{n}^{2} + \dots + \omega_{n}^{n-1} = 0.\]

3. Prove that \[(x + y\omega_{3} + z\omega_{3}^{2}) (x + y\omega_{3}^{2} + z\omega_{3}) = x^{2} + y^{2} + z^{2} – yz – zx – xy.\]

4. The \(n\)th roots of \(a\) are the products of the \(n\)th roots of unity by the principal value of \(\sqrt[n]{a}\).

5. It follows from Exs. XXI. 14 that the roots of \[z^{2} = \alpha + \beta i\] are \[\pm \sqrt{\tfrac{1}{2} \{\sqrt{\alpha^{2} + \beta^{2}} + \alpha\}} \pm i\sqrt{\tfrac{1}{2} \{\sqrt{\alpha^{2} + \beta^{2}} – \alpha\}},\] like or unlike signs being chosen according as \(\beta\) is positive or negative. Show that this result agrees with the result of § 48.

6. Show that \((x^{2m} – a^{2m})/(x^{2} – a^{2})\) is equal to \[\Bigl(x^{2} – 2ax\cos\frac{\pi}{m} + a^{2}\Bigr) \Bigl(x^{2} – 2ax\cos\frac{2\pi}{m} + a^{2}\Bigr) \dots \Bigl(x^{2} – 2ax\cos\frac{(m – 1)\pi}{m} + a^{2}\Bigr).\]

[The factors of \(x^{2m} – a^{2m}\) are \[(x – a),\quad (x – a\omega_{2m}),\quad (x – a\omega_{2m}^{2}),\ \dots\quad (x – a\omega_{2m}^{2m-1}).\] The factor \(x – a\omega_{2m}^{m}\) is \(x + a\). The factors \((x – a\omega_{2m}^{s})\), \((x – a\omega_{2m}^{2m-s})\) taken together give a factor \(x^{2} – 2ax \cos(s\pi/m) + a^{2}\).]

7. Resolve \(x^{2m+1} – a^{2m+1}\), \(x^{2m} + a^{2m}\), and \(x^{2m+1} + a^{2m+1}\) into factors in a similar way.

8. Show that \(x^{2n} – 2x^{n}a^{n} \cos\theta + a^{2n}\) is equal to \[\begin{gathered} \left(x^{2} – 2xa\cos\frac{\theta}{n} + a^{2}\right) \left(x^{2} – 2xa\cos\frac{\theta + 2\pi}{n} + a^{2}\right) \dots \\ \dots\left(x^{2} – 2xa\cos\frac{\theta + 2(n – 1)\pi}{n} + a^{2}\right).\end{gathered}\]

[Use the formula \[x^{2n} – 2x^{n}a^{n} \cos\theta + a^{2n} = \{x^{n} – a^{n}(\cos\theta + i\sin\theta)\} \{x^{n} – a^{n}(\cos\theta – i\sin\theta)\},\] and split up each of the last two expressions into \(n\) factors.]

9. Find all the roots of the equation \(x^{6} – 2x^{3} + 2 = 0\).

10. The problem of finding the accurate value of \(\omega_{n}\) in a numerical form involving square roots only, as in the formula \(\omega_{3} = \frac{1}{2}(-1 + i\sqrt{3})\), is the algebraical equivalent of the geometrical problem of inscribing a regular polygon of \(n\) sides in a circle of unit radius by Euclidean methods, i.e. by ruler and compasses. For this construction will be possible if and only if we can construct lengths measured by \(\cos(2\pi/n)\) and \(\sin(2\pi/n)\); and this is possible (Ch. II, Misc. Exs. II 22) if and only if these numbers are expressible in a form involving square roots only.

Euclid gives constructions for \(n = 3\), \(4\), \(5\), \(6\), \(8\), \(10\), \(12\), and \(15\). It is evident that the construction is possible for any value of \(n\) which can be found from these by multiplication by any power of \(2\). There are other special values of \(n\) for which such constructions are possible, the most interesting being \(n = 17\).

 

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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