47-49. Roots of complex numbers

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 $a = ρ (\cos ϕ + i \sin ϕ),$ where $ρ$ is positive and $ϕ$ is an angle such that $- π < ϕ \leq π$ . If we put $z = r (\cos θ + i \sin θ)$ , the equation takes the form $r^{n} (\cos n θ + i \sin n θ) = ρ (\cos ϕ + i \sin ϕ);$ so that $\begin{matrix} (1) & r^{n} = ρ, \cos n θ = \cos ϕ, \sin n θ = \sin ϕ . \end{matrix}$

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

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

The case in which $a = 1$ , $ρ = 1$ , $ϕ = 0$ is of particular interest. The $n$ roots of the equation $x^{n} = 1$ are $\cos (2 q π / n) + i \sin (2 q π / n), (q = 0, 1, \dots, n - 1) .$ These numbers are called the $n$ th roots of unity; the principal value is unity itself. If we write $ω_{n}$ for $\cos (2 π / n) + i \sin (2 π / n)$ , we see that the $n$ th roots of unity are $1, ω_{n}, ω_{n}^{2}, \dots, ω_{n}^{n - 1} .$

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 $\begin{aligned} 41, & \frac{1}{4} [ & \sqrt{5} - 1 + i \sqrt{10 + 2 \sqrt{5}}], & \frac{1}{4} [- & \sqrt{5} - 1 + i \sqrt{10 - 2 \sqrt{5}}], \\ \frac{1}{4} [- & \sqrt{5} - 1 - i \sqrt{10 - 2 \sqrt{5}}], & \frac{1}{4} [ & \sqrt{5} - 1 - i \sqrt{10 + 2 \sqrt{5}}] . \end{aligned}$

2. Prove that $1 + ω_{n} + ω_{n}^{2} + \dots + ω_{n}^{n - 1} = 0.$

3. Prove that $(x + y ω_{3} + z ω_{3}^{2}) (x + y ω_{3}^{2} + z ω_{3}) = x^{2} + y^{2} + z^{2} - y z - z x - x y .$

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 $z^{2} = α + β i$ are $\pm \sqrt{\frac{1}{2} {\sqrt{α^{2} + β^{2}} + α}} \pm i \sqrt{\frac{1}{2} {\sqrt{α^{2} + β^{2}} - α}},$ like or unlike signs being chosen according as $β$ is positive or negative. Show that this result agrees with the result of § 48.

6. Show that $(x^{2 m} - a^{2 m}) / (x^{2} - a^{2})$ is equal to $(x^{2} - 2 a x \cos \frac{π}{m} + a^{2}) (x^{2} - 2 a x \cos \frac{2 π}{m} + a^{2}) \dots (x^{2} - 2 a x \cos \frac{(m - 1) π}{m} + a^{2}) .$

[The factors of

x^{2 m} - a^{2 m}

are

(x - a), (x - a ω_{2 m}), (x - a ω_{2 m}^{2}), \dots (x - a ω_{2 m}^{2 m - 1}) .

The factor

x - a ω_{2 m}^{m}

x + a

. The factors

(x - a ω_{2 m}^{s})

(x - a ω_{2 m}^{2 m - s})

taken together give a factor

x^{2} - 2 a x \cos (s π / m) + a^{2}

7. Resolve $x^{2 m + 1} - a^{2 m + 1}$ , $x^{2 m} + a^{2 m}$ , and $x^{2 m + 1} + a^{2 m + 1}$ into factors in a similar way.

8. Show that $x^{2 n} - 2 x^{n} a^{n} \cos θ + a^{2 n}$ is equal to $\begin{matrix} (x^{2} - 2 x a \cos \frac{θ}{n} + a^{2}) (x^{2} - 2 x a \cos \frac{θ + 2 π}{n} + a^{2}) \dots \\ \dots (x^{2} - 2 x a \cos \frac{θ + 2 (n - 1) π}{n} + a^{2}) . \end{matrix}$

[Use the formula

x^{2 n} - 2 x^{n} a^{n} \cos θ + a^{2 n} = {x^{n} - a^{n} (\cos θ + i \sin θ)} {x^{n} - a^{n} (\cos θ - i \sin θ)},

and split up each of the last two expressions into

n

factors.]

9. Find all the roots of the equation $x^{6} - 2 x^{3} + 2 = 0$ .

10. The problem of finding the accurate value of $ω_{n}$ in a numerical form involving square roots only, as in the formula $ω_{3} = \frac{1}{2} (- 1 + i \sqrt{3})$ , 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 $\cos (2 π / n)$ and $\sin (2 π / n)$ ; 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 $(\cos θ + i \sin θ)^{1 / q}$ is $\cos (θ / q) + i \sin (θ / q) .$ 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 $(\cos θ + i \sin θ)^{p / q}$ is $\cos (p θ / q) + i \sin (p θ / q)$ , or that if $α$ is any rational number then one of the values of $(\cos θ + i \sin θ)^{α}$ is $\cos α θ + i \sin α θ .$ This is a generalised form of De Moivre’s Theorem (SecNo 45).

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46. Rational functions of a complex variable

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MISCELLANEOUS EXAMPLES ON CHAPTER III

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47. Roots of complex numbers.

48. Solution of the equation zn=a.

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

48. Solution of the equation $z^{n} = a$ .