5 The Imaginary. Complex Numbers

Chapter 5
The Imaginary. Complex Numbers

34. The Pure Imaginary. The other symbol which is needed to complete the number-system of algebra, unlike the irrational but like the negative and the fraction, admits of deﬁnition by a single equation of a very simple form, viz.,

x^{2} + 1 = 0

It is the symbol whose square is $- 1$ , the symbol $\sqrt{- 1}$ , now commonly written $i$ .¹³ It is called the unit of imaginaries.

In contradistinction to $i$ all the forms of number hitherto considered are called real. These names, “real” and “imaginary,” are unfortunate, for they suggest an opposition which does not exist. Judged by the only standards which are admissible in a pure doctrine of numbers $i$ is imaginary in the same sense as the negative, the fraction, and the irrational, but in no other sense; all are alike mere symbols devised for the sake of representing the results of operations even when these results are not numbers (positive integers). $i$ got the name imaginary from the diﬃculty once found in discovering some extra-arithmetical reality to correspond to it.

As the only property attached to $i$ by deﬁnition is that its square is $- 1$ , nothing stands in the way of its being “multiplied” by any real number $a$ ; the product, $i a$ , is called a pure imaginary.

An entire new system of numbers is thus created, coextensive with the system of real numbers, but distinct from it. Except $0$ , there is no number in the one which is at the same time contained in the other.¹⁴ Numbers in either system may be compared with each other by the deﬁnitions of equality and greater and lesser inequality (§ 30), $i a$ being called $⋚ i b$ , as $a ⋚ b$ ; but a number in one system cannot be said to be either greater than, equal to or less than a number in the other system.

35. Complex Numbers. The sum $a + i b$ is called a complex number. Its terms belong to two distinct systems, of which the fundamental units are $1$ and $i$ .

The general complex number $a + i b$ is deﬁned by a complex sequence

α_{1} + i β_{1}, α_{2} + i β_{2}, \dots, α_{μ} + i β_{μ}, \dots,

where

α_{1}, α_{2}, \dots

;

β_{1}, β_{2}, \dots

are regular sequences.

Since $a = a + i 0$ (§ 36, 3, Cor.) and $i b = 0 + i b$ , all real numbers, $a$ , and pure imaginaries, $i b$ , are contained in the system of complex numbers $a + i b$ .

$a + i b$ can vanish only when both $a = 0$ and $b = 0$ .

36. The Four Fundamental Operations on Complex Numbers. The assumption of the permanence of the fundamental laws leads immediately to the following deﬁnitions of the addition, subtraction, multiplication, and division of complex numbers.

\begin{matrix} 1 . & (a + i b) + (a^{'} + i b^{'}) & = & a + a^{'} + i (b + b^{'}) . \\ For (a + i b) + (a^{'} + i b^{'}) & = & a + i b + a^{'} + i b^{'}, Law II . \\ = & a + a^{'} + i b + i b^{'}, Law I . \\ = & a + a^{'} + i (b + b^{'}) . Laws II, V . \\ 2 . & (a + i b) - (a^{'} + i b^{'}) & = & a - a^{'} + i (b - b^{'}) . \end{matrix}

By deﬁnition of subtraction (VI) and § 36, 1.

COR. The necessary as well as the suﬃcient condition for the equality of two complex numbers $a + i b$ , $a^{'} + i b^{'}$ is that $a = a^{'}$ and $b = b^{'}$ .

\begin{matrix} For if (a + i b) - (a^{'} + i b^{'}) & = & a - a^{'} + i (b - b^{'}) = 0, \\ a - a^{'} = 0, b - b^{'} & = & 0 (§ 35), or a = a^{'}, b = b^{'} . \\ 3 . & (a + i b) (a^{'} + i b^{'}) & = & a a^{'} - b b^{'} + i (a b^{'} + b a^{'}) . \end{matrix}

\begin{matrix} For (a + i b) (a^{'} + i b^{'}) & = & (a + i b) a^{'} + (a + i b) i b^{'}, Law V . \\ = & a a^{'} + i b \cdot a^{'} + a \cdot i b^{'} + i b \cdot i b^{'}, Law V . \\ = & (a a^{'} - b b^{'}) + i (a b^{'} + b a^{'}) . Laws I–V . \end{matrix}

COR. If either factor of a product vanish, the product vanishes.

For i \times 0 = i (b - b) = i b - i b (§ 10, 5), = 0 (§ 14, 1) .

Hence (a + i b) 0 = a \times 0 + i b \times 0 = a \times 0 + i (b \times 0) = 0 .

Laws V, IV, § 28, § 29, 3.

4 . \frac{a + i b}{a^{'} + i b^{'}} = \frac{a a^{'} + b b^{'}}{a^{' 2} + b^{' 2}} + i \frac{b a^{'} - a b^{'}}{a^{' 2} + b^{' 2}} .

For let the quotient of $a + i b$ by $a^{'} + i b^{'}$ be $x + i y$ .

By the deﬁnition of division (VIII),

\begin{array}{l} (x + i y) (a^{'} + i b^{'}) = a + i b . \\ ∴ & x a^{'} - y b^{'} + i (x b^{'} + y a^{'}) = a + i b . § 36, 3 \\ ∴ & x a^{'} - y b^{'} = a, x b^{'} + y a^{'} = b . § 36, 2, C o r . \end{array}

Hence, solving for $x$ and $y$ between these two equations,

x = \frac{a a^{'} + b b^{'}}{a^{' 2} + b^{' 2}}, y = \frac{b a^{'} - a b^{'}}{a^{' 2} + b^{' 2}} .

Therefore, as in the case of real numbers, division is a determinate operation, except when the divisor is 0; it is then indeterminate. For $x$ and $y$ are determinate (by IX) unless $a^{' 2} + b^{' 2} = 0$ , that is, unless $a^{'} = b^{'} = 0$ , or $a^{'} + i b^{'} = 0$ ; for $a^{'}$ and $b^{'}$ being real, $a^{' 2}$ and $b^{' 2}$ are both positive, and one cannot destroy the other.¹⁵ Hence, by the reasoning in § 24,

COR. If a product of two complex numbers vanish, one of the factors must vanish.

37. Numerical Comparison of Complex Numbers. Two complex numbers, $a + i b$ , $a^{'} + i b^{'}$ , do not, generally speaking, admit of direct comparison with each other, as do two real numbers or two pure imaginaries; for $a$ may be greater than $a^{'}$ , while $b$ is less than $b^{'}$ .

They are compared numerically, however, by means of their moduli $\sqrt{a^{2} + b^{2}}$ , $\sqrt{a^{' 2} + b^{' 2}}$ ; $a + i b$ being said to be numerically greater than, equal to or less than $a^{'} + i b^{'}$ according as $\sqrt{a^{2} + b^{2}}$ is greater than, equal to or less than $\sqrt{a^{' 2} + b^{' 2}}$ . Compare § 47.

38. The Complex System Adequate. The system $a + i b$ is an adequate number-system for algebra. For, as will be shown (Chapter VII), all roots of algebraic equations are contained in this system.

But more than this, the system $a + i b$ is a closed system with respect to all existing mathematical operations, as are the rational system with respect to all ﬁnite combinations of the four fundamental operations and the real system with respect to these operations and regular sequence-building. For the results of the four fundamental operations on complex numbers are complex numbers (§ 36, 1, 2, 3, 4). Any other operation may be resolved into either a ﬁnite combination of additions, subtractions, multiplications, divisions or such combinations indeﬁnitely repeated. In either case the result, if determinate, is a complex number, as follows from the deﬁnitions 1, 2, 3, 4 of § 36, and the nature of the real number-system as developed in the preceding chapter (see Chapter VIII).

The most important class of these higher operations, and the class to which the rest may be reduced, consists of those operations which result in inﬁnite series (Chapter VIII); among which are involution, evolution, and the taking of logarithms (Chapter IX), sometimes included among the fundamental operations of algebra.

39. Fundamental Characteristics of the Algebra of Number. The algebra of number is completely characterized, formally considered, by the laws and deﬁnitions I–IX and the fact that its numbers are expressible linearly in terms of two fundamental units.¹⁶ It is a linear, associative, distributive, commutative algebra. Moreover, the most general linear, associative, distributive, commutative algebra, whose numbers are complex numbers of the form $x_{1} e_{1} + x_{2} e_{2} + \dots + x_{n} e_{n}$ , built from $n$ fundamental units $e_{1}, e_{2}, \dots, e_{n}$ , is reducible to the algebra of the complex number $a + i b$ . For Weierstrass¹⁷ has shown that any two complex numbers $a$ and $b$ of the form $x_{1} e_{1} + x_{2} e_{2} + \dots + x_{n} e_{n}$ , whose sum, diﬀerence, product, and quotient are numbers of this same form, and for which the laws and deﬁnitions I–IX hold good, may by suitable transformations be resolved into components $a_{1}, a_{2}, \dots a_{r}$ ; $b_{1}, b_{2}, \dots b_{r}$ , such that

\begin{array}{rcl} a & = & a_{1} + a_{2} + \dots + a_{r}, \\ b & = & b_{1} + b_{2} + \dots + b_{r}, \\ a \pm b & = & a_{1} \pm b_{1} + a_{2} \pm b_{2} + \dots + a_{r} \pm b_{r}, \\ a b & = & a_{1} b_{1} + a_{2} b_{2} + \dots + a_{r} b_{r}, \\ \frac{a}{b} & = & \frac{a_{1}}{b_{1}} + \frac{a_{2}}{b_{2}} + \dots + \frac{a_{r}}{b_{r}} . \end{array}

The components $a_{i}$ , $b_{i}$ are constructed either from one fundamental unit $g_{i}$ or from two fundamental units $g_{i}$ , $k_{i}$ .¹⁸

For components of the ﬁrst kind the multiplication formula is

(α g_{i}) (β g_{i}) = (α β) g_{i} .

For components of the second kind the multiplication formula is

(α g_{i} + β k_{i}) (α^{'} g_{i} + β^{'} k_{i}) = (α α^{'} - β β^{'}) g_{i} + (α β^{'} + β α^{'}) k_{i} .

And these formulas are evidently identical with the multiplication formulas

\begin{array}{l} (α 1) (β 1) = & (α β) 1, \\ (α 1 + β i) (α^{'} 1 + β^{'} i) = & (α α^{'} - β β^{'}) 1 + (α β^{'} + β α^{'}) i \end{array}

of common algebra.

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Chapter 5The Imaginary. Complex Numbers

Chapter 5
The Imaginary. Complex Numbers