CHAPTER XXI. IMAGINARY AND COMPLEX QUANTITIES. § 1. GENERAL THEOREMS. 589. It was pointed out, in the introduction to Chap. XI., that the square of every positive or negative quantity is positive. Hence, a quantity whose square is negative, is neither positive nor negative. Such a quantity is usually called Imaginary; while positive or negative quantities are called Real. In this chapter we shall exhibit the use of Imaginary quantities. By their means the results of the preceding ten chapters may be systematised and extended. We shall not interpret imaginary quantities at present, but shall merely use them symbolically: i.e. we shall deduce results by their use which involve only real quantities. To accomplish this it is necessary to postulate that Imaginary quantities shall be governed by the same laws of operation as real quantities. In the same way (as was indicated in the introduction to Chap. XI.) negative quantities may be introduced symbolically without interpretation, by assuming that they follow the same laws as positive quantities. It was afterwards shown that the relation of negative to positive could be used to represent oppositeness of direction. 590. In confining ourselves to positive and negative quantities, we confine ourselves to quantities which have the same or the opposite sign: i.e. the addition of two such quantities is always equivalent to addition or subtraction. So in introducing imaginary quantities, these may be either positive imaginary or negative imaginary. That is, we shall imply that the addition to any imaginary of a negative imaginary is equivalent to the subtraction of the corresponding positive. Thus we have four affections of quantity; viz. positive real, negative real, positive imaginary, and negative imaginary. 591. If a is any positive real quantity, (-a) is an imaginary quantity, to which we may attribute either sign. But since we postulate that all our symbols shall obey the same laws, we have (see Art. 233) √(− a) = √{a × (− 1)} = √a × √( − 1). Hence the square root of any negative number can be expressed as the product of a real number into the square root of – 1. Thus the only symbol we need use is √√(−1). The symbol i will be used as the equivalent of the positive value of the (-1); and i, therefore, as the negative value of the /(-1). 592. Wholly imaginary quantities. A real number xi is called a wholly imaginary quantity. Wholly imaginary quantities are incomparable with wholly real quantities. Thus, if a and b are both real quantities, the equation a=bi involves a contradiction, unless a = 0 and b = 0. Thus, the one equation a = bi is interpreted to mean the same as the two equations a=0 and b=0. This is the fundamental principle which enables us to make use of symbols of imaginary quantities. 593. Complex quantities. The sum of a wholly real and a wholly imaginary quantity is called a complex quantity. J. T. 27 Such a sum cannot be interpreted at present; but its use is explained by a corollary from the last article. Thus If a, b, c, d are wholly real, the equation a + bi = c + di is equivalent to a-c=(d-b)i. Hence it is also equivalent to the two equations a-c= =0 and d-b=0, i.e. a=c and b = d. Thus if one complex expression is equal to another, we may equate separately the real and imaginary terms. In the symbol a + bi, we may suppose that either a or b is zero: so that complex quantities include wholly imaginary and wholly real quantities. The remainder of this chapter deals mainly with complex quantities. Conjugates, Norms, and Moduli. 594. The complex quantities y + zi and y-zi are called Conjugates of one another. The quantity y2 + 22 is called the Norm of У + zi. The positive value of the square root of the Norm is called the Modulus. Thus the norm of either of two conjugate complex quantities is their product. And the modulus of either of two conjugate complex quantities is their geometric mean. 595. The necessary and sufficient condition that a complex quantity should vanish is that its norm [or modulus] should vanish. For, if y and z are real and not zero, y2 and z2 must both be positive. Hence y2 + z2 cannot vanish unless both y and z vanish. Thus, if y2+22 = 0, y + zi = 0. Conversely, if y + zi = 0, we must have y = 0 and z = 0; and .. y2+z2 = 0. 596. The norm [or modulus] of the product of two complex quantities is equal to the product of their norms [or moduli]. For and (ac-bd)2 + (ad + bc)2 = a2c2 + b2d2 + a2d2 + b2c2 COR. The norm [or modulus] of the nth power of a complex quantity is the nth power of its norm [or modulus]. 597. If the product of two complex quantities vanishes one of the factors must vanish. For, by Art. 595, in order that (a + bi) (c + di) may vanish, its norm must vanish. That is, by Art. 596, (a2 + b2) (c2 + d2) must vanish. .. either a2 + b2 or c2 + d2 must vanish. .. by Art. 595, either a + bi or c + di must vanish. Theorems on Equations. 598. The theorems on factors in Arts. 412, 413, 414 may be expressed as theorems on equations, and thus be applied to any complex (including real) quantities. Thus, if α, α, a... are (complex) roots of the equation f(x)=0 (α, α, a, being all different), then f(a) = 0, ƒ(a) = 0, f(a)=0, and so on. Thus, for all values of x, f(x)=(x) (x -α1) (x -α) (x-αg)...... If f(x) is of the nth degree, pan being its highest term, and if ɑ ɑë...ɑn are n roots of ƒ (x)=0; then, for all values of x, 599. f(x)= p(x-a1) (x -α)...(x-an). Hence, an equation of the nth degree cannot have more than n roots. For, if possible, let an+1 be an (n+1th) root of the above equation f(x) = 0: so that ƒ (an+1)=0. Then, putting xa, in the above identity, ƒ(an+1)=p (an+1— α2) (αn+1 — α2) ..... (An+1 — An). But the factor p is not zero, because, by hypothesis, the equation is of the nth degree. Hence one of the factors an+1 - α1, an+1 a,,... must be zero by Art. 597. That is, an+1 must be equal to one of the roots a1, a2,... An• 600. The complex roots of an equation with real coefficients enter in conjugate pairs. Let f(x) be any integral function of c, in which the coefficients of the several powers of x are all real. Then writing y + zi for x, the real terms must be those which involve even powers, and the imaginary terms those which involve odd powers of 2. Hence If ƒ (y + zi) = P+Qi, then f(y - zi) = P – Qi. Therefore, if y+zi is a root of f(x)=0; we have P=0 and Q=0; and . f(y-zi)=0. That is If y + zi is a root of ƒ (x) = 0, then y-zi is also a root of f(x) = 0. 601. To express the product of any number of factors of the form cos a + i sin a as a complex quantity. and We have shown geometrically, that cos (a + B) = cos a cos ẞ- sin a sin ß, Hence, since i2 == 1, (cos a + i sin a) (cos ẞ + i sin ẞ) = cos a cos ẞ+ i sin a cos ẞ + i cos a sin ẞ- sin a sin ß Hence again (cos a + i sin a) (cos ẞ + i sin ẞ) (cos y + i sin y) = = {cos (a + ß) + i sin (a + ẞ)} (cos y + i sin y) = cos (a + B + y) + i sin (a + ẞ + y). And so on. Thus the product of any number of factors (cos a + i sin a) (cos B + i sin ẞ)...(cos = cos (a + B + ... + §) + i sin (a + ß+ ... + i sin έ) + §). |