28. The Four Fundamental Operations Involving Complex Numbers are defined by assuming that the fundamental laws of algebra, as proved for real numbers, apply also for complex numbers.

29. Addition and Subtraction of Complex Numbers. The sum of two complex numbers is in general a complex number obtained by adding the real parts and the imaginary parts separately.

For, since complex numbers are assumed to obey the Fundamental Laws of Algebra, it may be seen that

(a + ib) ± (x + iy) = (a ± x) + i(b ± y).

The principle applies for three or more complex numbers. (3 + 5√− 1) + (6 − 2 √√/− 1) = (3 + 6) + (5 − 2)√—I

Ex. 1.

-

30. The sum of two conjugate complex numbers is a real number. (a + ib) + (a ib) = (a + a) + i(b − b) = 2a.

For,

31. Multiplication of Complex Numbers is defined by assuming that the Distributive Law for Multiplication (Chapter V. § 21) applies to complex numbers.

(a + ib)(x + iy) = (a + ib)x + (a + ib)iy

= ax + ibx+ aiy + ibiy

32. It may be shown, by applying the Associative Law (Chapter III. § 4), that in general the product of two or more complex numbers can be expressed as a complex number.

Ex. 2. (4+7i)(2 − 5 i) ≥ 8 + 14 i − 20 i — 35 i2 = 43 — 6 i.

Ex. 3. (a + ib)2 = a2 + 2 aib + i2b2 = a2 − b2 + 2 iab.

33. The product of two conjugate complex numbers is a number which is real and positive.

For, (a+ib)(a ib) = a2 — i2b2 = a2 + b2.

Ex. 4. (-5+2-3)(-5-21-3)=(-5)2-22(-3)2=25+12=37. 34. Division of Complex Numbers. The quotient obtained by dividing one complex number by another can be expressed as a complex number.

denominator of which is a complex number can be expressed as a

Complex Factors of Rational Integral Expressions.

36. The method of Chap. XX. § 48, may be applied to obtain factors of expressions of the form ax2 + bx + c which are the products of complex factors.

The following identity was obtained in Chapter XX. § 48:

If the expression b2-4ac be negative, √ 4 ac will be imaginary, and accordingly the factors of ax2 + bx + c, represented by the expressions in square brackets, will be complex.

Ex. 1. Factor x2 + 3x + 4.

We may complete the square with reference to x2 + 3x by using 3x as a finder term as follows:

We have

Hence,

3 x 3

2x

x2+3x+4= x2 + 3x + (1)2 -f +4,

= (x + 1)2+},

= (x + 1)2 - (√− 7)2,

= (x + § + 157)(x + § − 157).

Square Root of a Complex Number.

37.* Corresponding to the principle employed when finding the square root of a simple binomial surd (see Chap. XX. § 50 (vi.)), we have the following

Principle: If the square root of a complex number can be expressed as a complex number, then the square root of the conjugate complex number can also be expressed as a complex number.

* This section may be omitted when the chapter is read for the first time.

(1)

it follows that

√a-ib = √xi√ÿ.

(2)

If a, b, x, and y are real numbers, we obtain, by squaring both

members of (1),

a + ib = x − y + 2 i√xy.

Hence, by § 25, a = xy, and also b = 2√xy.

Accordingly we may construct the expression

38. It follows that the square root of a complex number can be expressed as a complex number.

or,

Hence, from (1) and (2) we obtain by multiplication,

√√(a + ib)(a — ib) = (√x + i√ÿ)(√x − i√ÿ),

√a2 + b2 = x + y.

Hence, by § 25, a = xy.

Solving equations (3) and (5) for x and y, we have

(3)

(4)

(5)

√a + ib = √√ √ a2 + b2 + a

2

Since a and b are real numbers, it follows that a2+b2 is positive; hence Va2+b2 is a real number, and accordingly the right member of (8) is a complex number.

* This section may be omitted when the chapter is read for the first time.