146. Complex numbers. The solution of the quadratic equation with negative discriminant (p. 71) affords us an expression which consists of a real number connected with an imaginary number by a+ or sign. Such an expression is called a complex number. It consists of two parts which are of different kinds, the real part and the imaginary part. Thus 6+ 4 i means 6 1's + 4 is. Obviously, to any pair of real numbers (x, y) corresponds a complex number xiy, and conversely.

4+14

6+13

147. Graphical representation of complex numbers. We have represented all real numbers on a single straight line. When we wished to represent two numbers simultaneously, we made use of the plane, and assumed a one-to-one correspondence between the points on the plane and the pairs of numbers (x, y). The general complex number x + iy depends on the values of the independent real numbers x and y, and may then properly be represented by a point on a plane. We represent real numbers on the X axis, imaginary numbers on the Y axis, and the complex number x + iy by the point (x, y) on the plane. Thus the complex numbers 6+i3,

2-14

7-15

− 4 + i 4, 7 − i 5, -2 i 4 are represented by points on the plane as indicated in the figure.

148. Equality of complex numbers. We define the two complex numbers a + ib and c + id to be equal when and only when a = c and b = d.

The definition seems reasonable, since 1 and i are different in kind, and we should not expect any real multiple of one to cancel any real multiple of the other.

Similarly, if we took not abstract expressions as 1 and i for units but concrete objects as trees and streets, we should say that a trees +b streets = c trees + d streets

when and only when a = c and b = d.

PRINCIPLE. When two numerical expressions involving imaginaries are equal to each other, we may equate real parts and imaginary parts separately.

The graphical interpretation of the definition of equality is that equal complex numbers are always represented by the same point on the plane.

From the definition given we see that a + ib = 0 when and only when a = = b = 0.

ASSUMPTION. We assume that complex numbers obey the commutative and associative laws and the distributive law given in § 10. We also assume the same rules for parentheses as given in § 15.

This assumption enables us to define the fundamental operations on complex numbers.

149. Addition and subtraction. By applying the assumptions just made we obtain the following symbolical expression for the operations of addition and subtraction of any two complex numbers aib and c + id:

a + ib ±(c+id) = a + c + i (b± d).

RULE. To add (subtract) complex numbers, add (subtract) the real and imaginary parts separately.

150. Graphical representation of addition. We now proceed to give the graphical interpretation of the operations of addition and subtraction.

THEOREM. The sum of two numbers A = a + ib and B

= c + id

is represented by the fourth vertex of the parallelogram formed on OA and OB as sides.

Let OASB be a parallelogram. Draw ESOE, AH | ES, BD (= d) | OE.

S

y

B

A ODB since their sides are

parallel, and OB = AS.

OD AH = c.

=

ES EH + HS = b + d,

OE OF FE = a + c,

and S has coördinates (a + c, b + d) and represents the sum of A and B, by § 149.

EXERCISES

1. The difference A -B of two numbers A = a + ib and B = c + id is represented by the extremity D of the line OD drawn from the origin parallel to the diagonal BA of the parallelogram formed on OB and OA as sides.

2. Represent graphically the following expressions.

151. Multiplication of complex numbers. The assumption of § 148 enables us to multiply complex numbers by the following RULE. To multiply the complex number a + ib by c + id, proceed as if they were real binomials, keeping in mind the laws for multiplying imaginaries.

ac + icb + iud + (i)2 bd = ac― bd + i(cb + ad).

152. Conjugate complex numbers. Complex numbers that differ only in the sign of their imaginary parts are called conjugate complex numbers, or conjugate imaginaries.

THEOREM. The sum and the product of conjugate complex numbers are real numbers.

Thus

a + iba ib = 2 a,

(a + ib) (ail) = a2 + b2.

153. Division of complex numbers. The quotient of two complex numbers may now be expressed as a single complex number.

RULE. To express the quotient rationalize the denominator, using the conjugate of the denominator.

a + ib
c + id

in the form x + iy,

as a rationalizing factor

We have now defined the fundamental operations on complex numbers and shall make frequent use of them. If the question remains in one's mind, "After all, what are they?" the answer is this: They are things for which we have defined the fundamental operations of numbers and, since they have the properties of numbers, must be called numbers, just as a flower that has all the characteristic properties of a known species is thereby determined to belong to that species. Furthermore, our operations have been so defined that if the imaginary parts of the complex numbers vanish and the numbers become real, the expression defining any operation on complex numbers reduces to one defining the same operation on the real part of the number. Thus in (1) above, if b d = 0, the expression reduces to

=