1.2 The Algebra of Complex Numbers
We have seen (in Section 1.1) that complex numbers came to be viewed as ordered pairs of real numbers. That is, a complex number is defined to be , where x and y are both real numbers.
The reason we say ordered pair is because we are thinking of a point in the plane. The point (2, 3), for example, isnot the same as (3, 2). The order in which we write and in the equation makes a difference. Clearly, then, two complex numbers are equal if and only if their x coordinates are and their y coordinates are equal. In other words, iff . (Throughout this text, iff means if and only if.)
A meaningful number system requires a method for combining ordered pairs. Thedefinition of algebraic operations must be consistent so that the sum, difference, product, and quotient of any two ordered pairs will again be an ordered pair. The key to defining how these numbers should be manipulated is to follow Gauss's lead and equate with . Then, if and are arbitrary complex numbers, we have
Thus, the following definitions should make sense.Definition 1.1, (Addition) Formula (1-8) .
Derivation of Formula (1-8).
Definition 1.2, (Subtraction) Formula (1-9) .
Derivation of Formula (1-9).
Example 1.1. Given . (a) Find and (b) . and . We can also use the notation and : and .
Explore Solution 1.1. (a).
Explore Solution 1.1. (b).
Given the rationale we devised for addition and subtraction, it is tempting to define the product as . It turns out, however, that this is not a good definition, and we ask you in the exercises for this section to explain why. How, then, should products be defined? Again, if we equate with and assume, for the moment, that makes sense (so that ), we have
Thus, itappears we are forced into the following definition. Definition 1.3, (Complex Multiplication) Formula (1-10) .
Derivation of Formula (1-10).
Example 1.2. Given . Find . We get the same answer by using the notation and : Of course, it makes sense that the answer came out as we expected because we used the notation x+iy as motivation for ourdefinition in the first place.
Explore Solution 1.2.
To motivate our definition for division, we will proceed along the same lines as we did for multiplication, assuming :
We need to figure out a way to write the preceding quantity in the form . To do this, we use a standard trick and multiply the numerator and denominator by , whichgives Thus, we finally arrive at a rather odd definition.
Definition 1.4, (Complex Division) Formula (1-11) , for .
Derivation of Formula (1-11).
Example 1.3. Given . Find . As with the example for multiplication, we also get this answer if we use the notation :
Explore Solution 1.3.
To perform operations on complex numbers, most mathematicians would use thenotation and engage in algebraic manipulations, as we did here, rather than apply the complicated-looking definitions we gave for those operations on ordered pairs. This procedure is valid because we used the notation as a guide for defining the operations in the first place. Remember, though, that the notation is nothing more than a convenient bookkeeping device for keeping track of howto manipulate ordered pairs. It is the ordered pair algebraic definitions that form the real foundation on which the complex number system is based. In fact, if you were to program a computer to do arithmetic on complex numbers, your program would perform calculations on ordered pairs, using exactly the definitions that we gave.
It turns out that our algebraic...