349CHAPTER21
Imaginary and Complex Numbers
All the algebra we’ve done so far has been of the first degree. Variables were never squared,
cubed, or raised to any higher power. Before we start dealing with equations of higher degree,
let’s study numbers a little more. This chapter introduces the imaginary numbers and the
complex numbers.The Square Root of − 1
Early mathematicians couldn’t represent fractions or ratios. Then the rational numbers were
developed. At one time, there was no such thing as the number zero. Then the idea was con-
ceived, and zero “found its place.” Negative numbers evolved. The distinction between the
rationals and irrationals followed. Finally, someone defined quantities that, when squared,
would produce negative numbers.A mystery number
If you square a positive real number, you get a positive real. If you square 0, you get 0. If you
square a negative real, you get a positive real. You can’t square any real number and get a nega-
tive real. Within the set of real numbers, expressions such as (−1)1/2 or (−3)1/2 or (−1/5)1/2 or
(−π)1/2 are undefined.
Before square roots of negative numbers were defined more than one mathematician
must have asked, “What if there actually are numbers that can be squared to produce negative
reals, but no one has found them yet?” They imagined that such numbers existed, and then
they explored how those numbers would behave in arithmetic. The outcome of their “mind
experiments” was the discovery of a new realm of numbers that they called “imaginary.” That
term has been used ever since.
Many mathematics texts use the lowercase English letter i to stand for the 1/2 power of −1,
which means the positive square root of −1. The choice of i is reasonable enough; it stands
for “imaginary.” In science, engineering, and applied mathematics, the letter j is often used,
becausei plays other roles, notably in the expressions for sequences and series. In this book,
we’ll call the unit imaginary number j, not i. This choice is based on my assumption thatCopyright © 2008 by The McGraw-Hill Companies, Inc. Click here for terms of use.