Chapter One
Complex Numbers
1.1 Introduction.
Let us hark back to the first grade when the only numbers you knew
were the ordinary everyday integers. You had no trouble solving problems in which you
were, for instance, asked to find a number
x
such that 3
x
6. You were quick to answer
”2”. Then, in the second grade, Miss Holt asked you to find a number
x
such that 3
x
8.
You were stumped—there was no such ”number”! You perhaps explained to Miss Holt that
3
2
6 and 3
3
9, and since 8 is between 6 and 9, you would somehow need a number
between 2 and 3, but there isn’t any such number. Thus were you introduced to ”fractions.”
These fractions, or rational numbers, were defined by Miss Holt to be ordered pairs of
integers—thus, for instance,
8,3
is a rational number. Two rational numbers
n
,
m
and
p
,
q
were defined to be equal whenever
nq
pm
. (More precisely, in other words, a
rational number is an equivalence class of ordered pairs,
etc.
) Recall that the arithmetic of
these pairs was then introduced: the sum of
n
,
m
and
p
,
q
was defined by
n
,
m
p
,
q
nq
pm
,
mq
,
and the product by
n
,
m
p
,
q
np
,
mq
.
Subtraction and division were defined, as usual, simply as the inverses of the two
operations.
In the second grade, you probably felt at first like you had thrown away the familiar
integers and were starting over. But no. You noticed that
n
,1
p
n
p
and
also
n
p
np
. Thus the set of all rational numbers whose second coordinate is
one behave just like the integers. If we simply abbreviate the rational number
n
by
n
,
there is absolutely no danger of confusion: 2
3
5 stands for
2,1
3,1
5,1
. The
equation 3
x
8 that started this all may then be interpreted as shorthand for the equation
u
,
v
8,1
, and one easily verifies that
x
u
,
v
is a solution. Now, if
someone runs at you in the night and hands you a note with 5 written on it, you do not
know whether this is simply the integer 5 or whether it is shorthand for the rational number
.
What we see is that it really doesn’t matter.
What we have ”really” done is
expanded the collection of integers to the collection of rational numbers. In other words,
we can think of the set of all rational numbers as including the integers–they are simply the
rationals with second coordinate 1.
One last observation about rational numbers. It is, as everyone must know, traditional to
1.1