2 1. From the Real Numbers to the Complex Numbers

inverse -n such that

(1) n + (-n) = (-n) + n = 0.

Of course 0 is the only number whose additive inverse is itself.

Let a,b be given integers. As usual we write a-b for the sum a+(-b). Consider

the equation a + x = b for an unknown x. We learn to solve this equation at a

young age; the idea is that subtraction is the inverse operation to addition. To solve

a + x = b for x, we ﬁrst add -a to both sides and use (1). We can then substitute

b for a + x to obtain the solution

x = 0 + x = (-a) + a + x = (-a) + b = b + (-a) = b - a.

This simple principle becomes a little more diﬃcult when we work with multi-

plication. It is not always possible, for example, to divide a collection of n objects

into two groups of equal size. In other words, the equation 2 * a = b does not have

a solution in Z unless b is an even number. Within Z, most integers (±1 are the

only exceptions) do not have multiplicative inverses.

To allow for division, we enlarge Z into the larger system Q of rational numbers.

We think of elements of Q as fractions, but the deﬁnition of Q is a bit subtle. One

reason for the subtlety is that we want

1

2

,

2

4

, and

50

100

all to represent the same

rational number, yet the expressions as fractions diﬀer. Several approaches enable

us to make this point precise. One way is to introduce the notion of equivalence

class and then to deﬁne a rational number to be an equivalence class of pairs of

integers. See [4] or [8] for this approach. A second way is to think of the rational

number system as known to us; we then write elements of Q as letters, x,y,u,v, and

so on, without worrying that each rational number can be written as a fraction in

inﬁnitely many ways. We will proceed in this second fashion. A third way appears

in Exercise 1.2 below. Finally we emphasize that we cannot divide by 0. Surely

the reader has seen alleged proofs that, for example, 1 = 2, where the only error is

a cleverly disguised division by 0.

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Exercise 1.1. Find an invalid argument that 1 = 2 in which the only invalid

step is a division by 0. Try to obscure the division by 0.

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Exercise 1.2. Show that there is a one-to-one correspondence between the set

Q of rational numbers and the following set L of lines. The set L consists of all

lines through the origin, except the vertical line x = 0, that pass through a nonzero

point (a,b) where a and b are integers. (This problem sounds sophisticated, but

one word gives the solution!)

The rational number system forms a ﬁeld. A ﬁeld consists of objects which can

be added and multiplied; these operations satisfy the laws we expect. We begin

our development by giving the precise deﬁnition of a ﬁeld.

Deﬁnition 2.1. A ﬁeld F is a mathematical system consisting of a collection of

objects and two operations, addition and multiplication, subject to the following

axioms.

1) For all x,y in F, we have x + y = y + x and x * y = y * x (the commutative

laws for addition and multiplication).