2. Number systems 5

x + y

yy-x

x

Figure 1.1. Diﬀerence of two squares.

We pause to make several remarks about square roots. The ﬁrst remark con-

cerns a notational convention; the discussion will help motivate the notion of or-

dered ﬁeld deﬁned below. The real number system will be deﬁned formally below,

and we prove that positive real numbers have square roots. Suppose t 0. We

write

√will

t to denote the positive x for which

x2

= t. Thus both x and -x are square

roots of t, but the notation

√

t means the positive square root. For the complex

numbers, things will be more subtle. We will prove that each nonzero complex

number z has two square roots, say ±w, but there is no sensible way to prefer one

to the other. We emphasize that the existence of square roots depends on more

than the ﬁeld axioms. Not all positive rational numbers have rational square roots,

and hence it must be proved that each positive real number has a square root. The

proof requires a limiting process. The quadratic formula, proved next, requires that

the expression

b2

-4ac be a square. In an arbitrary ﬁeld, the expression

√

t usually

means any x for which

x2

= t, but the ambiguity of signs can cause confusion. See

Exercise 1.4.

Theorem 2.1. Let F be a ﬁeld. Assume that 2 6 = 0 in F. For a 6 = 0 and arbitrary

b,c we consider the quadratic equation

(3)

ax2

+ bx + c = 0.

Then x solves (3) if and only if

(4) x =

-b ±

√

b2 - 4ac

2a

.

If

b2

- 4ac is not a square in F, then (3) has no solution.