2. Number systems 5
x + y
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
t to denote the positive x for which
= 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
-4ac be a square. In an arbitrary field, the expression

t usually
means any x for which
= 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
+ bx + c = 0.
Then x solves (3) if and only if
(4) x =
-b ±

b2 - 4ac
- 4ac is not a square in F, then (3) has no solution.
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