4 1. From the Real Numbers to the Complex Numbers
Multiplying both sides by 1 and using 2) yields
0 = (1)0 = (1)(1 + (1)) = (1)1 +
(1)2
= 1 +
(1)2.
Thus
(1)2
is an additive inverse to 1; of course 1 also is. By the uniqueness of
additive inverses, we see that
(1)2
= 1. The proof of 4) is similar. Start with
0 = 1 + (1) and multiply by x to get 0 = x + (1)x. Thus (1)x is an additive
inverse of x and the result follows by uniqueness of additive inverses. Finally, to
prove 5), we assume that xy = 0. If x = 0, the conclusion holds. If x 6 = 0, we can
multiply by
1
x
to obtain
y = (
1
x
x)y =
1
x
(xy) =
1
x
0 = 0.
Thus, if x 6 = 0, then y = 0, and the conclusion also holds.
We note a point of language, where mathematics usage may diﬀer with common
usage. For us, the phrase “either x = 0 or y = 0” allows the possibility that both
x = 0 and y = 0.
Example 2.1. A ﬁeld with two elements. Let F2 consist of the two elements
0 and 1. We put 1 + 1 = 0, but otherwise we add and multiply as usual. Then F2
is a ﬁeld.
This example illustrates several interesting things. For example, the object 2
(namely 1 + 1), can be 0 in a ﬁeld. This possibility will prevent the quadratic
formula from holding in a ﬁeld for which 2 = 0. In Theorem 2.1 we will derive the
quadratic formula when it is possible to do so.
First we make a simple observation. We have shown that
(1)2
= 1. Hence,
when 1 6 = 1, it follows that 1 has two square roots, namely ±1. Can an element
of a ﬁeld have more than two square roots? The answer is no.
Lemma 2.1. In a ﬁeld, an element t can have at most two square roots. If x is a
square root of t, then so is x, and there are no other possibilities.
Proof. If
x2
= t, then
(x)2
= t by 3) and 4) of Proposition 2.1. To prove that
there are no other possibilities, we assume that both x and y are square roots of t.
We then have
(2) 0 = t  t =
x2

y2
= (x  y)(x + y).
By 5) of Proposition 2.1, we obtain either x  y = 0 or x + y = 0. Thus y = ±x
and the result follows.
The diﬀerence of two squares law stating that x2  y2 = (x  y)(x + y) is a
gem of elementary mathematics. For example, suppose you are asked to multiply
88 times 92 in your head. You imagine 88*92 = (902)(90+2) = 81004 = 8096
and impress some audiences. One can also view this algebraic identity for positive
integers simply by removing a small square of dots from a large square of dots and
rearranging the dots to form a rectangle. The author once used this kind of method
when doing volunteer teaching of multiplication to third graders. See Figure 1.1
for a geometric interpretation of the identity in terms of area.