5. Alternative deﬁnitions of C 17

z = x + iy

w = a + ib

z + w

Figure 1.5. Addition of complex numbers.

to be the set of expressions of the form a + ib for real numbers a,b. We add and

multiply as expected, using the distributive law; then we set

i2

equal to -1. Thus

(15) (x + iy) + (a + ib) = (x + a) + i(y + b),

(16) (x + iy) * (a + ib) = xa + i(ya + xb) +

i2(yb)

= (xa - yb) + i(ya + xb).

Equations (15) and (16) give the same results as (12) and (13). While this new

approach is more elegant, it makes some readers feel uneasy. After all, we are

assuming the existence of an object, namely 0 + i1, whose square is -1. In the

ﬁrst approach we never assume the existence of such a thing, but such a thing does

exist: the square of (0, 1) is (-1, 0), which is the additive inverse of (1, 0).

The reader will be on safe logical ground if he or she regards the above para-

graph as an abbreviation for the previous discussion. In the next section we will

give two additional equivalent ways of deﬁning C.

I

Exercise 1.23. Prove the distributive law for addition and multiplication, as

deﬁned in (12) and (13). Do the same using (15) and (16). Compare.

The next lemma reveals a crucial diﬀerence between R and C.

Lemma 4.1. The complex numbers do not form an ordered ﬁeld.

Proof. Assume that a positive subset P exists. By Lemma 3.1, each nonzero

square is in P. Since 12 = 1 and i2 = -1, both 1 and -1 are squares and hence

must be positive, contradicting 2) of Deﬁnition 3.1.

5. Alternative deﬁnitions of C

In this section we discuss alternative approaches to deﬁning C. We use some

basic ideas from linear and abstract algebra that might be new to many students.

The primary purpose of this section is to assuage readers who ﬁnd the rules (12)

and (13) unappealing but who ﬁnd the rules (15) and (16) dubious, because we