1.5. THE COMPLEX NUMBERS 13

By this proposition again, (−1) (−1) = − (−1) = 1 and so x2 > 0 as claimed.Note this shows that 1 > 0 because 1 equals 12.Finally, consider 7. First, if x > 0 then if x−1 < 0, it would follow (−1)x−1 > 0 and so

x (−1)x−1 = (−1) 1 = −1 > 0. However, this would require

0 > 1 = 12 > 0

from what was just shown. Therefore, x−1 > 0. Now the assumption implies y+(−1)x > 0and so multiplying by x−1,

yx−1 + (−1)xx−1 = yx−1 + (−1) > 0

Now multiply by y−1, which by the above satisfies y−1 > 0, to obtain

x−1 + (−1) y−1 > 0

and sox−1 > y−1. ■

In an ordered field the symbols ≤ and ≥ have the usual meanings. Thus a ≤ b meansa < b or else a = b, etc.

1.5 The Complex Numbers

Just as a real number should be considered as a point on the line, a complex number isconsidered a point in the plane which can be identified in the usual way using the Cartesiancoordinates of the point. Thus (a, b) identifies a point whose x coordinate is a and whosey coordinate is b. In dealing with complex numbers, such a point is written as a + ib andmultiplication and addition are defined in the most obvious way subject to the conventionthat i2 = −1. Thus,

(a+ ib) + (c+ id) = (a+ c) + i (b+ d)

and(a+ ib) (c+ id) = ac+ iad+ ibc+ i2bd = (ac− bd) + i (bc+ ad) .

Every non zero complex number, a+ib, with a2+b2 ̸= 0, has a unique multiplicative inverse.

1

a+ ib=

a− ib

a2 + b2=

a

a2 + b2− i

b

a2 + b2.

You should prove the following theorem.

Theorem 1.5.1 The complex numbers with multiplication and addition defined as aboveform a field satisfying all the field axioms listed on Page 10.

Note that if x+ iy is a complex number, it can be written as

x+ iy =√x2 + y2

(x√

x2 + y2+ i

y√x2 + y2

)

Now

(x√

x2+y2, y√

x2+y2

)is a point on the unit circle and so there exists a unique θ ∈ [0, 2π)

such that this ordered pair equals (cos θ, sin θ) . Letting r =√x2 + y2, it follows that the

complex number can be written in the form

x+ iy = r (cos θ + i sin θ)