1.2 • THE ALGEBRA OF COMPLEX NUMBERS 11(P3) Addit ive identity: There is a complex number w such that z +w = z for
all complex numbers z. The number w is obviously the ordered pair (0, 0).(P4) Additive inver ses: For any complex number z, there is a unique complex
number 17 (depending on z) with the property that z+'I) = (0, 0). Obviously,if z = (x, y) = x + iy, the number "I will be (- x, -y) = -x -iy = -z.
(PS) Commutative la w for multiplica tion: z1z2 = z2z1.(P6) Associative law for multiplication: z1(z2z3) = (z1Zz)Z3.
(P7) Multiplica t ive identity: There is a complex number ( such that z( = z
for all complex numbers z. As you might expect, (1, 0) is the unique complex
number ( having this property. We ask you to verify this identity in the
exercises for this section.(PS) Multiplicat ive inverses: For any complex number z = (x, y) other
than the number (0, O), there is a complex number (depending on z), which
we denote z-1, having the property that zz-^1 = (1, O) = 1. Based on our
definition for division, it seems reasonable that the number z-i would be
Z - 1 -- .(!&). z --! -z - x+_ i_ iy -- x +y )-iv, -- x•+y• x +i~. - y -- ( x•+ii''~ x -y) · w e
ask you to confirm this result in the exercises for this section.(P9) The distributive law: zi(Zz + zs) = ziz2 + z 1 z3.
None of these properties is difficult to prove. Most of the proofs make use
of corresponding facts in the real number system. To illustrate, we give a proof
of property (Pl).
Proof of t he commutative la w fo r addition: Let z1 = (xi, Yi) and
Zz = (x2, vz) be arbitrary complex numbers. Then,
(xi, Yi)+ (x2, yz)
(Xi+ X2, Yi + y2) (by definition of addition of complex numbers)
(X2 + X1, Y2 + Yi ) (by the commut<itive Jaw for r<!G! numbers)
(x2, Y2) + (x1, Yi) (by definition of addition of complex numbers)= Z2 + Z1.
Actually, you can think of the real number system as a subset of the com-
plex number system. To see why, let's agree that , as any complex number of the
form (t, 0) is on the x-axis, we can identify it with the real number t. With this
correspondence, we can easily verify that our definitions for addition, subtrac-
tion, multiplication, and division of complex numbers are consistent with the
corresponding operations on real numbers. For example, if x i and x2 are real
numbers, then