40 I The Expanding Universe of Numbers
whereaandbare real numbers. The set of all complex numbers is customarily
denoted byC. We may define addition and multiplication inCto be matrix addition
and multiplication, sinceCis closed under these operations: if
B=
(
cd
−dc)
,
then
A+B=
(
a+cb+d
−(b+d) a+c)
, AB=
(
ac−bd ad+bc
−(ad+bc) ac−bd)
.
FurthermoreCcontains
0 =
(
00
00
)
, 1 =
(
10
01
)
,
andA∈Cimplies−A∈C.
It follows from the properties of matrix addition and multiplication that addition
and multiplication of complex numbers have the properties(A2)–(A5),(M2)–(M4)
and(AM1)–(AM2), with 0 and 1 as identity elements for addition and multiplication
respectively. The property(M5)also holds, since ifaandbare not both zero, and if
a′=a/(a^2 +b^2 ), b′=−b/(a^2 +b^2 ),then
A−^1 =
(
a′ b′
−b′ a′)
is a multiplicative inverse ofA. ThusCsatisfies the axioms for afield.
The setCalso contains the matrix
i=(
01
− 10
)
,
for whichi^2 =− 1 ,andanyA∈Ccan be represented in the form
A=a1+bi,wherea,b∈R. The multiplesa1,wherea∈R, form a subfield ofCisomorphic to
the real fieldR. By identifying the real numberawith the complex numbera1,we
may regardRitself as contained inC.
Thus we will now stop using matrices and use only the fact thatCis a field con-
tainingRsuch that everyz∈Ccan be represented in the form
z=x+iy,wherex,y∈Randi∈Csatisfiesi^2 =−1. The representation is necessarily unique,
sincei∈/R. We callxandytherealandimaginary partsofzand denote them byRz