Complex Numbers 19
same as the vector space addition, and the relations 1( ab) = ( 1a )b = a( 1b)
for all / E F and all a, b E A connect the ring multiplication with the
vector space multiplication. A is of dimension n if the vector space has
that dimension. Obviously, the system C of the complex numbers is also
an algebra of dimension two over the reals.
It is possible to reverse the procedure that we have followed and define
the complex number system as an algebra of dimension two over the reals,
which is also a field. Then it can be shown that the only possible choice
(within an isomorphism) for the product of two complex numbers is the
rule given in Definitions 1.1. Further details concerning this method can
be found in [19].
1.6 Absolute Value of a Complex Number
Definition 1.3 The absolute value or modulus of a complex number z =
x + iy, denoted lzl, is defined by
lzl = Vx^2 + y^2 (1.6-1)
where, as usual, the radical stands for the principal (nonnegative) square
root of x^2 + y^2 •
Example l1+2il = v'12+2^2 =.v'5.
Formula (1.6-1) defines a function f : C onto> ~+. Definition 1.3 extends
to the complex number system the concept of the absolute value of a real
number x. In fact, we have
l(x,O)I: ~ = { ~x
if x;::: 0
if x < 0Theorem 1.3 The following properties hold:
- lzl 2:: 0 and lzl = 0 iff z = 0
- lzl = 1-zl = lzl
- lzl^2 = zz
- 1~1':::; l~I + IYI
- lxl ':::; lzl and IYI :S lzl
- lz1z2I = lz1llz2I
- lzi ±z2I :S lz1I + lz2I
- lz1 ± z2I 2:: llz1l - lz2ll
9. lzi/z2I = lzil/lz2I, z2 -=f:. 0
- lznl = lzln, n an integer
Proofs The proofs of the properties 1, 2, and 3 follow easily from
Definition 1.3, and we omit them.