Just as a real number should be considered as a point on the line, a complex number is
considered a point in the plane which can be identified in the usual way using the Cartesian
coordinates of the point. Thus
(a,b)
identifies a point whose x coordinate is a and whose y
coordinate is b. In dealing with complex numbers, such a point is written as a + ib. For
example, in the following picture, I have graphed the point 3 + 2i. You see it corresponds to
the point in the plane whose coordinates are
(3,2)
.
PICT
Multiplication and addition are defined in the most obvious way subject to the convention
that i^{2} = −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 a^{2} + b^{2}≠0, has a unique multiplicative
inverse.
--1--= a−2-ib2 =-2-a-2 − i-2b--2.
a+ ib a + b a + b a + b
You should prove the following theorem.
Theorem 2.13.1The complex numbers with multiplication and additiondefined as above form a field satisfying all the field axioms listed on Page 10.
The field of complex numbers is denoted as ℂ. An important construction regarding
complex numbers is the complex conjugate denoted by a horizontal line above the number. It
is defined as follows.
a+-ib ≡ a− ib.
What it does is reflect a given complex number across the x axis. Algebraically, the following
formula is easy to obtain.
-----
(a+ ib)(a + ib) = a2 + b2.
Definition 2.13.2Define the absolute value of a complex number as follows.
∘ ------
|a +ib| ≡ a2 + b2.
Thus, denoting by z the complex number, z = a + ib,
|z| = (zz)1∕2.
With this definition, it is important to note the following. Be sure to verify this. It is not
too hard but you need to do it.
Remark 2.13.3: Let z = a+ib and w = c+id. Then
|z − w |
=
∘ -----2--------2-
(a − c) + (b − d)
.Thus the distance between the point in the plane determined by the ordered pair,
(a,b)
and the ordered pair
(c,d)
equals
|z − w|
where z and w are as just described.
For example, consider the distance between
(2,5)
and
(1,8)
. From the distance formula
which you should have seen in either algebra of calculus, this distance is defined
as
∘ ---------------- √--
(2− 1)2 + (5 − 8)2 = 10.
On the other hand, letting z = 2 + i5 and w = 1 + i8, z − w = 1 − i3 and so
-----
(z − w)(z − w ) = (1− i3)(1 +i3) = 10
so
|z − w|
=
√ --
10
, the same thing obtained with the distance formula.
Notation 2.13.4From now on I will sometimes use the symbol F to denote either ℂor ℝ, rather than fussing over which one is meant because it often does not make anydifference.
The triangle inequality holds for the complex numbers just like it does for the real
numbers.