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 and multiplication and
addition are defined in the most obvious way subject to the convention that i^{2} = −1.
Thus,
Theorem 1.5.1The complexnumbers with multiplication and addition defined as aboveform a field satisfying all the field axioms listed on Page 13.
Note that if x + iy is a complex number, it can be written as
( )
∘ -2---2- ---x----- ----y----
x +iy = x + y ∘x2-+-y2 + i∘x2-+-y2
Now
( )
√xx2+y2,√xy2+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,sinθ)
. Letting r =
∘ -2---2-
x +y
, it follows that the complex
number can be written in the form
x+ iy = r(cosθ + isinθ)
This is called the polar form of the complex number.
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 1.5.2Define the absolutevalue 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 1.5.3: 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)
andthe 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 this
distance equals
∘ ------2--------2
(2− 1) + (5− 8)
=
√ --
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.
Complex numbers, are often written in the so called polar form which is described next.
Suppose x + iy is a complex number. Then
where the quadratic polynomial, x^{2} + 3x + 9 cannot be factored without using complex
numbers.
The real and complex numbers both are fields satisfying the axioms on Page 13 and it is
usually one of these two fields which is used in linear algebra. The numbers are often called
scalars. However, it turns out that all algebraic notions work for any field and there are many
others. For this reason, I will often refer to the field of scalars as F although F will usually be
either the real or complex numbers. If there is any doubt, assume it is the field of complex
numbers which is meant.