2.13 The Complex Numbers
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
identifies a point whose
coordinate is a
and whose y
coordinate is b.
In dealing with complex numbers, such a point is written as a
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
Multiplication and addition are defined in the most obvious way subject to the convention
that i2 = −1. Thus,
Every non zero complex number, a
has a unique multiplicative
You should prove the following theorem.
Theorem 2.13.1 The complex numbers with multiplication and addition
defined 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.
What it does is reflect a given complex number across the x axis. Algebraically, the following
formula is easy to obtain.
Definition 2.13.2 Define the absolute value of a complex number as follows.
Thus, denoting by z the complex number, z = a + ib,
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
Thus the distance between the point in the plane determined by the ordered pair,
and the ordered pair
where z and w are as just described.
For example, consider the distance between
From the distance formula
which you should have seen in either algebra of calculus, this distance is defined
On the other hand, letting z = 2 + i5 and w = 1 + i8, z − w = 1 − i3 and so
the same thing obtained with the distance formula.
Notation 2.13.4 From now on I will sometimes use the symbol F to denote either ℂ
or ℝ, rather than fussing over which one is meant because it often does not make any
The triangle inequality holds for the complex numbers just like it does for the real
Theorem 2.13.5 Let z,w ∈ ℂ. Then
Proof: First note
Here is why: If z
Now look at the right side.
the same thing. Thus the rest of the proof goes just as before with real numbers. Using the
results of Problem 6 on Page 85, the following holds.
as claimed. The other inequality follows as before.
Now do the same argument switching the roles of z and w to conclude
which implies the desired inequality. This proves the theorem.