1.5 The Complex Numbers And Fields
Recall that a real number is a point on the real number line. 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
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
Multiplication and addition are defined in the most obvious way subject to the convention that i2 = −1.
Every non zero complex number a
has a unique multiplicative inverse.
You should prove the following theorem.
Theorem 1.5.1 The complex numbers with multiplication and addition defined as above form a
field satisfying all the field axioms. These are the following list of properties. In this list, F is the
symbol for a field.
- x + y = y + x, (commutative law for addition)
- There exists 0 such that x + 0 = x for all x, (additive identity).
- For each x ∈ F, there exists −x ∈ F such that x + = 0
, (existence of additive inverse).
z = x +
,(associative law for addition).
- xy = yx,(commutative law for multiplication). You could write this as x × y = y × x.
z = x
,(associative law for multiplication).
- There exists 1 such that 1x = x for all x,(multiplicative identity).
- For each x≠0, there exists x−1 such that xx−1 = 1.(existence of multiplicative inverse).
- x =
xy + xz.(distributive law).
When you have a field F some things follow right away from the above axioms.
Theorem 1.5.2 Let F be a field. This means it satisfies the axioms of the above theorem. Then the
- 0 is unique.
- −x is unique
- 0x = 0
x = −x
- x−1 is unique
Proof: Consider the first claim. Suppose
is another additive identity. Then
and so sure enough, there is only one such additive identity. Consider uniqueness of −x next. Suppose y is
also an additive inverse. Then
so the additive inverse is unique also.
Now add −0x to both sides to conclude that 0 = 0x. Next
and by uniqueness of −x, this implies
as claimed. Finally, if x≠
0 and y
is a multiplicative
so y = x−1. ■
Something which satisfies these axioms is called a field. Linear algebra is all about fields, although in
this book, the field of most interest will be the field of complex numbers or the field of real numbers. You
have seen in earlier courses that the real numbers also satisfy the above axioms. The field of complex
numbers is denoted as ℂ and the field of real numbers is denoted as ℝ. 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 1.5.3 Define the absolute value of a complex number as follows.
Thus, denoting by z the complex number z = a + ib,
Also from the definition, if z = x + iy and w = u + iv are two complex numbers, then
You should verify this.
Notation 1.5.4 Recall the following notation.
There is also a notation which is used to denote a product.
The triangle inequality holds for the absolute value for complex numbers just as it does for the ordinary
Proposition 1.5.5 Let z,w be complex numbers. Then the triangle inequality holds.
Proof: Let z = x + iy and w = u + iv. First note that
so this shows the first version of the triangle inequality. To get the second,
and so by the first form of the inequality
and so both
are no larger than
and this proves the second version because
is one of
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 1.5.6 : 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 this distance
On the other hand, letting z
= 2 + i
5 and w
= 1 + i
8, z − w
= 1 − i
= 10 so
the same thing obtained with the