1.12 The Complex Numbers
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
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
and addition are defined in the most obvious way subject to the
convention that i2
Every non zero complex number a
has a unique multiplicative
You should prove the following theorem.
Theorem 1.12.1 The complex numbers with multiplication and addition
defined as above form a field satisfying all the field axioms. These are the following
- x + y = y + x, (commutative law for addition)
- x + 0 = x, (additive identity).
- For each x ∈ ℝ, there exists −x ∈ ℝ such that x + = 0
, (existence of
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).
- 1x = x,(multiplicative identity).
- For each x≠0, there exists x−1 such that xx−1 = 1.(existence of multiplicative
- x =
xy + xz.(distributive law).
Something which satisfies these axioms is called a field. In this book, the field of most
interest will be the field of real numbers. You have seen in earlier courses that the
set of real numbers with the usual operations also satisfies 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
What it does is reflect a given complex number across the x axis. Algebraically, the
following formula is easy to obtain.
Observation 1.12.2 The conjugate of a sum of complex numbers equals the sum of
the complex conjugates and the conjugate of a product of complex numbers equals the
product of the conjugates. To illustrate, consider the claim about the product.
Showing the claim works for a sum is left for you. Of course this means the conclusion
holds for any finite product or finite sum. Indeed, for zk a complex number, the
associative law of multiplication above gives
Now by induction, the first product in the above can be split up into the product of the
conjugates. Similar observations hold for sums.
Definition 1.12.3 Define the absolute value of a complex number as
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 1.12.4 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 absolute value.
Proposition 1.12.5 Let z,w be complex numbers. Then the triangle inequality
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
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 1.12.6 : 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 equals
On the other hand, letting
= 2 + i
5 and w
= 1 + i
8, z − w
= 1 − i
3 and so
the same thing obtained with the distance formula.