1.3 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
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
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 inverse.
You should prove the following theorem.
Theorem 1.3.1 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.
- 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 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).
- 1x = 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).
Something which satisfies these axioms is called a field. 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 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 1.3.2 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
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 1.3.3 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 1.3.4 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 1.3.5 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 1.3.6 : 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