1.5 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.
dealing with complex numbers, such a point is written as a
and multiplication and
addition are defined in the most obvious way subject to the convention that i2
Every non zero complex number, a + ib, with a2 + b2≠0, has a unique multiplicative
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 listed on Page 13.
Note that if x + iy is a complex number, it can be written as
is a point on the unit circle and so there exists a unique
that this ordered pair equals
it follows that the complex
number can be written in the form
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
What it does is reflect a given complex number across the x axis. Algebraically, the following
formula is easy to obtain.
Definition 1.5.2 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 1.5.3 : Let z = a + ib and w = c + id. Then
Thus the distance between the point in the plane determined by the ordered pair,
the ordered pair
where z and w are as just described.
For example, consider the distance between
From the distance formula this
On the other hand, letting z
= 2 + i
5 and w
= 1 + i
= 1 −i
3 and so
= 10 so
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
Now note that
is a point on the unit circle. Therefore, there exists a unique angle, θ ∈ [0,2π) such
The polar form of the complex number is then
where θ is this angle just described and r =
A fundamental identity is the formula of De Moivre which follows.
Theorem 1.5.4 Let r > 0 be given. Then if n is a positive integer,
Proof: It is clear the formula holds if n = 1. Suppose it is true for n.
which by induction equals
by the formulas for the cosine and sine of the sum of two angles. ■
Corollary 1.5.5 Let z be a non zero complex number. Then there are always exactly k
kth roots of z in ℂ.
Proof: Let z = x + iy and let z =
be the polar form of the complex
number. By De Moivre’s theorem, a complex number,
is a kth root of z if and only if
This requires rk =
and also both cos
can only happen if
for l an integer. Thus
and so the kth roots of z are of the form
Since the cosine and sine are periodic of period 2π, there are exactly k distinct numbers which
result from this formula. ■
Example 1.5.6 Find the three cube roots of i.
First note that i = 1
Using the formula in the proof of the above
corollary, the cube roots of i
where l = 0,1,2. Therefore, the roots are
Thus the cube roots of i are
The ability to find kth roots can also be used to factor some polynomials.
Example 1.5.7 Factor the polynomial x3 − 27.
First find the cube roots of 27. By the above procedure using De Moivre’s theorem, these cube
roots are 3,3
+ 27 =
+ 9 and so
where the quadratic polynomial, x2 + 3x + 9 cannot be factored without using complex
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.