1Complex numbers

IA Vectors and Matrices

1 Complex numbers
In
R
, not every polynomial equation has a solution. For example, there does
not exist any
x
such that
x
2
+ 1 = 0, since for any
x
,
x
2
is non-negative, and
x
2
+ 1 can never be 0. To solve this problem, we introduce the “number”
i
that
satisfies
i
2
=
1. Then
i
is a solution to the equation
x
2
+ 1 = 0. Similarly,
i
is also a solution to the equation.
We can add and multiply numbers with
i
. For example, we can obtain
numbers 3 +
i
or 1 + 3
i
. These numbers are known as complex numbers. It turns
out that by adding this single number
i
, every polynomial equation will have a
root. In fact, for an
n
th order polynomial equation, we will later see that there
will always be
n
roots, if we account for multiplicity. We will go into details in
Chapter 5.
Apart from solving equations, complex numbers have a lot of rather important
applications. For example, they are used in electronics to represent alternating
currents, and form an integral part in the formulation of quantum mechanics.