Complex Numbers 1

1
Complex Numbers 1
Introdution to Complex Numbers.
Consider the following equations
x 2 0
In order to deal with each new type of
equation we have had to expand our number
system.
The solution to this is a Natural Number these
are the numbers
. . . . . . . .


With each new type of equation, we have had to
invent/discover the numbers that solve them.
1
2
3
4
5
6
x 2 0
We now think about how we might exapnd our number
system further still in order to cope with the
following, previously unsolvable, equation
The solution to this equation requires the idea
of a negative number the natural numbers alone
cannot deal with equations like this.
. . . . . . . . . . . . . . . . .


1
2
3
4
5
6
-5
-4
-3
-2
-1
0
x2 1 0
2x 3 0
The solution this time requires the idea of a
fractional (rational) number. The previous set
of numbers cannot cope with this equation.
which the real numbers are unable to deal with.
x2 - 2 0
The solution this time requires the idea of an
irrational number (one that cannot be wriiten
as a fraction). The previous set of numbers
cannot cope with equations like this. The idea of
a real number is required and the gaps in the
rational number line are filled in
2
How are we to deal with the equation
We know that no real number solves it.
x2 1 0
Let us (at least for the moment) take a leap
of faith (!) and assume there is such a thing
as a solution
This equation is solved by the non-real number
called i, thus the number i has the following
property
and that whenever we have previously said that
there is no solution, what we really meant to say
is that there is no solution within the
real number system
i2 -1
then we need to give it a name in order to
think about the implications of its existence!
A complex number is a number of the form a
bi, a is called the real part and b is
called the i-part.
On the previous slide, we saw how we
expanded each number system to cope with each new
type of equation.
The following are all examples of complex numbers
We now create a new number system,
which incorporates the real numbers, but which
is capable of dealing with equations like this
0 1i (i)
6 0i (6)
3 - 9i
0 0i (0)
-32 pi
-54 -76i
3
We will look at the algebra of these numbers in
the next presentation. Here we will look at the
very basic properties of complex numbers
i
x i
The multiples of i are
i,
-1
1
Notice how the powers of i cycle round.
i2 -1,
Now that we have a symbol to represent the square
root of -1, we can take the square root of any
negative number
-i
i3 i x i2 i x -1 -i,
i4 i x i3 i x -i - i2 -(-1) 1,
The basic rule in the game of complex
numbers whenever you see i2, replace it with -1.
i5 i, i6 -1, etc
v(-1 x 9)
v-1 x v9
i x 3 3i
v-9
v-8
v(-1 x 8)
v-1 x v(2 x 4)
i x v2 x v4
2 v2 i