www.mathcentre.ac.uk/.../01-Motivating-StudyF61Mb.mp4

In this video I'm going to try
to motivate the study of complex

numbers by explaining how we can
find the square root of a

negative number. Before we do
that, let's record some facts

about real numbers.

On the diagram here, I've drawn
what we call a real number line.

And every real number has its
place on this line. Now I've

marked the whole real numbers
from minus nine up to plus nine,

so all the positive numbers to
the right hand side. The

negative numbers are to the left
hand side. Every real number has

its place on this line, so the
integers, positive integers,

negative inches, integers are
here. We could also put the

fractions on as well. So for
example the real number minus

1/2 would lie somewhere in here.

Decimal numbers, like 3.5 would
be somewhere in there.

And even numbers like pie with
some, some place someone here as

well. So as pie is going to be
in there somewhere. So the point

is that all real numbers have
their place on this real number

line. Let's look at what happens
when we square any real number.

Suppose we take the number 3 and
we square it.

When we square 3, remember we're
multiplying it by itself, so 3 *

3 the answer is 9.

What about if we take the number
minus three and square that?

Again, when we square it, we
multiplying the number by

itself, so it's minus 3
multiplied by minus three.

And here, if you recall that
multiplying a negative number by

a negative number yields a
positive result, the answer

minus three times minus three is
plus 9 positive 9.

Now the point I'm trying to make
is, whenever you Square a

number, be it a positive number
or a negative number. The answer

is never negative.

In fact, unless the answer,
unless the number we started

with zero, the answer is always
going to be positive. You can't

get a negative answer by
squaring a real number.

Now, over the years,
mathematicians found this

shortcoming a problem and they
decided that will try and work

around that by introducing a new
number, and we're going to give

this new number at the symbol I,
and I'm going to be a special

number that has this property
that when you square it, the

answer is minus one.

So I is a special number such
that the square of I I squared

is minus one. Now that clearly
is a very special number because

I've just explained that when
you square any positive or

negative real number, the answer
can never be negative. So

clearly this number I can't be a

real number. What it is, it's an
imaginary number. We say I is an

imaginary number. Now that might
seem rather strange. When you

first meet, it's starting to
deal with imaginary numbers, but

it turns out that when we
progress a little further and we

do some calculations with this
imaginary number, I lots of

problems in engineering and
physics and applied mathematics

can be solved using this
imaginary number I.

Now using I, we can formally
write down the square root of

any negative number at all. So
supposing we want to write down

an expression for the square
root of minus nine square root

of a negative number.

What we do is we write the minus
nine in the following way. We

write it as plus 9 multiplied by

minus one. And then we split
this product as follows. We

split it as the square root of

9. Multiplied by the square root
of minus one.

So the square root of 9. We do
know the square root of 9 is 3

and the square root of minus one
is going to be I because I

squared is minus one. So I is
the square root of minus one.

The words now we can formally
write down the square root of

minus nine is 3 times I.

Let's give you another example.
Suppose we wanted the square

root. Of minus Seven, we do it
in exactly the same way we split

minus 7 into 7 times minus one,
and we write it as the square

root of 7 multiplied by the
square root of minus one.

Now the square root of 7. We
can't simplify, will just leave

that in this so called surd form
square root of 7 and the square

root of minus one.

We know is I.

So the square root of minus
Seven. We can write as the

square root of plus Seven times.
I, so the introduction of this

imaginary number I allows us to
formally write down an

expression for the square root
of any negative number.

Now using this imaginary number
I, we can do various algebraic

calculations, just as we would
with normal algebra. Let me show

you a couple of examples.
Supposing were asked to

calculate I cubed.

Or I cubed we can write as I
squared multiplied by I.

We already know that I squared
is minus one, so I squared

becomes minus 1 multiplied by I.

Which is just minus one times I
or minus I so we can simplify

the expression I cubed in this
way just to get the answer.

Minus I let's look at another
one, supposing we have an

expression I to the four, we
could write that as I squared

multiplied by I squared.

And in each case I squared is
minus one. So here we have minus

1 multiplied by minus one and
minus one times minus one is

plus one. So I to the four is
plus one and in the same way we

can start to simplify any
expression that involves I and

powers of I. You'll see
this imaginary number I

in lots more calculations
in the following videos.