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

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

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

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about real numbers.

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

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

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

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

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

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

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

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

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1/2 would lie somewhere in here.

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Decimal numbers, like 3.5 would
be somewhere in there.

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

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

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

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

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Suppose we take the number 3 and
we square it.

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

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3 the answer is 9.

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

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

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

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

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a negative number yields a
positive result, the answer

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minus three times minus three is
plus 9 positive 9.

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

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

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is never negative.

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

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

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get a negative answer by
squaring a real number.

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Now, over the years,
mathematicians found this

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shortcoming a problem and they
decided that will try and work

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

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

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

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answer is minus one.

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

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

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

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

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clearly this number I can't be a

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

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

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

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

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do some calculations with this
imaginary number, I lots of

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problems in engineering and
physics and applied mathematics

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can be solved using this
imaginary number I.

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

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

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an expression for the square
root of minus nine square root

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of a negative number.

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

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write it as plus 9 multiplied by

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

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split it as the square root of

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9. Multiplied by the square root
of minus one.

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

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

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

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

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minus nine is 3 times I.

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

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

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

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

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

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

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root of minus one.

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We know is I.

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

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

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imaginary number I allows us to
formally write down an

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expression for the square root
of any negative number.

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

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

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you a couple of examples.
Supposing were asked to

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calculate I cubed.

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

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

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becomes minus 1 multiplied by I.

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

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

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

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

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multiplied by I squared.

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

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

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

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can start to simplify any
expression that involves I and

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

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in lots more calculations
in the following videos.