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

• 0:03 - 0:07
In this video I'm going to try
to motivate the study of complex
• 0:07 - 0:11
numbers by explaining how we can
find the square root of a
• 0:11 - 0:16
negative number. Before we do
that, let's record some facts
• 0:16 - 0:17
• 0:18 - 0:23
On the diagram here, I've drawn
what we call a real number line.
• 0:24 - 0:28
And every real number has its
place on this line. Now I've
• 0:28 - 0:32
marked the whole real numbers
from minus nine up to plus nine,
• 0:32 - 0:36
so all the positive numbers to
the right hand side. The
• 0:36 - 0:40
negative numbers are to the left
hand side. Every real number has
• 0:40 - 0:44
its place on this line, so the
integers, positive integers,
• 0:44 - 0:47
negative inches, integers are
here. We could also put the
• 0:47 - 0:51
fractions on as well. So for
example the real number minus
• 0:51 - 0:53
1/2 would lie somewhere in here.
• 0:55 - 0:59
Decimal numbers, like 3.5 would
be somewhere in there.
• 1:00 - 1:03
And even numbers like pie with
some, some place someone here as
• 1:03 - 1:08
well. So as pie is going to be
in there somewhere. So the point
• 1:08 - 1:11
is that all real numbers have
their place on this real number
• 1:11 - 1:16
line. Let's look at what happens
when we square any real number.
• 1:16 - 1:20
Suppose we take the number 3 and
we square it.
• 1:20 - 1:27
When we square 3, remember we're
multiplying it by itself, so 3 *
• 1:27 - 1:29
• 1:31 - 1:36
What about if we take the number
minus three and square that?
• 1:37 - 1:41
Again, when we square it, we
multiplying the number by
• 1:41 - 1:45
itself, so it's minus 3
multiplied by minus three.
• 1:45 - 1:49
And here, if you recall that
multiplying a negative number by
• 1:49 - 1:52
a negative number yields a
• 1:52 - 1:55
minus three times minus three is
plus 9 positive 9.
• 1:56 - 2:00
Now the point I'm trying to make
is, whenever you Square a
• 2:00 - 2:05
number, be it a positive number
or a negative number. The answer
• 2:05 - 2:06
is never negative.
• 2:06 - 2:09
unless the number we started
• 2:09 - 2:13
with zero, the answer is always
going to be positive. You can't
• 2:13 - 2:15
squaring a real number.
• 2:16 - 2:18
Now, over the years,
mathematicians found this
• 2:18 - 2:22
shortcoming a problem and they
decided that will try and work
• 2:22 - 2:26
around that by introducing a new
number, and we're going to give
• 2:26 - 2:30
this new number at the symbol I,
and I'm going to be a special
• 2:30 - 2:34
number that has this property
that when you square it, the
• 2:34 - 2:35
• 2:36 - 2:41
So I is a special number such
that the square of I I squared
• 2:41 - 2:45
is minus one. Now that clearly
is a very special number because
• 2:45 - 2:49
I've just explained that when
you square any positive or
• 2:49 - 2:52
can never be negative. So
• 2:52 - 2:55
clearly this number I can't be a
• 2:55 - 3:00
real number. What it is, it's an
imaginary number. We say I is an
• 3:00 - 3:04
imaginary number. Now that might
seem rather strange. When you
• 3:04 - 3:08
first meet, it's starting to
deal with imaginary numbers, but
• 3:08 - 3:12
it turns out that when we
progress a little further and we
• 3:12 - 3:16
do some calculations with this
imaginary number, I lots of
• 3:16 - 3:19
problems in engineering and
physics and applied mathematics
• 3:19 - 3:22
can be solved using this
imaginary number I.
• 3:23 - 3:27
Now using I, we can formally
write down the square root of
• 3:27 - 3:31
any negative number at all. So
supposing we want to write down
• 3:31 - 3:34
an expression for the square
root of minus nine square root
• 3:34 - 3:35
of a negative number.
• 3:36 - 3:41
What we do is we write the minus
nine in the following way. We
• 3:41 - 3:43
write it as plus 9 multiplied by
• 3:43 - 3:47
minus one. And then we split
this product as follows. We
• 3:47 - 3:50
split it as the square root of
• 3:50 - 3:55
9. Multiplied by the square root
of minus one.
• 3:55 - 4:00
So the square root of 9. We do
know the square root of 9 is 3
• 4:00 - 4:05
and the square root of minus one
is going to be I because I
• 4:05 - 4:09
squared is minus one. So I is
the square root of minus one.
• 4:09 - 4:12
The words now we can formally
write down the square root of
• 4:12 - 4:14
minus nine is 3 times I.
• 4:15 - 4:19
Let's give you another example.
Suppose we wanted the square
• 4:19 - 4:25
root. Of minus Seven, we do it
in exactly the same way we split
• 4:25 - 4:31
minus 7 into 7 times minus one,
and we write it as the square
• 4:31 - 4:36
root of 7 multiplied by the
square root of minus one.
• 4:36 - 4:39
Now the square root of 7. We
can't simplify, will just leave
• 4:39 - 4:43
that in this so called surd form
square root of 7 and the square
• 4:43 - 4:44
root of minus one.
• 4:44 - 4:46
We know is I.
• 4:47 - 4:51
So the square root of minus
Seven. We can write as the
• 4:51 - 4:54
square root of plus Seven times.
I, so the introduction of this
• 4:54 - 4:57
imaginary number I allows us to
formally write down an
• 4:57 - 5:00
expression for the square root
of any negative number.
• 5:01 - 5:05
Now using this imaginary number
I, we can do various algebraic
• 5:05 - 5:09
calculations, just as we would
with normal algebra. Let me show
• 5:09 - 5:13
you a couple of examples.
• 5:13 - 5:14
calculate I cubed.
• 5:14 - 5:19
Or I cubed we can write as I
squared multiplied by I.
• 5:19 - 5:26
We already know that I squared
is minus one, so I squared
• 5:26 - 5:29
becomes minus 1 multiplied by I.
• 5:29 - 5:34
Which is just minus one times I
or minus I so we can simplify
• 5:34 - 5:38
the expression I cubed in this
way just to get the answer.
• 5:38 - 5:41
Minus I let's look at another
one, supposing we have an
• 5:41 - 5:45
expression I to the four, we
could write that as I squared
• 5:45 - 5:47
multiplied by I squared.
• 5:47 - 5:52
And in each case I squared is
minus one. So here we have minus
• 5:52 - 5:55
1 multiplied by minus one and
minus one times minus one is
• 5:55 - 6:00
plus one. So I to the four is
plus one and in the same way we
• 6:00 - 6:03
can start to simplify any
expression that involves I and
• 6:03 - 6:07
powers of I. You'll see
this imaginary number I
• 6:07 - 6:10
in lots more calculations
in the following videos.
Title:
www.mathcentre.ac.uk/.../01-Motivating-StudyF61Mb.mp4
Video Language:
English
 mathcentre edited English subtitles for www.mathcentre.ac.uk/.../01-Motivating-StudyF61Mb.mp4