www.mathcentre.ac.uk/.../alg-ext-1-new.mp4

My special subject for tonight
is the number 8 and you may ask

what has that got to do with the
pound in your pocket?

Imagine you have found a

wonderful bank. With.

100% interest. And you put your
pound in the bank and come back

12 months later. You have two

pounds. But suppose you do a

special deal. You come back
after six months.

And you get 50% interest for six
months. You take it out, put it

back in, and six months later
you have another 50% interest on

your £1.50. And that gives you
£2.25. You've improved the deal.

Now suppose that you compound
the interest like that every

three months. You'll find you
get even more. So why don't we?

Compound the interest every 2nd.

Or every microsecond one would

imagine. The principle total in
your account would get higher

and higher. But surprisingly.

Your pound at the end of the
year with continuous interest,

compounding all the time, is
only two pounds.

71.8 pounds and that number
is the magical number E.

A lot of our well known numbers
go back to Greek times.

The number one, yes, the Greek
certainly had the number one. I

went to hot on note, but they
were quite interested in pie.

Because they like geometry.

They certainly weren't familiar
with the number I.

The mysterious square root of

minus one. And they
didn't know about E.

A didn't come into being
until something like the

beginning of the 1600s.

At that time, the
mathematician called Napier

was experimenting with
logarithms.

And you may have heard that the
number is in fact the base of

what we now call the natural
logarithms. Napier didn't know

he was playing with his number.

And the number of Mathematicians
who dealt with logarithms after

him, it didn't know that either.
And they certainly didn't call

this number E. However.

Newton, who was in lots of
things like this.

Had an infinite series for a
around the middle of the 1600s.

He knew that if you added one to

one. To one over 2 factorial and
add wanna pon 3 factorial and

add one appan 4 factorial.

And you continue like that. You
have a serious which converges.

To a real number and that real
number is E.

Much later are
mathematician called Euler,

a German mathematician.

In early days of the next

century. I live by the way,
gave us much of our

mathematical notation.

He decided to study this number
more closely and he was the

first one to call it E, not
because his name began with he.

But because he was.

Using vows he already had a

in operation. So the next
foul was E.

He has some amazing properties.

It defines a function called the
exponential function, which is

another good reason for it being
a A for exponential.

18X

It's a function for every real
value of X. You've got a number,

and that's related to this

infinite series. One plus X.

Plus X squared up on 2.

Plus X cubed appan sex.

Plus X to the 4th Appan 24 and
so on for every value of X using

code that infinite series

converges. And give you a value
of the function E to the X.

Now.

If you look at other well known

functions like. Cossacks of

Cynex. They also have infinite
series, and if you write E

to the IX.

I being the square root of minus
one, and you write out that

infinite series with IX in place

of X. And gather the real terms
and the imaginary terms.

Surprise, surprise, you find the

ether Dix. Is costs X
plus I sine X.

Now if you put X equals Π.

The ratio of the circumference
to the diameter of a circle you

get E to the I π.

Is caused by, which is minus one
plus I sine π which is 0.

So. E to the I Pi is equal
to minus one, and if you write

that equation in another way.

Each of the I π plus one equals

0. And you bring together
all those five magical

mathematical numbers 01.

Pie I and a well brought
together in one beautiful

formula.