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.