0:00:01.550,0:00:06.139
My special subject for tonight[br]is the number 8 and you may ask

0:00:06.139,0:00:10.375
what has that got to do with the[br]pound in your pocket?

0:00:11.050,0:00:13.160
Imagine you have found a

0:00:13.160,0:00:15.480
wonderful bank. With.

0:00:16.110,0:00:21.703
100% interest. And you put your[br]pound in the bank and come back

0:00:21.703,0:00:24.566
12 months later. You have two

0:00:24.566,0:00:27.850
pounds. But suppose you do a

0:00:27.850,0:00:31.390
special deal. You come back[br]after six months.

0:00:31.990,0:00:37.422
And you get 50% interest for six[br]months. You take it out, put it

0:00:37.422,0:00:42.078
back in, and six months later[br]you have another 50% interest on

0:00:42.078,0:00:47.460
your £1.50. And that gives you[br]£2.25. You've improved the deal.

0:00:48.080,0:00:53.330
Now suppose that you compound[br]the interest like that every

0:00:53.330,0:00:59.630
three months. You'll find you[br]get even more. So why don't we?

0:01:00.270,0:01:02.260
Compound the interest every 2nd.

0:01:03.220,0:01:06.085
Or every microsecond one would

0:01:06.085,0:01:10.600
imagine. The principle total in[br]your account would get higher

0:01:10.600,0:01:14.000
and higher. But surprisingly.

0:01:14.650,0:01:18.995
Your pound at the end of the[br]year with continuous interest,

0:01:18.995,0:01:22.155
compounding all the time, is[br]only two pounds.

0:01:22.680,0:01:29.510
71.8 pounds and that number[br]is the magical number E.

0:01:30.740,0:01:34.940
A lot of our well known numbers[br]go back to Greek times.

0:01:35.870,0:01:39.950
The number one, yes, the Greek[br]certainly had the number one. I

0:01:39.950,0:01:44.030
went to hot on note, but they[br]were quite interested in pie.

0:01:44.590,0:01:47.390
Because they like geometry.

0:01:48.270,0:01:52.894
They certainly weren't familiar[br]with the number I.

0:01:53.640,0:01:55.295
The mysterious square root of

0:01:55.295,0:01:58.526
minus one. And they[br]didn't know about E.

0:01:59.900,0:02:04.535
A didn't come into being[br]until something like the

0:02:04.535,0:02:06.595
beginning of the 1600s.

0:02:07.930,0:02:12.270
At that time, the[br]mathematician called Napier

0:02:12.270,0:02:14.750
was experimenting with[br]logarithms.

0:02:15.790,0:02:20.942
And you may have heard that the[br]number is in fact the base of

0:02:20.942,0:02:24.622
what we now call the natural[br]logarithms. Napier didn't know

0:02:24.622,0:02:26.830
he was playing with his number.

0:02:28.750,0:02:32.010
And the number of Mathematicians[br]who dealt with logarithms after

0:02:32.010,0:02:35.596
him, it didn't know that either.[br]And they certainly didn't call

0:02:35.596,0:02:37.910
this number E. However.

0:02:38.560,0:02:42.232
Newton, who was in lots of[br]things like this.

0:02:42.920,0:02:48.620
Had an infinite series for a[br]around the middle of the 1600s.

0:02:48.620,0:02:52.420
He knew that if you added one to

0:02:52.420,0:02:58.626
one. To one over 2 factorial and[br]add wanna pon 3 factorial and

0:02:58.626,0:03:00.741
add one appan 4 factorial.

0:03:01.410,0:03:06.195
And you continue like that. You[br]have a serious which converges.

0:03:06.790,0:03:10.450
To a real number and that real[br]number is E.

0:03:11.370,0:03:15.654
Much later are[br]mathematician called Euler,

0:03:15.654,0:03:17.796
a German mathematician.

0:03:18.860,0:03:21.026
In early days of the next

0:03:21.026,0:03:25.400
century. I live by the way,[br]gave us much of our

0:03:25.400,0:03:26.002
mathematical notation.

0:03:27.190,0:03:31.786
He decided to study this number[br]more closely and he was the

0:03:31.786,0:03:36.765
first one to call it E, not[br]because his name began with he.

0:03:37.540,0:03:38.888
But because he was.

0:03:39.770,0:03:43.676
Using vows he already had a

0:03:43.676,0:03:47.610
in operation. So the next[br]foul was E.

0:03:48.760,0:03:52.070
He has some amazing properties.

0:03:52.750,0:03:56.750
It defines a function called the[br]exponential function, which is

0:03:56.750,0:04:00.750
another good reason for it being[br]a A for exponential.

0:04:01.450,0:04:04.220
18X

0:04:05.240,0:04:09.322
It's a function for every real[br]value of X. You've got a number,

0:04:09.322,0:04:10.892
and that's related to this

0:04:10.892,0:04:13.820
infinite series. One plus X.

0:04:14.490,0:04:16.788
Plus X squared up on 2.

0:04:17.620,0:04:19.770
Plus X cubed appan sex.

0:04:20.380,0:04:26.908
Plus X to the 4th Appan 24 and[br]so on for every value of X using

0:04:26.908,0:04:28.540
code that infinite series

0:04:28.540,0:04:35.062
converges. And give you a value[br]of the function E to the X.

0:04:36.360,0:04:36.860
Now.

0:04:37.930,0:04:40.387
If you look at other well known

0:04:40.387,0:04:43.592
functions like. Cossacks of

0:04:43.592,0:04:50.990
Cynex. They also have infinite[br]series, and if you write E

0:04:50.990,0:04:52.745
to the IX.

0:04:53.280,0:04:57.765
I being the square root of minus[br]one, and you write out that

0:04:57.765,0:04:59.835
infinite series with IX in place

0:04:59.835,0:05:05.590
of X. And gather the real terms[br]and the imaginary terms.

0:05:05.590,0:05:07.890
Surprise, surprise, you find the

0:05:07.890,0:05:13.719
ether Dix. Is costs X[br]plus I sine X.

0:05:15.480,0:05:17.454
Now if you put X equals Π.

0:05:18.430,0:05:22.558
The ratio of the circumference[br]to the diameter of a circle you

0:05:22.558,0:05:24.622
get E to the I π.

0:05:25.250,0:05:30.290
Is caused by, which is minus one[br]plus I sine π which is 0.

0:05:31.030,0:05:37.492
So. E to the I Pi is equal[br]to minus one, and if you write

0:05:37.492,0:05:39.332
that equation in another way.

0:05:40.130,0:05:42.546
Each of the I π plus one equals

0:05:42.546,0:05:48.862
0. And you bring together[br]all those five magical

0:05:48.862,0:05:50.794
mathematical numbers 01.

0:05:51.350,0:05:57.170
Pie I and a well brought[br]together in one beautiful

0:05:57.170,0:05:57.752
formula.