WEBVTT 00:00:01.550 --> 00:00:06.139 My special subject for tonight is the number 8 and you may ask 00:00:06.139 --> 00:00:10.375 what has that got to do with the pound in your pocket? 00:00:11.050 --> 00:00:13.160 Imagine you have found a 00:00:13.160 --> 00:00:15.480 wonderful bank. With. 00:00:16.110 --> 00:00:21.703 100% interest. And you put your pound in the bank and come back 00:00:21.703 --> 00:00:24.566 12 months later. You have two 00:00:24.566 --> 00:00:27.850 pounds. But suppose you do a 00:00:27.850 --> 00:00:31.390 special deal. You come back after six months. 00:00:31.990 --> 00:00:37.422 And you get 50% interest for six months. You take it out, put it 00:00:37.422 --> 00:00:42.078 back in, and six months later you have another 50% interest on 00:00:42.078 --> 00:00:47.460 your £1.50. And that gives you £2.25. You've improved the deal. 00:00:48.080 --> 00:00:53.330 Now suppose that you compound the interest like that every 00:00:53.330 --> 00:00:59.630 three months. You'll find you get even more. So why don't we? 00:01:00.270 --> 00:01:02.260 Compound the interest every 2nd. 00:01:03.220 --> 00:01:06.085 Or every microsecond one would 00:01:06.085 --> 00:01:10.600 imagine. The principle total in your account would get higher 00:01:10.600 --> 00:01:14.000 and higher. But surprisingly. 00:01:14.650 --> 00:01:18.995 Your pound at the end of the year with continuous interest, 00:01:18.995 --> 00:01:22.155 compounding all the time, is only two pounds. 00:01:22.680 --> 00:01:29.510 71.8 pounds and that number is the magical number E. 00:01:30.740 --> 00:01:34.940 A lot of our well known numbers go back to Greek times. 00:01:35.870 --> 00:01:39.950 The number one, yes, the Greek certainly had the number one. I 00:01:39.950 --> 00:01:44.030 went to hot on note, but they were quite interested in pie. 00:01:44.590 --> 00:01:47.390 Because they like geometry. 00:01:48.270 --> 00:01:52.894 They certainly weren't familiar with the number I. 00:01:53.640 --> 00:01:55.295 The mysterious square root of 00:01:55.295 --> 00:01:58.526 minus one. And they didn't know about E. 00:01:59.900 --> 00:02:04.535 A didn't come into being until something like the 00:02:04.535 --> 00:02:06.595 beginning of the 1600s. 00:02:07.930 --> 00:02:12.270 At that time, the mathematician called Napier 00:02:12.270 --> 00:02:14.750 was experimenting with logarithms. 00:02:15.790 --> 00:02:20.942 And you may have heard that the number is in fact the base of 00:02:20.942 --> 00:02:24.622 what we now call the natural logarithms. Napier didn't know 00:02:24.622 --> 00:02:26.830 he was playing with his number. 00:02:28.750 --> 00:02:32.010 And the number of Mathematicians who dealt with logarithms after 00:02:32.010 --> 00:02:35.596 him, it didn't know that either. And they certainly didn't call 00:02:35.596 --> 00:02:37.910 this number E. However. 00:02:38.560 --> 00:02:42.232 Newton, who was in lots of things like this. 00:02:42.920 --> 00:02:48.620 Had an infinite series for a around the middle of the 1600s. 00:02:48.620 --> 00:02:52.420 He knew that if you added one to 00:02:52.420 --> 00:02:58.626 one. To one over 2 factorial and add wanna pon 3 factorial and 00:02:58.626 --> 00:03:00.741 add one appan 4 factorial. 00:03:01.410 --> 00:03:06.195 And you continue like that. You have a serious which converges. 00:03:06.790 --> 00:03:10.450 To a real number and that real number is E. 00:03:11.370 --> 00:03:15.654 Much later are mathematician called Euler, 00:03:15.654 --> 00:03:17.796 a German mathematician. 00:03:18.860 --> 00:03:21.026 In early days of the next 00:03:21.026 --> 00:03:25.400 century. I live by the way, gave us much of our 00:03:25.400 --> 00:03:26.002 mathematical notation. 00:03:27.190 --> 00:03:31.786 He decided to study this number more closely and he was the 00:03:31.786 --> 00:03:36.765 first one to call it E, not because his name began with he. 00:03:37.540 --> 00:03:38.888 But because he was. 00:03:39.770 --> 00:03:43.676 Using vows he already had a 00:03:43.676 --> 00:03:47.610 in operation. So the next foul was E. 00:03:48.760 --> 00:03:52.070 He has some amazing properties. 00:03:52.750 --> 00:03:56.750 It defines a function called the exponential function, which is 00:03:56.750 --> 00:04:00.750 another good reason for it being a A for exponential. 00:04:01.450 --> 00:04:04.220 18X 00:04:05.240 --> 00:04:09.322 It's a function for every real value of X. You've got a number, 00:04:09.322 --> 00:04:10.892 and that's related to this 00:04:10.892 --> 00:04:13.820 infinite series. One plus X. 00:04:14.490 --> 00:04:16.788 Plus X squared up on 2. 00:04:17.620 --> 00:04:19.770 Plus X cubed appan sex. 00:04:20.380 --> 00:04:26.908 Plus X to the 4th Appan 24 and so on for every value of X using 00:04:26.908 --> 00:04:28.540 code that infinite series 00:04:28.540 --> 00:04:35.062 converges. And give you a value of the function E to the X. 00:04:36.360 --> 00:04:36.860 Now. 00:04:37.930 --> 00:04:40.387 If you look at other well known 00:04:40.387 --> 00:04:43.592 functions like. Cossacks of 00:04:43.592 --> 00:04:50.990 Cynex. They also have infinite series, and if you write E 00:04:50.990 --> 00:04:52.745 to the IX. 00:04:53.280 --> 00:04:57.765 I being the square root of minus one, and you write out that 00:04:57.765 --> 00:04:59.835 infinite series with IX in place 00:04:59.835 --> 00:05:05.590 of X. And gather the real terms and the imaginary terms. 00:05:05.590 --> 00:05:07.890 Surprise, surprise, you find the 00:05:07.890 --> 00:05:13.719 ether Dix. Is costs X plus I sine X. 00:05:15.480 --> 00:05:17.454 Now if you put X equals Π. 00:05:18.430 --> 00:05:22.558 The ratio of the circumference to the diameter of a circle you 00:05:22.558 --> 00:05:24.622 get E to the I π. 00:05:25.250 --> 00:05:30.290 Is caused by, which is minus one plus I sine π which is 0. 00:05:31.030 --> 00:05:37.492 So. E to the I Pi is equal to minus one, and if you write 00:05:37.492 --> 00:05:39.332 that equation in another way. 00:05:40.130 --> 00:05:42.546 Each of the I π plus one equals 00:05:42.546 --> 00:05:48.862 0. And you bring together all those five magical 00:05:48.862 --> 00:05:50.794 mathematical numbers 01. 00:05:51.350 --> 00:05:57.170 Pie I and a well brought together in one beautiful 00:05:57.170 --> 00:05:57.752 formula.