Welcome everybody. So my name is John Chapman, Professor of Applied Maths here in Oxford, and it's my pleasure to introduce today's speaker.
I'm going to try not to wander because this Mike is picking me up before I do that little bit of housekeeping.
We're not expecting a fire alarm test. So if there is a fire alarm, it's genuine.
And you should leave in an orderly fashion following the green man either frontier by the stage or through the doors that you came in back.
Okay. So Robyn Wilson is a Emeritus Professor of pure mathematics from the Open University and also a former fellow of Keble College here in Oxford.
His interests lie in graph theory, but also in the history of mathematics.
In fact, he's a former president of the British Society for the History of Mathematics.
He's written a number of books, including popular books on Sudoku.
You might be familiar with the Full Colour Theorem, and his title today is Euler's Pioneering Equation,
and after the lecture, there'll be a chance to buy his book of the same name and have it signed by Robin.
So I'm just hearing that that unfortunately, our speaker has been unavoidably delayed by the snow, but by a stroke of luck.
It turns out that Euler himself is in Oxford today and at extremely short notice as agreed to give today's lecture.
You must be gentle with him. He is over 300 years old. Please welcome Leonhard Euler.
Wow. Good afternoon.
In this talk, I'd like to tell you about my pioneering equation called The Most Beautiful Theorem in Mathematics.
But first, let me introduce myself. Leonard Euler I was born in Switzerland but spent many years in the Imperial Courts of
Petersburg and Berlin and have been called the most prolific mathematician of all time.
Having published over 800 books and papers in over 70 volumes ranging across almost all branches of mathematics and physics.
At the time, these amounted to about one third of all the 18th century publications in these subjects.
The most beautiful theorem in mathematics. Why is it the most beautiful theorem in mathematics?
This description came from a poll run by the Mathematical Intelligencer, an American mathematics magazine.
When my equation tops the list. But such polls aren't restricted to mathematicians.
A similar poll for the greatest equation ever was taken by Physics World.
With my equation appearing in the top two and where head of such equations as Einsteins equals M.C. squared and Newton's laws of motion,
other people have been equally impressed.
Indeed, when only 40 in the future, Nobel Prize winning physicist Richard Feynman called my equation the most dramatic, remarkable formula in math.
While Fields medal winner Sir Michael Otchere, who used to be at this institute,
has described it as the mathematical equivalent of Hamlet's to be or not to be very succinct.
But at the same time, very deep.
And the mathematical populariser Keith Devlin waxed even more eloquent, saying like a Shakespearean sonnet that captures the very essence of love,
or a painting that brings out the beauty of the human form that is far more than just skin deep.
Euler's equation. My equation reaches down into the very depths of existence.
It has even featured in two episodes of The Simpsons, whatever that is.
And it was crucial in a criminal court case when an American physics graduate student was sentenced to
eight years in prison after vandalising 100 luxury sports cars by spray painting slogans onto the.
He was identified after spraying my equation, which had just popped into his head onto a mitsubishi Montero and as he announced his trial,
I have known Euler's equation since I was five. Everyone should know Euler's equation.
So what is this equation of mind that everyone should know?
My equation is important because it combines five of the most important constants in mathematics.
One The basis of our counting system. Zero The number that expresses nothingness.
Pi The basis of circle measurement. E The number linked to exponential growth and I an imaginary number, the square root of minus one.
And it also involves the fundamental mathematical operations of addition, multiplication and taking powers.
So if you take e and raise it to the power I times pi and then add one, we get zero or equivalently each of the API is minus one.
As one participant in the physics world, physics world pole remarked,
what could be more mystical than an imaginary number, interacting with real numbers to produce nothing.
And the numbers have even featured in a nursery rhyme. Leonard Euler had a farm E-I-E-I-O.
Zero. And on that farm he had one pig.
E i e i0. As you'll see, my equation is a special case of a more general result the type published in 1748.
This beautifully relates the exponential function and the trigonometric functions cos x plus codex and signings.
But why should the exponential function, each of which goes shooting off to infinity as x becomes large?
Have anything to do with sign and cosine which oscillate forever between the values one and minus one.
Indeed, there is no real reason why there should be such a relationship.
No real reason, but there is a complex reason.
Introducing the complex number. I lead to such connections and realising this was one of my greatest achievements and my
result has even appeared on a Swiss postage stamp where it appears up the left hand side.
Although my results may seem rather abstract, they're also fundamental importance in physics and engineering.
This is because exponentials of the form, each of the key described things that grow.
If K is positive or decay, if K is negative, well, those are the form each of the I case he described circular motion,
but by my identity each of the I Katie is made up of cos Katie and sine Katie and so can be used to represent things that oscillate.
For example, each of the omega t refers to an alternating electric current with angular frequency omega.
And these imaginary exponentials are much easier to deal with mathematically than science and cosines.
And indeed, for more advanced topics such as quantum mechanics or image processing, many calculations cannot be done without them.
So in this talk, I'm going to introduce the five councils, one at a time before showing you how they combine to give what we've called my equation.
So start with, one, the basis of our accounting system.
It's been said that there are three types of people, those that can count and those that can't.
But how do we count? We use a decimal system using only the ten digits one, two, nine and zero.
But it's also a place value system with a place to give each number determining its value.
For example, the number 5157 means 5105 tens and seven ones.
And here the number five plays two different roles, depending on its position as five thousands and has five tens.
And the advantage of such a place value system is that we can carry out our number calculations column by column.
Another example is the binary system used in computing, which is based on two rather than ten.
It's been said that there are ten types of people, those that can count in binary and those account.
So binary numbers such as one 1 to 1 means one lot of two Q plus one not a two squared
plus no loss of two to the one plus one unit corresponding to our decimal number 13.
In fact, it's as easy as one 1011. So how did our accounting systems arise?
How did early civilisations count? Let's look at some of them.
Around 1800 B.C., the Egyptians who wrote on papyrus used a decimal system.
But it wasn't a place value system because they used different symbols for one, ten, one and so on, repeating them as often as necessary.
So the number below. Reading from right to left is to lotus flowers, six cold ropes, five heel bones and eight rolls or 2658.
And around the same time, the Mesopotamians or Babylonians were imprinting their numbers on clay tablets.
They did use a place value system. But it was based on 60, not on ten.
A method of counting we still use when we measure time. 60 seconds in a minute.
60 minutes in an hour. And see a vertical symbol for one and a horizontal one for ten.
The number one 1237 shown here means one north of 60 squared plus 12, lots of 60 plus 37 units, which add up to our decimal number, 4357.
Moving forward by over a thousand years takes us to classical Greece and Rome.
We're all familiar with Roman numerals, a decimal system with letters representing numbers.
But it's not a place value system because different letters are used for 110, 100 and a thousand, and also for 550 and 500.
And because calculating with these letters isn't easy. They used a counting board or abacus for their everyday calculations.
But the Greek system is seems even more confusing.
It's also a decimal system, but again, it's not a place value system because different Greek letters represent the units from 1 to 9,
then the tens from 10 to 90, the hundreds from 100 to 900.
So a number like 888 would be written as 800 plus 80 plus eight.
Or Omega Pi eater. Meanwhile, in China, they use counting boards for their arithmetic,
placing small bamboo rods into separate compartments four units, tens, hundreds and so on.
And this was a decimal place value system, one of the first.
But here, as you can see, each number comes in two forms, vertical and horizontal, which alternate.
So you can tell them easily apart. So 1713 is a horizontal one, a vertical seven, another horizontal one and a vertical three.
And you notice that for the number 6036, the zero gives us an empty box and the two forms of six are different.
A different method of counting was used for the calendar.
The calendar calculations of the Mayans of Mexico and Central America.
And these survive in a small number of codices drawn on tree bark and then folded.
So here counting was based mainly on 20, combining dots and lines.
As you can see on the left, to give all the numbers from 1 to 19 and for larger numbers,
as you can see in the middle, they piled these numbers on top of each other.
So here you can see 1220s. Plus 13 corresponding to our decimal number 273.
And a rather attractive feature of Mayan counting is that each number also had a pictorial head form like the ones you can see below.
And interestingly, if you look at bottom left, they also had a symbol for zero.
The shell like symbol you can see there. So this leads us to our second number, zero.
In India, around 250 B.C. The edicts of Ashoka, the first Buddhist monarch, were carved in pillars around the kingdom.
Some of these contained early examples of Indian based ten numerals as a decimal
place value system began to emerge using only the digits 1 to 9 and later on zero.
So how did zero arise? We've seen how the Chinese left spaces in their counting boards,
while other civilisations left spaces in the sand to distinguish between numbers like 35 and 305.
But gradually, special symbols began to emerge.
And at the top, you can see a cave in Gwalior in India, where the number 270 ringed in blue is clearly seen on the wall.
And you can see the little circle representing zero.
But there's great excitement last autumn when some birch boxes and lain undiscovered 400 years was found in the Bodleian Library here in Oxford,
and it had hundreds of blobs on it, each representing zero.
You can see one of those blobs in the bottom line there with the arrow pointing to it.
And this representation of zero predated all other known appearances of zero by some 400 years or so.
Quite remarkable. Notice that Zero can play two roles as a placeholder as we've seen,
but also as a number to calculate with positive and negative numbers were already used in that time in money markets for profits and debts.
And around the 600 rules for calculating them and with zero were given by the Indian mathematician Brahma Gupta.
And here are some examples you can see on the right. Adding zero and a negative number gives a negative number.
A negative number taken from zero becomes positive.
And so. And the only meaningless one was the last one relating to division by zero.
This is strictly forbidden. Forbidden.
Because if you take an equation like four times zero, because nine times zero and cancel the zeros, then you get four equals nine, which is nonsense.
And you can do that for any two numbers you like.
So this picture shows how our number systems developed over the centuries with the Brahma brahmi numerals at the top,
leading eventually to the numerals at the very bottom, which we can clearly recognise similar to our own.
Also developing, as you can see on the very right, were the Arabic numerals which are still used in the Middle East.
But it took many centuries for what we now call the Hindu Arabic numerals to become fully established.
And here on the left is a 16th century picture representing arithmetic and contrasting the modern
Elgar ist with his Hindu Hindu Arabic numerals with the old fashion obsessed with his counting board.
Meanwhile, arithmetic books promoting the Hindu Arabic numerals began to appear by Fibonacci in 1202 Luca Pacioli in 1494 and Robert Record in 1543.
And the drawing on the right comes from Pad Share this summer and it shows you how to calculate on your fingers.
Let's now turn to our third number, PI, which arises in two ways as the ratio of the circumference of a circle to its diameter pi is C over.
So C is pi or two. Pi are where R is the radius and this ratio is the same for circles of any size from a pizza to the moon.
But PI is also the ratio of the area of a circle to a square of its radius.
A is a pie is overall squared, so a is Pyle squared and this ratio is also the same for all circles, as proved by Euclid in the third century BCE.
And you might think about why do you get the same number pi coming in the formula for two pi are using the circumference or pi squared for the area.
Why is it the same constant? I'll leave that for you to think about.
We can never actually write down pi. Exactly. It's decimal expansion goes on forever.
But if my six figures that you saw earlier weren't enough for you, here are a few more.
But if you happen to live in the Kohl's plus area of Vienna and have forgotten any of this, don't worry.
You'll find them all at your local metro stop. And if they're not enough for you.
And here are a few more. But the point is we can never write out PI in full.
Although PI has actually been memorised to 100,000 decimal places.
What a way to spend a life and calculated to 20 trillion places.
Even. That's only a beginning. There's still a very, very long way to go.
But you can easily remember the first few digits from these mnemonics.
How I wish I could calculate pi. Count the letters in each word.
You get 3141592. And similarly for the question, may I have a large container of coffee?
So the number of letters in each word spell out the first eight digits.
And from the second sentence. How I need to drink.
Alcoholic, of course. After. After all these lectures involving quantum mechanics, that gives you 14 decimal places.
And below is even one in Greek, which gives us 22 decimal places.
Actually there is a poem which gives you 300 decimal places if you count the number of letters in each word.
But I haven't included that today. So when did people start to measure measure circles?
Several early civilisations needed to estimate the circumference or area,
and although they had no conception of PI as a number, their results yield approximations to it.
The Mesopotamian Clay tablet relates the perimeter of a regular hexagon.
So there's a conference of the surrounding circle, as the sex suggests.
Symbol number zero 5736.
Now, if the radius of the circle is all, then each side of the hexagon is.
Also, Arthur is a collector of triangles, and so this ratio of six R over two pi r or three of a pi is,
as it says, 50 zero, 57, 36, 57 over 60 plus 36 over 3600.
And if you work it out, this gives you pi is three and one eighth or 3.125 in decimals, which is a lower estimate that is within 1% of the true value.
And around the same time, an Egyptian papyrus asked the following question Given a round fields of diameter, nine cat, that's a unit of length.
What is its area? And the answers given in steps take away one ninth of the diameter, which is one leaving eight.
Then multiply eight by eight. And that gives you 64.
See tat of land as the area. So to find the area, they reduced the diameter by one ninth and squared the results.
This method, they probably found it by experience. And in terms of the radius, the area turns out to be 256 over 81 R squared.
So the pi is about 3.160, an upper estimate.
Thus also within 1% of the true value. So 4000 years ago they knew pi to within 1%.
A much more convenient, very convenient, but much less accurate value appeared about a thousand years later in the Old Testament.
If you read in One Kings or two Corinthians, you will learn to work in bronze.
Named Hiram made a molten sea with diameter, ten cubits and circumference, 30 cubits.
And that gives you pi is equal to three. A much better method for finding PI was introduced by the Greeks and would be used for almost 2000 years.
Often credited to Archimedes. It actually dates back to the fifth century B.C.,
when the Greek Sophist Antiphon and Bryson approximated an area by regular polygons and then tried to obtain better
and better estimates by repeatedly doubling the number of sides until the polygons eventually became the circle.
So Antiphon on the top left. He first took a square inside a unit circle and found this area to be two.
He then doubled the number of size to an octagon giving area to Route two or 2.828.
Much better, but still some way away from 3.14.
Bryson's approach was similar, except he also considered polygons outside the circle,
getting upper bounds of four for the square and 3.32 for the Octagon.
200 years later, Archimedes adopted the same area, the same idea,
but he worked with perimeters rather than areas, starting with hexagons inside and outside the circle.
He doubled the number of size to 12 to 24, 48 and 96, eventually obtaining these balance.
You can see here, balance of pi. Pi is a little bit more than three and ten over 71 and a little bit less than three in the seventh.
PI's a little bit less than 22 over seven. And this actually gives pi to two decimal places.
What was happening elsewhere in China around the year 263.
The Huawei also used polygons to approximate pi, starting with hexagons and dodecanese.
He found simple methods for calculating the successive areas and pyramids.
Whenever he doubled the number of sides and four polygons with 192 sides, that's twice 96.
He obtained balance of about 3.14. He then did four more doublings, and that led to polygons of 3072 sides.
They didn't actually draw them, but he did the calculations on them and found that PI is 3.14159.
Even more impressively, around the year five hundreds,
Xu Zhang and his son doubled the number of size three more times to over 24,000 size and obtained pi to six decimal places.
Quite remarkable. And they also improved Archimedes fractional value of 22 over 72355 over 113, which also gives pi to six decimal places.
And this lesser approximation wasn't rediscovered in Europe for another thousand years.
Well, after this, everyone got on the game, got in on the game as the number of sides continued to double with corresponding increase in accuracy,
leading eventually to the remarkable Dutchman Leo Lugo from Colon, who went up to polygons with over 500 billion sides.
You can see the number written out up there and that gives pi to 20 decimal places.
And if you look carefully below his portrait, you can see two numbers there.
Those are his upper and lower estimates for pi to 20 decimal places.
But not content with this. He then used polygons with two to the power 62 size.
That's that big number there. And he found pi to 35 decimal places.
He also this latter value took a pair on his tombstone in Leiden.
For many years, PI was known in Germany as the blue dolphin number.
I'm not going to get a little bit more technical for a few minutes, but if you lose it, don't worry.
Just hang in there and we'll get back to you later on. So this is a new and highly productive method for estimating PI,
which was used extensively in the 18th and 19th centuries and involves the tangents of angles you're familiar from trigonometry.
The ten theta is A over B,
but it also involves the inverse tangent where we turn things backwards and we write this as ten to the minus one, or sometimes as arc ten.
The point is that if ten of theta is A over B, then we say that theta is ten to the minus one over B,
so ten to the minus one simply undoes whatever ten does.
For example, ten of pi by four. That's ten, 45 degrees is one.
So tens of minus one of one is pi over four. Instead of pi by six.
That's 1030 is one of three. So 10th and minus one of one of three is pi over six.
I'm using radian measures here, which some of you will have come across.
Basically PI means 180 degrees. Pi by two is 90 degrees.
And so. Now the thing is that you can actually combine different values of ten to the minus one.
For example, if you add ten to the minus one a half, ten to the minus one, a third, you get pi over four.
And you can see that from the picture on the left with the angles, ten, four minus one and a half and ten from us from the third marks there.
And together they give 45 degrees. But you can also prove it by simple geometry.
And in general, we can combine any two inverse tangents simply by using the formula at the bottom of the frame.
Well, many functions can be written as infinite series.
And some of you will have come across this, for example,
ten to the minus one of X is the infinite series shown at the top X minus the third x cubed, plus a fifth, x the fifth, and so on.
So it's only odd powers of X that appear.
And in the denominators you have one, three, five, seven. The odd numbers.
Now this result was actually known in 15th century India, but it's usually named after the Scotsman,
James Gregory, that you can see here who rediscovered it 300 years later.
But if you now put X equals one into this, you have ten to the minus one of one, which is pi over four.
Is equal to one minus a third plus a fifth, minus seventh, and so on.
And this is surely and this result is was also found in India, but is usually credited to liveness.
And it's surely one of the most amazing results in the whole of mathematics since by just adding and subtracting numbers of the full one over n,
we get a result involving the circle number pi.
Why should PI appear there? Unfortunately, the lively theories that you see here converges so slowly that we cannot use it to find pie in practice.
For example, if you take the first 300 terms of the series, well, that gives you pi to only two decimal places.
If you want five decimal places, you've got to take the first half a million terms.
So not the sort of thing to do in practice, but you can still use Gregory's series above to estimate pie if you substitute values other than one.
Because remember the terms of the minus one half and ten to the minus one, a third add up to pi over four.
So we can substitute x is a is a half an x is a third into that series.
And that gives you the two series below. And the point is that because of the increasing powers of two and three and the denominators,
these converge much faster and they yield good estimates for PI.
Indeed, in 1861, a gentleman from Potsdam used these very series to find PI to 261 decimal places.
All the other series of pi which converge even faster when in 1706 the Englishman John Machin repeatedly used the addition form that
I showed you to show that PI is equal to 16 times ten to the minus one one fifth minus four times ten to the minus one 101 over 239.
Don't worry how those appear. They just emerge out of the formulas.
And then he wrote out the 210 to the minus one series that you can see,
and these series converge rapidly because of the powers of five and 239 in the denominators.
For example, if you just take the first three terms of each series and you already get the value 3.14, just three terms.
It's also useful because five is an easy number to divide by.
And as a result, Machin was able to calculate PI by hand to 100 decimal places.
A great improvement on anything that had come before.
Well, 76 is a good year for PI because as well as mentions result,
a Welsh maths teacher called William Jones wrote a new introduction to the mathematics
in which he introduced for the first time the symbol PI for the circle number.
I'm always surprised that the symbol PI for the circle number doesn't go back a lot further, but it first appeared in 1706.
And here are two extracts from his book. In the Upper One.
In the middle, you can see matching series with the fives and two, three nines and then immediately below it.
Is the first ever appearance in print of PI.
You can see it in the penultimate line. And below is maintenance value for pay in full.
You can see the 100 decimal places and describe is true to above 100 places as
computed by the accurate and ready pen of the truly ingenious Mr. John Machine.
Well, such results can be used to obtain improved values of PI.
As I said, and most notorious of all was one obtained by William Shanks,
who in 1873 use medicines formula that you just seen to calculate PI to an impressive 707 decimal places.
And these were later inscribed in the ceiling frieze in the room of the Palace of Discovery in Paris, where they can still be seen.
Unfortunately for Shanks and for the Palace, it was later found that only the 500, the first 527 of these decimal places are correct.
But they weren't going to redecorate their city. Let's look at a very different way to find pie.
In 1777, the country performed, describing an experiment for estimating it.
Suppose you throw a large number of needles or matchsticks of length l onto a grid of parallel lines at a distance de apart.
It's not difficult to show that this proportion is to over pi times lfg from which you can calculate a value for pi.
Now in this particular case, eleventy, it turns out, before over five.
And exactly five of those ten needles crossed lines and this gives us pi is 3.2, which isn't too bad for just ten needles.
Incidentally, in 1901, an Italian mathematician called Mario Lazzarini carried out such an experiment,
which 11 was five over six, performing 3408 trials and claiming 1808 crossings.
And this gave pies 355113, which, as we saw, gives pi to six decimal places.
He was lucky if just one needle had landed differently, his result would have been correct to only two decimal places.
In 1997, a bizarre event took place in the American state of Indiana,
where the House of Representatives unanimously passed a bill introducing a new mathematical truth.
This attempted to legislate an incorrect value for PI proposed by a local physician who then allowed
the state to use his value freely but would expect royalties from anyone out of state who used it.
A bill for an act, introducing a new mathematical truth and offered as a contribution to education to be used only in the
state of Indiana free of cost by paying any royalties whatsoever on the same according to the proposal.
This physician the ratio of the diameter of circumference is as 5/4 to four, which gives pi is 3.2.
For some strange reason. The bill was then passed on to the House Committee on Swamp Lands, who in turn passed it onto the Committee of Education.
It should been found as a circular area. It's the quadrant of the circumference is the area of an equilateral rectangle.
Is the square on one side. Well, that makes no sense at all.
But even so, the bill proceeded to the Committee on Temperance, of all things, who recommended its passage.
Fortunately,
a mathematician from Purdue University happened to visit the state house on the very day when the bill when the bill was about to be finally ratified,
and he persuaded the senators to stop it just in time. As far as we know, is still with the Committee on Temperance.
The 20th century saw several new discoveries, many of them completely bizarre.
And I'm just going to mention three of them briefly. You're not expected to be able to prove them.
In 1914, the Indian mathematician Ramanujan found some remarkable exact formulas for one over pi, including the one at the top.
This is amazing. It's an infinite series in which strange numbers such as 1103 and 26,398 seem to appear from nowhere.
But such series converge extremely rapidly and form the basis of some of today's fastest algorithms for calculating PI.
Many years later, in 1989,
David and Gregory should know the twins from New York produced a similar result one in the middle with even larger numbers, as you can see.
Remarkably, the third set an example simpler, but it caused much surprise at the time.
If you see how much simpler it is. But the importance is that if you work in base 16 rather than base ten,
then you can calculate your digits of pi from your series one at a time without having to recalculate any or all of the preceding digits first.
Why pay 16 should come in there. I'll leave you to think about.
Well, to end my discussion of pie, let's look at a simple puzzle that dates back to 1702.
It appeared in a book called Nucleus Elements by the Cambridge mathematician William Whiston.
If you haven't seen it before, you may find the answer surprising. So the circumference of the earth is about 25,000 miles or 132 million feet.
Now, assuming the earth to be a perfect sphere, supposing you tie a long piece of string tightly around it.
Don't try this at home so you tie it tightly around and then extend the string by two pi or just about six, just over six feet.
So you've got the whole circumference, you're only adding six feet and you then prop the string equally all around the earth.
How high above the ground is the string? And most people think the resulting gap must be extremely small, perhaps a tiny fraction of an inch.
But the answer is one foot. Which to many people is quite surprising.
In fact, you get the same answer. Whether you tie the string around the earth,
a tennis ball or any other sphere through the sphere has radius of feet than the original string has to offer.
When you extend it by two pi feet, the new circumference is two plus two pi, which is two pi times R plus one.
So the new radius is R plus one. One foot more than before.
Let's now move on to our next number, the exponential number E, which is 2.71828 and so on.
It's another number whose decimal expansion goes on forever. Here we're concerned with how quickly things grow.
We often use the phrase exponential growth to indicate something that grows very fast.
But how fast is this? Incidentally, the letter E for this number first appeared in print in 1736.
In my mechanical hair, you can see an extract from my mechanics, a book on the mathematics of motion and the first appearance of E appears.
And I was the one that named it E, by the way, in the penultimate line, where E denotes the number whose hyperbolic logarithm is one.
So that's the first appearance of the symbol letter E in that context.
Well, to show you what we mean by exponential growth, here's a story about the origins of the game of chess.
A wealthy king was so impressed by this new game that he offered the wise men who invented it any reward he wished to which the wise man replied,
My price is for you to give me one grain of wheat for the first square of the chessboard.
Two grains for the sake of the square. Four for the third square and so on.
Doubling the number of grains on each successive square until the chessboard is filled.
The King was amazed to be offered such a such a tiny reward.
Or so he believed. You can see how the number of grains is growing there until his Treasurer's calculated that the total number of grains of wheat.
This works out at two to the 64 minus one grains enough wheat to form a pile the size of Mt. Everest.
If you place them end to end, they'd reach the nearest star, Alpha Centauri, and back again.
Let's see how quickly other sequences can grow. Look at the top left.
A very simple form of growth is linear growth, as in the counting numbers and is one, two, three, four, five and so on.
And somewhat quicker it's the way the perfect squares and squared one, four, nine, 16 and so on growth and even faster is that of the cubes.
And these are all examples of polynomial growth because they involve paths of N.
But alternatively, we could look at powers of two or any other number as we saw in the chessboard story.
The numbers two to the end, the powers of to start slowly but soon gather pace because each successive number
is twice the previous one and the powers of three grow even more quickly.
These are examples of exponential growth where n is the appears as the exponent.
Let's compare these types of growth. In fact, let's look at the running times of some polynomials and exponentials when PN is 1030 and 50.
For computer performing, say, a million operations, a second for polynomial growth such as and cubed.
That's the third blue line there. Such a computer would take about one eighth of a second when an is 50.
But exponential growth such as to to the end is much greater as we've seen.
When N is 50, the computer would take over 35 years and would be vastly greater than this for three to the end.
So in the long run, exponential growth tends to exceed polynomial growth, often by a huge, huge margin.
Algorithms that run in polynomial time are generally thought of as efficient,
while those that run in exponential time nor may take much longer to implement and are regarded as inefficient.
But returning to E. What exactly is this number and how is it to rise?
In 1683, the Swiss mathematician Jakob Bernoulli was calculating compound interest, given a sum of money to invest at a given rate of interest.
How does it grow? The answer depends on how often we calculate the interest.
How much is earned if we calculate it once a year or twice a year, or every month or every week or every day,
or even continuously as an example, to keep the calculations fairly simple.
Let's see what happens if we invest £1 at the rather unlikely annual rate of 100%.
After one year, our pounds is double to £2. But if you calculate the interest twice a year, that's 50% each time.
Then after six months, our per pound is multiplied by one and a half to give you £1.50,
and after another six months, that amount is multiplied by one and a half.
Again, to give you £2.25, which is more than you had before.
And if you calculate every three months, then there are four periods, and after each one the amount is multiplied by one and one fourth,
first to £1.25, then to about £1.56, then to 195 by the end of the year to £2.44.
That's £1 times one and a quarter to the power four.
So again, the final amount is even larger and as superiors get shorter, what happens to the final amounts increase without bound?
Wouldn't that be good? Or do they settle down to a limiting value?
And the results are shown in this table here to five decimal places and to find them notice of the years divide into end periods.
Then after each period the result is multiplied by one plus one over n.
So the final month at the end of the year is one plus one over end to the power.
And I mean, don't worry about the details, but you'll see the numbers there.
You can see from the tables and increases this these final amounts do approach a limiting value or seem
to the corresponds to when you calculate the interest continuously and this limiting amount 2.71828.
That's just the exponential number e.
Now, the greatest, greatest advances in understanding exponentials were actually were made in the early 18th century.
And after Bernoulli, the main figure in the story actually was myself.
I investigated the number E and the related exponential function each of the X and in 1748 my introduction to the analysis of Infinite,
which I claim is one of the most important mathematics books ever written, brought together many of my results from earlier works.
And here are some of my main findings, which will be familiar to some of you.
We've just seen that is the limit of the number is one plus one over into the power and NZ increases indefinitely
and similarly can show that each of the x is the limit of OnePlus X over end to the end for any number x.
But as Isaac Newton had already discovered, is also the sum of the infinite series.
As shown here, the denominators are the factorial.
One factorial is one. Two factorial is two times one.
Three factor is three times two times one, and so on.
More generally, there's a similar series, as you can see, for each of the X, which converges for all values of X.
And in fact, all of these series converge extremely fast because the factorial in the denominator is increase so rapidly.
For example, just the first ten terms of that series already give you the correct value of E to five decimal places.
And on the right is the curve. The graph of y is each.
The X and the important features are that each point x, the slope of the graph is also each of the x.
That is the slope at any point is the y value. So the curve becomes steeper and steeper as X increases this exponential growth.
So I'd like to end my discussion of exponential growth by returning by by looking at.
In 1798, Thomas Malthus wrote his essay on population,
where he contrasted the steady linear growth of food supplies available with the exponential growth as he saw in population.
And he concluded that however one may cope in the short term, exponential growth would win in the long term and there be severe food shortages.
A conclusion that was indeed borne out in practice.
So how fast does a population grow? This is a little bit of calculus.
If you don't know calculus, it won't last long.
Even if t is the size of the population at time t, and if the population grows at a fixed rate k proportional to its size,
then you get what's called a differential equation.
DNA of a g t is k times in and this can be solved to give nft is the constant times each of the key where the constant is the initial population.
So this is an example of exponential growth, as you can see.
And in the same way we can model exponential decay as for example, in the decay of radium or in the cooling of a cup of tea.
So we come now to the last of our constants, the imaginary square root of minus one, which can be traced back to the 16th century.
When Cardano, one of the Italian mathematicians who first wrote about cubic equations, was trying to solve a number puzzle.
Can you divide ten into two parts whose product is 40?
Well, the two paths he took to be X and ten minus X, they add up to ten.
So X times ten minus X is 40. And solving this quadratic equation, he found the solutions to be five,
plus the square root of -15 and five, minus the square root of -15, which seemed meaningless.
Communism that nevertheless we will operate putting aside the mental tortures involved.
He checked these answers, actually worked at them. You get ten, multiply them, you get 40.
But he complained that so progresses arithmetic, arithmetic, subtlety, the end of which is as refined as it is useless.
We're trying to take the square to a negative number. Doesn't seem to make sense for one time.
One is one minus one times. Minus one is also one.
As a Victorian Augustus two, Morgan remarked, 300 years later,
we have shown the symbol scroll of minus eight to be void of meaning all rather self-contradictory and absurd.
While his contemporary, the astronomer Royal George Arie, commented,
I have not the smallest confidence in any result, which is essentially obtained by the use of imaginary symbols.
However, live news had been more encouraging, claiming that the imaginary numbers are a wonderful flight of God's spirit.
They are almost an amphibian between being and not be.
I must confess that to my shame, even I who use complex numbers so effectively did criticise them at the time.
Well, for many purposes, our ordinary numbers are real. Numbers we call them are enough.
But suppose we now agree to allow this mysterious object called the square root of minus one.
Or I, as I named it, we can then form many more numbers, such as one plus three I or two plus I.
And if we ignore for the moment what these actually mean, we can still carry out some simple calculations.
Adding is straightforward one plus three, four, two plus.
I want us to is three, three. If I size four, I get three plus four I.
Multiplication is also simple. I won't go through the details.
But the point is that whenever you see an I squared you just replace it by minus one.
And we can also represent these numbers geometrically, as was first done by Casper, Vessel of Norway and later by Gauss and by Joel Rebbe.
Algol organises is often called so it's often called the Gaussian plane or Argon diagram,
but neither name is historically correct, so it's much better to call it the complex plane.
Even here in Oxford, they call it the arc and diagram, and they should be ashamed of themselves.
So the point is we represent each number A-plus. B by the point of coordinates.
A, B, so the first picture on the left shows four points, such as one plus two II and three plus I represented in this way.
And if you want to add to complex numbers such as those two numbers, you use what's called the parallelogram law,
you just draw the parallelogram and at the top you'll see the sum is three feels four I.
In the bottom. If you want to multiply by pictorially, you just rotate through 90 degrees.
In fact, as a telephone operator said to me the other day, the number you have dialled is purely imaginary.
Please rotate your throne, your phone through 90 degrees and try again.
And I did. And it worked brilliantly. And doing this again, you're multiplying it done to lots of 90.
So you multiply by squared or minus one. And that just gives you a rotation through 180 degrees.
Well, as we've seen, there was much suspicion in Victorian times about these imaginary numbers.
The Irish mathematician and astronomer William Rowan Hamilton largely ended the
suspicion by defining the complex numbers as pairs ab of ordinary real numbers.
And these combine according to two special rules. You can see that.
So if you take them as pairs and you do that,
then you get these numbers and you can actually see that how to get one zero corresponds to one and zero one corresponds to I.
So since complex numbers can be represented in the plane, a natural question is can you extend this idea to three dimensions with numbers
the form A plus B plus S.J. where I squared and just J squared at both minus one.
Now here addition is still okay. But multiplication isn't because you try to multiply them.
You get a term involving eight times J. And what's that equal to Hamilton?
Try putting it equal to minus one or two. I also j also zero.
Nothing works. Everything fails.
And as he later wrote to his son Archibald, every morning on my coming down to breakfast,
your brother William, Edwin and yourself used to ask me, Well, Papa, can you multiply?
Triplets were true. I was always obliged to reply with a sad shake of the head.
No, I can only add and subtract them.
Well, after struggling with his troublesome triplets for many, many years, Hamilton had his moment of glory on the 16th of October 1843,
while walking with his wife along Dublin's Royal Canal, when an electric current seemed to close and the spark splashed forth.
I pulled out on the spot a pocketbook, that one you can see on the left and made an entry there.
And then. Nor could I resist the impulse to cut with a knife on a stone on Brougham Bridge.
As he passed it, the fundamental formula with the symbols I, j and K namely I squared is j square.
This case great is minus one. What is also equal as it happens to eject k that.
That's the inscription. And there is a postage stamp that commemorates that.
So these are called the quaternions with four terms A plus B, plus S.J. plus the UK.
I squared just in case they're all minus one.
But unlike our ordinary multiplication, which is commutative three times four is four times three, and so on.
This one isn't eight times j, is not j times I but minus j, times i j k is minus k k is minus.
Okay, but then you can multiply any two of these you like as long as you stick to those rules.
Well, for the last few minutes. Now we've got our five constants. Let's return to my equation and my identity.
Recall that my identity connects the exponential function which goes shooting off to
infinity with the functions cosine and sine which oscillate between one and two minus one.
To show this connection. Recall that these functions can all be expanded as infinite series values for all values of X.
You can see those in the middle. What now happens if we allow ourselves to to introduce the complex number I the square root of minus one.
As I did in 1737. So to do so, you take that series there for each of the four, each of the x.
And you replace eggs everywhere by eggs, as I've done.
So this gives you each of the x is one plus I x over one factorial plus i x squared
or all squared of over two factorial plus i x or cubed over three factorial.
And so. But ice credits minus one, which means that IQ does minus ice of the fourth is one.
And if you still stir it all around and simplify it, you will actually collect terms together.
You find that you get the series for cosine x plus I times a series for sign x that is each of the axis x plus I sign x my identity.
And as I said before, one of the most remarkable equations in the whole of mathematics.
In fact, I gave more than one proof of my identity.
Here's the version that appeared in the introduction here in 1748 as part of a different approach in which I made use of so-called infinitesimal,
which I won't go into today. And the first ever appearance of my identity is in the penultimate line.
You can see E to the V times. The scrotum minus one to the IV is called the pulse.
The scrotum minus one times sine V. That's the first appearance of Euler's identity.
My identity as I myself collected at the time.
From these equations, we can just understand how complex exponential exponentials can be expressed by real signs and cosines.
Moreover, from my identity, we can deduce my equation.
We just let X is pi. That's the radian for 180 degrees.
Then you get course. Pi is minus one sign pi zero.
So each of the pi is cause pi minus one plus zero I and this minus one.
So each of the IP plus one equals zero.
And although I certainly made that deduction, it doesn't actually appear explicitly in any of my published works.
I can't remember why I didn't put it in. Can we can we illustrate this pictorially?
Yes. Here's a pictorial representation. In 1959, an English schoolteacher called Mr. Hull showed how to do it.
He took the power series for each of the x and put x is a pi.
This gives one plus I pi plus IP squared over two factorial and so on.
Now let's trace these values. Start off with a one on the right.
You then add IP. That means you go upwards and then you subtract a half pi squared.
And then you go down and and you eventually are tracing a spiral pattern.
And what does it converge to? It converges to the point minus one, the sum of the series.
Which is what you expecting? Well, we've almost finished.
But I just very quickly like to mention two near misses to discovering my equation.
In 1702, Johann Bernoulli was investigating the area of his brother, the Jacobi met earlier.
He was investigating the area of a sector of a circle of radius a that's the shaded bit on the right,
above the axis and below the line from the origin to the point x, y.
And he found it to be that complicated expression given there involving the logarithm of a complex number.
And leaving aside what that means, I later said that if you put X equals one,
then you get a quarter circle and the expression simplifies to a square to four times the log of minus one.
But because it's a quarter circle, it's areas pi squared over four and if you do some counselling you find that log of minus one is ei pi.
Can you take the logarithm of a negative number? Yes. If you allow yourself complex numbers to come in, and that's a quite a remarkable result.
And although I wrote it down explicitly, stupidly, I didn't take exponentials to deduce my equation in the form.
Each of the I pi is each log of minus one is minus one.
And indeed, I often credited Bernoulli with discovering this value for a log of minus one.
And the last very brief near Miss was the work of the Cambridge mathematician Roger Coates.
He worked closely with Isaac Newton on the second edition of Kepler Mathematica.
And he's been credited with introducing radian measures for angles.
Now, the point was in 1712, he was investigating the surface area of the ellipsoid egg shaped region obtained by rotating loops around an axis.
The details are complicated. The point is, he managed to find two different expressions from the area for the area one involving
logarithms and the other involving chicken and trigonometry and both involving an angle phi.
So he first prove that the surface area is a certain multiple of log, of course, plus high sign.
He also proved to be the same multiple of PHI. So say don't worry about how he got there.
The point is that if you equate these results, then you get this result that you can see at the bottom.
If he had now taken exponentials, he'd have found my identity in the form.
Each of the phi is each of the log cos plus I that's cause five plus I signed Phi if you take an exponential C to found that but he didn't.
Another near miss. So to end with.
What should we call the equation? Each of the five plus one equals zero.
We just see how it follows easily from results of Johan Bernoulli and Roger Coates.
But neither of them seems to have done so.
Even I never wrote it down explicitly, although I certainly realise that it follows immediately from my identity.
In fact, we don't know who first published the equation explicitly, though there is an early appearance in a French journal from about 1813.
But almost everybody nowadays attributes the result to me.
So are surely justified in naming it. Euler's equation to honour the achievements of myself has been called a truly great mathematical pioneer.
A word that I say modestly prevents describes me so well, and which appropriately includes.
Among its letters are five constants PI, one, O, zero and E.
Thank you very much.