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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.
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I'm going to try not to wander because this Mike is picking me up before I do that little bit of housekeeping.
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We're not expecting a fire alarm test. So if there is a fire alarm, it's genuine.
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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.
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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.
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His interests lie in graph theory, but also in the history of mathematics.
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In fact, he's a former president of the British Society for the History of Mathematics.
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He's written a number of books, including popular books on Sudoku.
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You might be familiar with the Full Colour Theorem, and his title today is Euler's Pioneering Equation,
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and after the lecture, there'll be a chance to buy his book of the same name and have it signed by Robin.
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So I'm just hearing that that unfortunately, our speaker has been unavoidably delayed by the snow, but by a stroke of luck.
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It turns out that Euler himself is in Oxford today and at extremely short notice as agreed to give today's lecture.
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You must be gentle with him. He is over 300 years old. Please welcome Leonhard Euler.
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Wow. Good afternoon.
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In this talk, I'd like to tell you about my pioneering equation called The Most Beautiful Theorem in Mathematics.
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But first, let me introduce myself. Leonard Euler I was born in Switzerland but spent many years in the Imperial Courts of
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Petersburg and Berlin and have been called the most prolific mathematician of all time.
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Having published over 800 books and papers in over 70 volumes ranging across almost all branches of mathematics and physics.
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At the time, these amounted to about one third of all the 18th century publications in these subjects.
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The most beautiful theorem in mathematics. Why is it the most beautiful theorem in mathematics?
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This description came from a poll run by the Mathematical Intelligencer, an American mathematics magazine.
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When my equation tops the list. But such polls aren't restricted to mathematicians.
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A similar poll for the greatest equation ever was taken by Physics World.
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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,
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other people have been equally impressed.
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Indeed, when only 40 in the future, Nobel Prize winning physicist Richard Feynman called my equation the most dramatic, remarkable formula in math.
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While Fields medal winner Sir Michael Otchere, who used to be at this institute,
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has described it as the mathematical equivalent of Hamlet's to be or not to be very succinct.
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But at the same time, very deep.
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And the mathematical populariser Keith Devlin waxed even more eloquent, saying like a Shakespearean sonnet that captures the very essence of love,
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or a painting that brings out the beauty of the human form that is far more than just skin deep.
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Euler's equation. My equation reaches down into the very depths of existence.
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It has even featured in two episodes of The Simpsons, whatever that is.
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And it was crucial in a criminal court case when an American physics graduate student was sentenced to
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eight years in prison after vandalising 100 luxury sports cars by spray painting slogans onto the.
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He was identified after spraying my equation, which had just popped into his head onto a mitsubishi Montero and as he announced his trial,
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I have known Euler's equation since I was five. Everyone should know Euler's equation.
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So what is this equation of mind that everyone should know?
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My equation is important because it combines five of the most important constants in mathematics.
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One The basis of our counting system. Zero The number that expresses nothingness.
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Pi The basis of circle measurement. E The number linked to exponential growth and I an imaginary number, the square root of minus one.
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And it also involves the fundamental mathematical operations of addition, multiplication and taking powers.
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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.
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As one participant in the physics world, physics world pole remarked,
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what could be more mystical than an imaginary number, interacting with real numbers to produce nothing.
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And the numbers have even featured in a nursery rhyme. Leonard Euler had a farm E-I-E-I-O.
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Zero. And on that farm he had one pig.
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E i e i0. As you'll see, my equation is a special case of a more general result the type published in 1748.
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This beautifully relates the exponential function and the trigonometric functions cos x plus codex and signings.
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But why should the exponential function, each of which goes shooting off to infinity as x becomes large?
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Have anything to do with sign and cosine which oscillate forever between the values one and minus one.
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Indeed, there is no real reason why there should be such a relationship.
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No real reason, but there is a complex reason.
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Introducing the complex number. I lead to such connections and realising this was one of my greatest achievements and my
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result has even appeared on a Swiss postage stamp where it appears up the left hand side.
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Although my results may seem rather abstract, they're also fundamental importance in physics and engineering.
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This is because exponentials of the form, each of the key described things that grow.
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If K is positive or decay, if K is negative, well, those are the form each of the I case he described circular motion,
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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.
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For example, each of the omega t refers to an alternating electric current with angular frequency omega.
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And these imaginary exponentials are much easier to deal with mathematically than science and cosines.
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And indeed, for more advanced topics such as quantum mechanics or image processing, many calculations cannot be done without them.
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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.
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So start with, one, the basis of our accounting system.
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It's been said that there are three types of people, those that can count and those that can't.
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But how do we count? We use a decimal system using only the ten digits one, two, nine and zero.
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But it's also a place value system with a place to give each number determining its value.
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For example, the number 5157 means 5105 tens and seven ones.
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And here the number five plays two different roles, depending on its position as five thousands and has five tens.
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And the advantage of such a place value system is that we can carry out our number calculations column by column.
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Another example is the binary system used in computing, which is based on two rather than ten.
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It's been said that there are ten types of people, those that can count in binary and those account.
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So binary numbers such as one 1 to 1 means one lot of two Q plus one not a two squared
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plus no loss of two to the one plus one unit corresponding to our decimal number 13.
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In fact, it's as easy as one 1011. So how did our accounting systems arise?
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How did early civilisations count? Let's look at some of them.
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Around 1800 B.C., the Egyptians who wrote on papyrus used a decimal system.
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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.
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So the number below. Reading from right to left is to lotus flowers, six cold ropes, five heel bones and eight rolls or 2658.
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And around the same time, the Mesopotamians or Babylonians were imprinting their numbers on clay tablets.
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They did use a place value system. But it was based on 60, not on ten.
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A method of counting we still use when we measure time. 60 seconds in a minute.
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60 minutes in an hour. And see a vertical symbol for one and a horizontal one for ten.
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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.
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Moving forward by over a thousand years takes us to classical Greece and Rome.
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We're all familiar with Roman numerals, a decimal system with letters representing numbers.
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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.
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And because calculating with these letters isn't easy. They used a counting board or abacus for their everyday calculations.
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But the Greek system is seems even more confusing.
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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,
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then the tens from 10 to 90, the hundreds from 100 to 900.
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So a number like 888 would be written as 800 plus 80 plus eight.
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Or Omega Pi eater. Meanwhile, in China, they use counting boards for their arithmetic,
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placing small bamboo rods into separate compartments four units, tens, hundreds and so on.
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And this was a decimal place value system, one of the first.
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But here, as you can see, each number comes in two forms, vertical and horizontal, which alternate.
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So you can tell them easily apart. So 1713 is a horizontal one, a vertical seven, another horizontal one and a vertical three.
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And you notice that for the number 6036, the zero gives us an empty box and the two forms of six are different.
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A different method of counting was used for the calendar.
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The calendar calculations of the Mayans of Mexico and Central America.
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And these survive in a small number of codices drawn on tree bark and then folded.
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So here counting was based mainly on 20, combining dots and lines.
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As you can see on the left, to give all the numbers from 1 to 19 and for larger numbers,
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as you can see in the middle, they piled these numbers on top of each other.
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So here you can see 1220s. Plus 13 corresponding to our decimal number 273.
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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.
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And interestingly, if you look at bottom left, they also had a symbol for zero.
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The shell like symbol you can see there. So this leads us to our second number, zero.
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In India, around 250 B.C. The edicts of Ashoka, the first Buddhist monarch, were carved in pillars around the kingdom.
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Some of these contained early examples of Indian based ten numerals as a decimal
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place value system began to emerge using only the digits 1 to 9 and later on zero.
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So how did zero arise? We've seen how the Chinese left spaces in their counting boards,
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while other civilisations left spaces in the sand to distinguish between numbers like 35 and 305.
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But gradually, special symbols began to emerge.
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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.
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And you can see the little circle representing zero.
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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,
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and it had hundreds of blobs on it, each representing zero.
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You can see one of those blobs in the bottom line there with the arrow pointing to it.
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And this representation of zero predated all other known appearances of zero by some 400 years or so.
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Quite remarkable. Notice that Zero can play two roles as a placeholder as we've seen,
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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.
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And around the 600 rules for calculating them and with zero were given by the Indian mathematician Brahma Gupta.
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And here are some examples you can see on the right. Adding zero and a negative number gives a negative number.
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A negative number taken from zero becomes positive.
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And so. And the only meaningless one was the last one relating to division by zero.
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This is strictly forbidden. Forbidden.
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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.
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And you can do that for any two numbers you like.
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So this picture shows how our number systems developed over the centuries with the Brahma brahmi numerals at the top,
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leading eventually to the numerals at the very bottom, which we can clearly recognise similar to our own.
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Also developing, as you can see on the very right, were the Arabic numerals which are still used in the Middle East.
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But it took many centuries for what we now call the Hindu Arabic numerals to become fully established.
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And here on the left is a 16th century picture representing arithmetic and contrasting the modern
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Elgar ist with his Hindu Hindu Arabic numerals with the old fashion obsessed with his counting board.
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Meanwhile, arithmetic books promoting the Hindu Arabic numerals began to appear by Fibonacci in 1202 Luca Pacioli in 1494 and Robert Record in 1543.
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And the drawing on the right comes from Pad Share this summer and it shows you how to calculate on your fingers.
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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.
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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.
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But PI is also the ratio of the area of a circle to a square of its radius.
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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.
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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.
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Why is it the same constant? I'll leave that for you to think about.
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We can never actually write down pi. Exactly. It's decimal expansion goes on forever.
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But if my six figures that you saw earlier weren't enough for you, here are a few more.
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But if you happen to live in the Kohl's plus area of Vienna and have forgotten any of this, don't worry.
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You'll find them all at your local metro stop. And if they're not enough for you.
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And here are a few more. But the point is we can never write out PI in full.
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Although PI has actually been memorised to 100,000 decimal places.
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What a way to spend a life and calculated to 20 trillion places.
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Even. That's only a beginning. There's still a very, very long way to go.
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But you can easily remember the first few digits from these mnemonics.
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How I wish I could calculate pi. Count the letters in each word.
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You get 3141592. And similarly for the question, may I have a large container of coffee?
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So the number of letters in each word spell out the first eight digits.
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And from the second sentence. How I need to drink.
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Alcoholic, of course. After. After all these lectures involving quantum mechanics, that gives you 14 decimal places.
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And below is even one in Greek, which gives us 22 decimal places.
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Actually there is a poem which gives you 300 decimal places if you count the number of letters in each word.
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But I haven't included that today. So when did people start to measure measure circles?
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Several early civilisations needed to estimate the circumference or area,
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and although they had no conception of PI as a number, their results yield approximations to it.
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The Mesopotamian Clay tablet relates the perimeter of a regular hexagon.
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So there's a conference of the surrounding circle, as the sex suggests.
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Symbol number zero 5736.
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Now, if the radius of the circle is all, then each side of the hexagon is.
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Also, Arthur is a collector of triangles, and so this ratio of six R over two pi r or three of a pi is,
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as it says, 50 zero, 57, 36, 57 over 60 plus 36 over 3600.
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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.
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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.
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What is its area? And the answers given in steps take away one ninth of the diameter, which is one leaving eight.
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Then multiply eight by eight. And that gives you 64.
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See tat of land as the area. So to find the area, they reduced the diameter by one ninth and squared the results.
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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.
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So the pi is about 3.160, an upper estimate.
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Thus also within 1% of the true value. So 4000 years ago they knew pi to within 1%.
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A much more convenient, very convenient, but much less accurate value appeared about a thousand years later in the Old Testament.
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If you read in One Kings or two Corinthians, you will learn to work in bronze.
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Named Hiram made a molten sea with diameter, ten cubits and circumference, 30 cubits.
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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.
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Often credited to Archimedes. It actually dates back to the fifth century B.C.,
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when the Greek Sophist Antiphon and Bryson approximated an area by regular polygons and then tried to obtain better
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and better estimates by repeatedly doubling the number of sides until the polygons eventually became the circle.
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So Antiphon on the top left. He first took a square inside a unit circle and found this area to be two.
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He then doubled the number of size to an octagon giving area to Route two or 2.828.
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Much better, but still some way away from 3.14.
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Bryson's approach was similar, except he also considered polygons outside the circle,
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getting upper bounds of four for the square and 3.32 for the Octagon.
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200 years later, Archimedes adopted the same area, the same idea,
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but he worked with perimeters rather than areas, starting with hexagons inside and outside the circle.
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He doubled the number of size to 12 to 24, 48 and 96, eventually obtaining these balance.
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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.
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PI's a little bit less than 22 over seven. And this actually gives pi to two decimal places.
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What was happening elsewhere in China around the year 263.
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The Huawei also used polygons to approximate pi, starting with hexagons and dodecanese.
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He found simple methods for calculating the successive areas and pyramids.
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Whenever he doubled the number of sides and four polygons with 192 sides, that's twice 96.
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He obtained balance of about 3.14. He then did four more doublings, and that led to polygons of 3072 sides.
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They didn't actually draw them, but he did the calculations on them and found that PI is 3.14159.
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Even more impressively, around the year five hundreds,
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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.
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Quite remarkable. And they also improved Archimedes fractional value of 22 over 72355 over 113, which also gives pi to six decimal places.
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And this lesser approximation wasn't rediscovered in Europe for another thousand years.
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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,
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leading eventually to the remarkable Dutchman Leo Lugo from Colon, who went up to polygons with over 500 billion sides.
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You can see the number written out up there and that gives pi to 20 decimal places.
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And if you look carefully below his portrait, you can see two numbers there.
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Those are his upper and lower estimates for pi to 20 decimal places.
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But not content with this. He then used polygons with two to the power 62 size.
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That's that big number there. And he found pi to 35 decimal places.
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He also this latter value took a pair on his tombstone in Leiden.
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For many years, PI was known in Germany as the blue dolphin number.
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I'm not going to get a little bit more technical for a few minutes, but if you lose it, don't worry.
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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,
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which was used extensively in the 18th and 19th centuries and involves the tangents of angles you're familiar from trigonometry.
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The ten theta is A over B,
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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.
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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,
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so ten to the minus one simply undoes whatever ten does.
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For example, ten of pi by four. That's ten, 45 degrees is one.
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So tens of minus one of one is pi over four. Instead of pi by six.
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That's 1030 is one of three. So 10th and minus one of one of three is pi over six.
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I'm using radian measures here, which some of you will have come across.
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Basically PI means 180 degrees. Pi by two is 90 degrees.
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And so. Now the thing is that you can actually combine different values of ten to the minus one.
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For example, if you add ten to the minus one a half, ten to the minus one, a third, you get pi over four.
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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.
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And together they give 45 degrees. But you can also prove it by simple geometry.
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And in general, we can combine any two inverse tangents simply by using the formula at the bottom of the frame.
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Well, many functions can be written as infinite series.
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And some of you will have come across this, for example,
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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.
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So it's only odd powers of X that appear.
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And in the denominators you have one, three, five, seven. The odd numbers.
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Now this result was actually known in 15th century India, but it's usually named after the Scotsman,
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James Gregory, that you can see here who rediscovered it 300 years later.
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But if you now put X equals one into this, you have ten to the minus one of one, which is pi over four.
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Is equal to one minus a third plus a fifth, minus seventh, and so on.
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And this is surely and this result is was also found in India, but is usually credited to liveness.
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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,
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we get a result involving the circle number pi.
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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.
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For example, if you take the first 300 terms of the series, well, that gives you pi to only two decimal places.
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If you want five decimal places, you've got to take the first half a million terms.
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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.
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Because remember the terms of the minus one half and ten to the minus one, a third add up to pi over four.
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So we can substitute x is a is a half an x is a third into that series.
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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,
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these converge much faster and they yield good estimates for PI.
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Indeed, in 1861, a gentleman from Potsdam used these very series to find PI to 261 decimal places.
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All the other series of pi which converge even faster when in 1706 the Englishman John Machin repeatedly used the addition form that
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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.
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Don't worry how those appear. They just emerge out of the formulas.
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And then he wrote out the 210 to the minus one series that you can see,
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and these series converge rapidly because of the powers of five and 239 in the denominators.
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For example, if you just take the first three terms of each series and you already get the value 3.14, just three terms.
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It's also useful because five is an easy number to divide by.
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And as a result, Machin was able to calculate PI by hand to 100 decimal places.
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A great improvement on anything that had come before.
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Well, 76 is a good year for PI because as well as mentions result,
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a Welsh maths teacher called William Jones wrote a new introduction to the mathematics
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in which he introduced for the first time the symbol PI for the circle number.
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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.
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And here are two extracts from his book. In the Upper One.
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In the middle, you can see matching series with the fives and two, three nines and then immediately below it.
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Is the first ever appearance in print of PI.
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You can see it in the penultimate line. And below is maintenance value for pay in full.
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You can see the 100 decimal places and describe is true to above 100 places as
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computed by the accurate and ready pen of the truly ingenious Mr. John Machine.
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Well, such results can be used to obtain improved values of PI.
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As I said, and most notorious of all was one obtained by William Shanks,
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who in 1873 use medicines formula that you just seen to calculate PI to an impressive 707 decimal places.
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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.
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Unfortunately for Shanks and for the Palace, it was later found that only the 500, the first 527 of these decimal places are correct.
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But they weren't going to redecorate their city. Let's look at a very different way to find pie.
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In 1777, the country performed, describing an experiment for estimating it.
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Suppose you throw a large number of needles or matchsticks of length l onto a grid of parallel lines at a distance de apart.
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It's not difficult to show that this proportion is to over pi times lfg from which you can calculate a value for pi.
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Now in this particular case, eleventy, it turns out, before over five.
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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.
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Incidentally, in 1901, an Italian mathematician called Mario Lazzarini carried out such an experiment,
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which 11 was five over six, performing 3408 trials and claiming 1808 crossings.
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And this gave pies 355113, which, as we saw, gives pi to six decimal places.
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He was lucky if just one needle had landed differently, his result would have been correct to only two decimal places.
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In 1997, a bizarre event took place in the American state of Indiana,
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where the House of Representatives unanimously passed a bill introducing a new mathematical truth.
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This attempted to legislate an incorrect value for PI proposed by a local physician who then allowed
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the state to use his value freely but would expect royalties from anyone out of state who used it.
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A bill for an act, introducing a new mathematical truth and offered as a contribution to education to be used only in the
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state of Indiana free of cost by paying any royalties whatsoever on the same according to the proposal.
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This physician the ratio of the diameter of circumference is as 5/4 to four, which gives pi is 3.2.
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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.
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It should been found as a circular area. It's the quadrant of the circumference is the area of an equilateral rectangle.
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Is the square on one side. Well, that makes no sense at all.
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But even so, the bill proceeded to the Committee on Temperance, of all things, who recommended its passage.
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Fortunately,
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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,
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and he persuaded the senators to stop it just in time. As far as we know, is still with the Committee on Temperance.
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The 20th century saw several new discoveries, many of them completely bizarre.
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And I'm just going to mention three of them briefly. You're not expected to be able to prove them.
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In 1914, the Indian mathematician Ramanujan found some remarkable exact formulas for one over pi, including the one at the top.
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This is amazing. It's an infinite series in which strange numbers such as 1103 and 26,398 seem to appear from nowhere.
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But such series converge extremely rapidly and form the basis of some of today's fastest algorithms for calculating PI.
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Many years later, in 1989,
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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.
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Remarkably, the third set an example simpler, but it caused much surprise at the time.
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If you see how much simpler it is. But the importance is that if you work in base 16 rather than base ten,
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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.
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Why pay 16 should come in there. I'll leave you to think about.
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Well, to end my discussion of pie, let's look at a simple puzzle that dates back to 1702.
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It appeared in a book called Nucleus Elements by the Cambridge mathematician William Whiston.
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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.
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Now, assuming the earth to be a perfect sphere, supposing you tie a long piece of string tightly around it.
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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.
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So you've got the whole circumference, you're only adding six feet and you then prop the string equally all around the earth.
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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.
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But the answer is one foot. Which to many people is quite surprising.
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In fact, you get the same answer. Whether you tie the string around the earth,
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a tennis ball or any other sphere through the sphere has radius of feet than the original string has to offer.
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When you extend it by two pi feet, the new circumference is two plus two pi, which is two pi times R plus one.
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So the new radius is R plus one. One foot more than before.
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Let's now move on to our next number, the exponential number E, which is 2.71828 and so on.
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It's another number whose decimal expansion goes on forever. Here we're concerned with how quickly things grow.
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We often use the phrase exponential growth to indicate something that grows very fast.
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But how fast is this? Incidentally, the letter E for this number first appeared in print in 1736.
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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.
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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.
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So that's the first appearance of the symbol letter E in that context.
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Well, to show you what we mean by exponential growth, here's a story about the origins of the game of chess.
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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,
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My price is for you to give me one grain of wheat for the first square of the chessboard.
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Two grains for the sake of the square. Four for the third square and so on.
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Doubling the number of grains on each successive square until the chessboard is filled.
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The King was amazed to be offered such a such a tiny reward.
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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.
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This works out at two to the 64 minus one grains enough wheat to form a pile the size of Mt. Everest.
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If you place them end to end, they'd reach the nearest star, Alpha Centauri, and back again.
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Let's see how quickly other sequences can grow. Look at the top left.
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A very simple form of growth is linear growth, as in the counting numbers and is one, two, three, four, five and so on.
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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.
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And these are all examples of polynomial growth because they involve paths of N.
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But alternatively, we could look at powers of two or any other number as we saw in the chessboard story.
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The numbers two to the end, the powers of to start slowly but soon gather pace because each successive number
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is twice the previous one and the powers of three grow even more quickly.
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These are examples of exponential growth where n is the appears as the exponent.
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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.
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For computer performing, say, a million operations, a second for polynomial growth such as and cubed.
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That's the third blue line there. Such a computer would take about one eighth of a second when an is 50.
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But exponential growth such as to to the end is much greater as we've seen.
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When N is 50, the computer would take over 35 years and would be vastly greater than this for three to the end.
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So in the long run, exponential growth tends to exceed polynomial growth, often by a huge, huge margin.
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Algorithms that run in polynomial time are generally thought of as efficient,
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while those that run in exponential time nor may take much longer to implement and are regarded as inefficient.
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But returning to E. What exactly is this number and how is it to rise?
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In 1683, the Swiss mathematician Jakob Bernoulli was calculating compound interest, given a sum of money to invest at a given rate of interest.
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How does it grow? The answer depends on how often we calculate the interest.
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How much is earned if we calculate it once a year or twice a year, or every month or every week or every day,
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or even continuously as an example, to keep the calculations fairly simple.
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Let's see what happens if we invest £1 at the rather unlikely annual rate of 100%.
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After one year, our pounds is double to £2. But if you calculate the interest twice a year, that's 50% each time.
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Then after six months, our per pound is multiplied by one and a half to give you £1.50,
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and after another six months, that amount is multiplied by one and a half.
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Again, to give you £2.25, which is more than you had before.
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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,
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first to £1.25, then to about £1.56, then to 195 by the end of the year to £2.44.
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That's £1 times one and a quarter to the power four.
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So again, the final amount is even larger and as superiors get shorter, what happens to the final amounts increase without bound?
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Wouldn't that be good? Or do they settle down to a limiting value?
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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.
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Then after each period the result is multiplied by one plus one over n.
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So the final month at the end of the year is one plus one over end to the power.
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And I mean, don't worry about the details, but you'll see the numbers there.
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You can see from the tables and increases this these final amounts do approach a limiting value or seem
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to the corresponds to when you calculate the interest continuously and this limiting amount 2.71828.
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That's just the exponential number e.
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Now, the greatest, greatest advances in understanding exponentials were actually were made in the early 18th century.
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And after Bernoulli, the main figure in the story actually was myself.
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I investigated the number E and the related exponential function each of the X and in 1748 my introduction to the analysis of Infinite,
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which I claim is one of the most important mathematics books ever written, brought together many of my results from earlier works.
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And here are some of my main findings, which will be familiar to some of you.
381
00:46:38,810 --> 00:46:45,050
We've just seen that is the limit of the number is one plus one over into the power and NZ increases indefinitely
382
00:46:45,530 --> 00:46:51,560
and similarly can show that each of the x is the limit of OnePlus X over end to the end for any number x.
383
00:46:54,140 --> 00:46:58,580
But as Isaac Newton had already discovered, is also the sum of the infinite series.
384
00:46:58,880 --> 00:47:02,310
As shown here, the denominators are the factorial.
385
00:47:02,720 --> 00:47:06,620
One factorial is one. Two factorial is two times one.
386
00:47:06,620 --> 00:47:08,900
Three factor is three times two times one, and so on.
387
00:47:09,990 --> 00:47:15,380
More generally, there's a similar series, as you can see, for each of the X, which converges for all values of X.
388
00:47:15,770 --> 00:47:22,700
And in fact, all of these series converge extremely fast because the factorial in the denominator is increase so rapidly.
389
00:47:23,480 --> 00:47:29,600
For example, just the first ten terms of that series already give you the correct value of E to five decimal places.
390
00:47:31,580 --> 00:47:35,440
And on the right is the curve. The graph of y is each.
391
00:47:35,460 --> 00:47:41,960
The X and the important features are that each point x, the slope of the graph is also each of the x.
392
00:47:42,710 --> 00:47:51,800
That is the slope at any point is the y value. So the curve becomes steeper and steeper as X increases this exponential growth.
393
00:47:53,630 --> 00:48:00,320
So I'd like to end my discussion of exponential growth by returning by by looking at.
394
00:48:01,850 --> 00:48:06,440
In 1798, Thomas Malthus wrote his essay on population,
395
00:48:07,280 --> 00:48:15,440
where he contrasted the steady linear growth of food supplies available with the exponential growth as he saw in population.
396
00:48:16,190 --> 00:48:24,500
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.
397
00:48:24,890 --> 00:48:27,920
A conclusion that was indeed borne out in practice.
398
00:48:31,310 --> 00:48:35,000
So how fast does a population grow? This is a little bit of calculus.
399
00:48:35,000 --> 00:48:37,700
If you don't know calculus, it won't last long.
400
00:48:39,980 --> 00:48:47,420
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,
401
00:48:48,020 --> 00:48:49,999
then you get what's called a differential equation.
402
00:48:50,000 --> 00:49:01,310
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.
403
00:49:02,030 --> 00:49:05,450
So this is an example of exponential growth, as you can see.
404
00:49:06,380 --> 00:49:14,330
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.
405
00:49:17,610 --> 00:49:26,730
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.
406
00:49:27,000 --> 00:49:34,500
When Cardano, one of the Italian mathematicians who first wrote about cubic equations, was trying to solve a number puzzle.
407
00:49:35,550 --> 00:49:40,500
Can you divide ten into two parts whose product is 40?
408
00:49:42,820 --> 00:49:49,000
Well, the two paths he took to be X and ten minus X, they add up to ten.
409
00:49:49,810 --> 00:49:57,100
So X times ten minus X is 40. And solving this quadratic equation, he found the solutions to be five,
410
00:49:57,100 --> 00:50:04,330
plus the square root of -15 and five, minus the square root of -15, which seemed meaningless.
411
00:50:06,490 --> 00:50:11,270
Communism that nevertheless we will operate putting aside the mental tortures involved.
412
00:50:11,920 --> 00:50:17,530
He checked these answers, actually worked at them. You get ten, multiply them, you get 40.
413
00:50:18,490 --> 00:50:25,360
But he complained that so progresses arithmetic, arithmetic, subtlety, the end of which is as refined as it is useless.
414
00:50:28,740 --> 00:50:33,710
We're trying to take the square to a negative number. Doesn't seem to make sense for one time.
415
00:50:33,750 --> 00:50:37,440
One is one minus one times. Minus one is also one.
416
00:50:38,100 --> 00:50:41,730
As a Victorian Augustus two, Morgan remarked, 300 years later,
417
00:50:42,660 --> 00:50:48,720
we have shown the symbol scroll of minus eight to be void of meaning all rather self-contradictory and absurd.
418
00:50:49,740 --> 00:50:54,450
While his contemporary, the astronomer Royal George Arie, commented,
419
00:50:54,720 --> 00:51:00,120
I have not the smallest confidence in any result, which is essentially obtained by the use of imaginary symbols.
420
00:51:02,570 --> 00:51:09,320
However, live news had been more encouraging, claiming that the imaginary numbers are a wonderful flight of God's spirit.
421
00:51:09,710 --> 00:51:13,820
They are almost an amphibian between being and not be.
422
00:51:15,410 --> 00:51:21,980
I must confess that to my shame, even I who use complex numbers so effectively did criticise them at the time.
423
00:51:25,690 --> 00:51:29,740
Well, for many purposes, our ordinary numbers are real. Numbers we call them are enough.
424
00:51:30,250 --> 00:51:35,170
But suppose we now agree to allow this mysterious object called the square root of minus one.
425
00:51:35,740 --> 00:51:45,070
Or I, as I named it, we can then form many more numbers, such as one plus three I or two plus I.
426
00:51:46,390 --> 00:51:51,670
And if we ignore for the moment what these actually mean, we can still carry out some simple calculations.
427
00:51:52,120 --> 00:51:55,989
Adding is straightforward one plus three, four, two plus.
428
00:51:55,990 --> 00:52:00,940
I want us to is three, three. If I size four, I get three plus four I.
429
00:52:01,850 --> 00:52:05,950
Multiplication is also simple. I won't go through the details.
430
00:52:05,950 --> 00:52:11,290
But the point is that whenever you see an I squared you just replace it by minus one.
431
00:52:15,700 --> 00:52:23,920
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.
432
00:52:23,980 --> 00:52:32,230
Algol organises is often called so it's often called the Gaussian plane or Argon diagram,
433
00:52:32,920 --> 00:52:37,870
but neither name is historically correct, so it's much better to call it the complex plane.
434
00:52:38,650 --> 00:52:42,520
Even here in Oxford, they call it the arc and diagram, and they should be ashamed of themselves.
435
00:52:45,840 --> 00:52:50,520
So the point is we represent each number A-plus. B by the point of coordinates.
436
00:52:50,520 --> 00:52:58,770
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.
437
00:52:59,940 --> 00:53:06,060
And if you want to add to complex numbers such as those two numbers, you use what's called the parallelogram law,
438
00:53:06,690 --> 00:53:12,060
you just draw the parallelogram and at the top you'll see the sum is three feels four I.
439
00:53:14,030 --> 00:53:20,030
In the bottom. If you want to multiply by pictorially, you just rotate through 90 degrees.
440
00:53:21,410 --> 00:53:27,260
In fact, as a telephone operator said to me the other day, the number you have dialled is purely imaginary.
441
00:53:27,470 --> 00:53:31,310
Please rotate your throne, your phone through 90 degrees and try again.
442
00:53:32,570 --> 00:53:39,350
And I did. And it worked brilliantly. And doing this again, you're multiplying it done to lots of 90.
443
00:53:39,350 --> 00:53:44,660
So you multiply by squared or minus one. And that just gives you a rotation through 180 degrees.
444
00:53:48,910 --> 00:53:53,650
Well, as we've seen, there was much suspicion in Victorian times about these imaginary numbers.
445
00:53:54,910 --> 00:53:59,049
The Irish mathematician and astronomer William Rowan Hamilton largely ended the
446
00:53:59,050 --> 00:54:06,190
suspicion by defining the complex numbers as pairs ab of ordinary real numbers.
447
00:54:07,030 --> 00:54:10,900
And these combine according to two special rules. You can see that.
448
00:54:12,240 --> 00:54:14,459
So if you take them as pairs and you do that,
449
00:54:14,460 --> 00:54:23,130
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.
450
00:54:25,680 --> 00:54:32,790
So since complex numbers can be represented in the plane, a natural question is can you extend this idea to three dimensions with numbers
451
00:54:32,850 --> 00:54:38,550
the form A plus B plus S.J. where I squared and just J squared at both minus one.
452
00:54:39,390 --> 00:54:45,710
Now here addition is still okay. But multiplication isn't because you try to multiply them.
453
00:54:45,920 --> 00:54:51,230
You get a term involving eight times J. And what's that equal to Hamilton?
454
00:54:51,260 --> 00:54:55,160
Try putting it equal to minus one or two. I also j also zero.
455
00:54:55,850 --> 00:54:58,999
Nothing works. Everything fails.
456
00:54:59,000 --> 00:55:03,530
And as he later wrote to his son Archibald, every morning on my coming down to breakfast,
457
00:55:03,530 --> 00:55:07,729
your brother William, Edwin and yourself used to ask me, Well, Papa, can you multiply?
458
00:55:07,730 --> 00:55:12,690
Triplets were true. I was always obliged to reply with a sad shake of the head.
459
00:55:13,130 --> 00:55:15,170
No, I can only add and subtract them.
460
00:55:16,850 --> 00:55:25,100
Well, after struggling with his troublesome triplets for many, many years, Hamilton had his moment of glory on the 16th of October 1843,
461
00:55:25,640 --> 00:55:33,620
while walking with his wife along Dublin's Royal Canal, when an electric current seemed to close and the spark splashed forth.
462
00:55:33,980 --> 00:55:38,930
I pulled out on the spot a pocketbook, that one you can see on the left and made an entry there.
463
00:55:38,930 --> 00:55:44,720
And then. Nor could I resist the impulse to cut with a knife on a stone on Brougham Bridge.
464
00:55:44,900 --> 00:55:51,020
As he passed it, the fundamental formula with the symbols I, j and K namely I squared is j square.
465
00:55:51,020 --> 00:55:56,660
This case great is minus one. What is also equal as it happens to eject k that.
466
00:55:57,550 --> 00:56:01,690
That's the inscription. And there is a postage stamp that commemorates that.
467
00:56:02,410 --> 00:56:07,930
So these are called the quaternions with four terms A plus B, plus S.J. plus the UK.
468
00:56:08,500 --> 00:56:11,050
I squared just in case they're all minus one.
469
00:56:11,470 --> 00:56:17,740
But unlike our ordinary multiplication, which is commutative three times four is four times three, and so on.
470
00:56:18,250 --> 00:56:28,240
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.
471
00:56:28,490 --> 00:56:33,400
Okay, but then you can multiply any two of these you like as long as you stick to those rules.
472
00:56:35,660 --> 00:56:41,210
Well, for the last few minutes. Now we've got our five constants. Let's return to my equation and my identity.
473
00:56:43,730 --> 00:56:47,420
Recall that my identity connects the exponential function which goes shooting off to
474
00:56:47,420 --> 00:56:53,480
infinity with the functions cosine and sine which oscillate between one and two minus one.
475
00:56:55,670 --> 00:57:02,440
To show this connection. Recall that these functions can all be expanded as infinite series values for all values of X.
476
00:57:02,450 --> 00:57:11,660
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.
477
00:57:12,500 --> 00:57:19,580
As I did in 1737. So to do so, you take that series there for each of the four, each of the x.
478
00:57:22,380 --> 00:57:25,470
And you replace eggs everywhere by eggs, as I've done.
479
00:57:25,530 --> 00:57:31,049
So this gives you each of the x is one plus I x over one factorial plus i x squared
480
00:57:31,050 --> 00:57:35,820
or all squared of over two factorial plus i x or cubed over three factorial.
481
00:57:35,820 --> 00:57:44,230
And so. But ice credits minus one, which means that IQ does minus ice of the fourth is one.
482
00:57:44,470 --> 00:57:50,080
And if you still stir it all around and simplify it, you will actually collect terms together.
483
00:57:50,260 --> 00:58:00,420
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.
484
00:58:00,430 --> 00:58:04,990
And as I said before, one of the most remarkable equations in the whole of mathematics.
485
00:58:05,980 --> 00:58:08,500
In fact, I gave more than one proof of my identity.
486
00:58:09,550 --> 00:58:18,040
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,
487
00:58:18,100 --> 00:58:25,520
which I won't go into today. And the first ever appearance of my identity is in the penultimate line.
488
00:58:26,030 --> 00:58:31,879
You can see E to the V times. The scrotum minus one to the IV is called the pulse.
489
00:58:31,880 --> 00:58:36,980
The scrotum minus one times sine V. That's the first appearance of Euler's identity.
490
00:58:37,220 --> 00:58:40,340
My identity as I myself collected at the time.
491
00:58:40,670 --> 00:58:47,840
From these equations, we can just understand how complex exponential exponentials can be expressed by real signs and cosines.
492
00:58:50,490 --> 00:58:53,660
Moreover, from my identity, we can deduce my equation.
493
00:58:54,320 --> 00:58:57,980
We just let X is pi. That's the radian for 180 degrees.
494
00:58:58,700 --> 00:59:03,110
Then you get course. Pi is minus one sign pi zero.
495
00:59:03,440 --> 00:59:09,410
So each of the pi is cause pi minus one plus zero I and this minus one.
496
00:59:09,700 --> 00:59:12,350
So each of the IP plus one equals zero.
497
00:59:13,520 --> 00:59:19,110
And although I certainly made that deduction, it doesn't actually appear explicitly in any of my published works.
498
00:59:19,130 --> 00:59:26,980
I can't remember why I didn't put it in. Can we can we illustrate this pictorially?
499
00:59:28,030 --> 00:59:34,720
Yes. Here's a pictorial representation. In 1959, an English schoolteacher called Mr. Hull showed how to do it.
500
00:59:35,140 --> 00:59:39,210
He took the power series for each of the x and put x is a pi.
501
00:59:40,090 --> 00:59:45,910
This gives one plus I pi plus IP squared over two factorial and so on.
502
00:59:46,580 --> 00:59:50,470
Now let's trace these values. Start off with a one on the right.
503
00:59:50,680 --> 00:59:56,589
You then add IP. That means you go upwards and then you subtract a half pi squared.
504
00:59:56,590 --> 01:00:00,820
And then you go down and and you eventually are tracing a spiral pattern.
505
01:00:01,780 --> 01:00:06,819
And what does it converge to? It converges to the point minus one, the sum of the series.
506
01:00:06,820 --> 01:00:11,379
Which is what you expecting? Well, we've almost finished.
507
01:00:11,380 --> 01:00:15,940
But I just very quickly like to mention two near misses to discovering my equation.
508
01:00:18,310 --> 01:00:25,240
In 1702, Johann Bernoulli was investigating the area of his brother, the Jacobi met earlier.
509
01:00:25,780 --> 01:00:31,450
He was investigating the area of a sector of a circle of radius a that's the shaded bit on the right,
510
01:00:31,930 --> 01:00:36,790
above the axis and below the line from the origin to the point x, y.
511
01:00:37,330 --> 01:00:43,360
And he found it to be that complicated expression given there involving the logarithm of a complex number.
512
01:00:44,290 --> 01:00:48,429
And leaving aside what that means, I later said that if you put X equals one,
513
01:00:48,430 --> 01:00:56,320
then you get a quarter circle and the expression simplifies to a square to four times the log of minus one.
514
01:00:57,100 --> 01:01:05,680
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.
515
01:01:06,840 --> 01:01:14,940
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.
516
01:01:16,250 --> 01:01:21,600
And although I wrote it down explicitly, stupidly, I didn't take exponentials to deduce my equation in the form.
517
01:01:21,600 --> 01:01:25,440
Each of the I pi is each log of minus one is minus one.
518
01:01:26,370 --> 01:01:30,360
And indeed, I often credited Bernoulli with discovering this value for a log of minus one.
519
01:01:32,400 --> 01:01:38,490
And the last very brief near Miss was the work of the Cambridge mathematician Roger Coates.
520
01:01:38,910 --> 01:01:43,140
He worked closely with Isaac Newton on the second edition of Kepler Mathematica.
521
01:01:43,710 --> 01:01:47,070
And he's been credited with introducing radian measures for angles.
522
01:01:48,600 --> 01:01:59,100
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.
523
01:01:59,610 --> 01:02:06,389
The details are complicated. The point is, he managed to find two different expressions from the area for the area one involving
524
01:02:06,390 --> 01:02:12,360
logarithms and the other involving chicken and trigonometry and both involving an angle phi.
525
01:02:13,080 --> 01:02:18,510
So he first prove that the surface area is a certain multiple of log, of course, plus high sign.
526
01:02:19,230 --> 01:02:24,090
He also proved to be the same multiple of PHI. So say don't worry about how he got there.
527
01:02:24,090 --> 01:02:29,400
The point is that if you equate these results, then you get this result that you can see at the bottom.
528
01:02:30,840 --> 01:02:36,360
If he had now taken exponentials, he'd have found my identity in the form.
529
01:02:36,360 --> 01:02:45,780
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.
530
01:02:47,010 --> 01:02:50,130
Another near miss. So to end with.
531
01:02:52,690 --> 01:02:56,200
What should we call the equation? Each of the five plus one equals zero.
532
01:02:58,040 --> 01:03:02,210
We just see how it follows easily from results of Johan Bernoulli and Roger Coates.
533
01:03:02,840 --> 01:03:05,000
But neither of them seems to have done so.
534
01:03:06,950 --> 01:03:12,200
Even I never wrote it down explicitly, although I certainly realise that it follows immediately from my identity.
535
01:03:13,430 --> 01:03:20,450
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.
536
01:03:23,560 --> 01:03:26,740
But almost everybody nowadays attributes the result to me.
537
01:03:27,910 --> 01:03:37,300
So are surely justified in naming it. Euler's equation to honour the achievements of myself has been called a truly great mathematical pioneer.
538
01:03:38,170 --> 01:03:43,059
A word that I say modestly prevents describes me so well, and which appropriately includes.
539
01:03:43,060 --> 01:03:49,030
Among its letters are five constants PI, one, O, zero and E.
540
01:03:50,950 --> 01:03:51,730
Thank you very much.