Euler's Equation: 'The Most Beautiful Theorem in Mathematics' - Professor Robin Wilson

good evening in this talk I'd like to tell you about my pioneering equation 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 st. Petersburg and Berlin having published over 800 books and papers in over 70 volumes I've been called the most prolific mathematician of all time ranging across almost all branches of mathematics and physics at the time these are mounted to about one-third of all the maths and physics publications of the 18th century so why is it the most beautiful theorem in mathematics well this comes from when my equation top to pole run by the mathematical intelligence Intelligencer an american mathematics magazine but such poles 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 way ahead of such equations as Einstein's e equals mc-squared and Newton's laws of motion other people have been equally impressed indeed when only 14 the future Nobel prize-winning physicist Richard Feinman called my equation the most remarkable formula in math while Fields Medal winner so Michael Watts here has described it as the mathematical equivalent of Hamlet's to be or not to be very sick synced but at the same time very deep and the mathematical popularizer Keith Devlin waxed even more eloquent saying like a Shakespearean sonnet that covers the captures the very essence of love or a painting that brings out the beauty of the human forms is far far more than just skin deep Oilers equation my equation reaches down into the very depths of existence it even featured in two episodes of The Simpsons and was crucial in the Criminal Court case when an American physics graduate student was sentenced to eight years in prison after vandalizing 100 luxury sports cars by spray painting slogans on to them he was identified after spraying my equation which had just popped into his head onto a mitsubishi montero whatever that is and as he announces his trial I've known Oilers equations since I was five everyone should know Oilers equation so what is this result of mine 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 1 it also involves the fundamental of mathematical operations of addition multiplication and taking powers so if we take E and raise it to the power I times pi and then add 1 we get zero or equivalently e to the I pi is minus 1 and there's one participant in the physics world polls moved to remark 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 eieio and on that farm he had one pig e-i-e-i-o i so see my equation is a special case of a more general result that i published in 1748 the celebrated result relates the exponential function and the trigonometric functions cos x and sine x but why should the exponential function e to the x which goes shooting off to infinity as X becomes large have anything to do with these trig functions which forever oscillate between values 1 and minus 1 indeed there's no real reason why there should be such a relationship but there are complex reasons introducing the complex number I leads to such connections and realizing this with one of my gracious achievements and Myra sult has even appeared on a Swiss postage stamps where it appear appears up the left-hand side well that my results may seem rather abstract they're also fundamental importance to physicists and engineers this is because Exponential's of the form e to the KT describe things that grow if K is greater than 0 or decay if K is negative well those are the form e to the I KT describe circular motion but by my identity e to the ikj is made up from cos K T and sine K T and therefore can be used to represent things that oscillate for example e to the I Omega T refers to an alternating electric current with angular frequency Omega mathematically these imaginary Exponential's are much easier to deal with than cosines and sines and indeed for more advanced topics such as quantum mechanics and image processing many calculations cannot be carried out without them but today I'm going to introduce four five constants one at a time before showing you how to combine them into what we've called my equation so let's start with one the basis of our counting system it's been said that there are three types of mat of 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 because the placing of each number determines its value for example the number five one five seven means five thousand one hundred five tens and 7 ones or if you like five times 10 cubed plus one times 10 squared plus five times 10 to the 1 plus seven and here for number five plays two different roles are pending on its position as five thousands and as 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 a binary system using computing which is based on 2 rather than 10 it's been said that there are 10 types of people those that can count in binary and those that can't so a binary number such as 1 1 0 1 means 1 lots of 2 cubed plus 1 not a 2 squared no lots of 2 to the 1 and 1 unit corresponding to our decimal number 13 in fact it's as easy as 1 10 11 so how did our counting systems arise how did early civilizations count let's look at some around 1800 BC the Egyptians who wrote on papyrus use a decimal system but it wasn't a place value system because they use different symbols for 1/10 a hundred and so on repeating them as often as necessary so the number below reading from right to left is to lotus flowers 6 coiled ropes 5 heel bones and 8 rods or 2658 and around the same time the Mesopotamians or Babylonians we're imprinting their numbers on clay tablets they use the place value system but it was based on 60 not on 10 a method of counting that survives in our measurement of time 60 seconds in a minute 60 minutes in an hour and using a vertical symbol for one and a horizontal one for 10 the number one 1237 shown here represents one loss 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 that uses letters to represent numbers but it's not a place value system because different letters are used for one ten hundred and a thousand and also for 550 and 580 with these letters is difficult they use the counting ball or abacus for their calculations the Greek system seems even more confusing it's also a decimal system but again it's not a place value system because they use different Greek letters for a letter from the units from one to nine tens from 10 to 90 and the hundreds from 100 to 900 so a number like 888 would be written as 800 plus 80 plus 8 or Omega pi ETA meanwhile in China they use counting balls for their arithmetic placing small bamboo rods into separate compartments for units tens hundreds and so on this was a decimal place value system one of the first and notice of each number comes in two forms vertical and horizontal which alternate so 1713 is a horizontal one a vertical seven another horizontal one and a vertical three they're alternating horizontal and vertical and notice two 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 calculations the Mayans of Mexico and Central America these survived in a small number of codices drawn on tea tree dark tree bark and then fold it and here counting was mainly based on 20 combining dots and lines to give all the numbers from 1 to 19 as you see on the left and for larger number they are two numbers they pile these numbers on top of each other so in the middle you can see 12 20s plus 30 corresponding to our number 273 a rather attractive feature of their counting was that each number also had a pictorial head form like the ones at the bottom and notice that they also had a symbol for 0 the shell or I like symbol at the bottom so this leads us to our second number 0 in India king ahsoka became the first Buddhist monarch around 250 BC and his edicts were carved on pillars around the kingdom some of these contains early examples of Indian base 10 numerals as a decimal place value system began to emerge using only the numbers 1 to 9 and later 0 so how did 0 arise we've seen how the Chinese left a space in their counting boards while other civilizations use spaces in the sand to distinguish a number like 305 from 35 but gradually special symbols began to emerge here above is a cave in Gwalior in India where the number 270 ringed in blue is clearly seen on the wall but there was great excitement last autumn when some birch bark was found in the borderland library in Oxford that had been undiscovered for over a hundred years you can see it below and it had hundreds of blobs on it each representing zero like the one you can see arid and amazingly it predated all other known appearances of zero by hundreds of years notice how Sarah plays two roles as a placeholder as we've seen and also as a number as a calculator with both positive and negative numbers were already used in money transactions for profits and debts and around the year 600 rules for calculating with them calculating with them were given by the Indian mathematician Brahma Gupta here are some examples adding 0 and a negative number gives a negative number a negative number taken from 0 gives a positive 1 and so on of all these rules the only meaningless one was his last one relating to division by zero division by zero is forbidden because if you have an equation like 4 times zero equals nine times zero if you can calculate the zero that'll tell you that 4 is equal to nine which is nonsense and that works for any two numbers you like this time we've got here shows how our number systems developed over the centuries leading to the new rules at the bottom which are clearly recognizable to us but also developing this time on the right you can see the Arabic numerals which are still used in the Middle East but it took very many centuries for what we now call the hindu-arabic numerals to become fully established here on the left is a 16th century picture representing arithmetic which contrasts a modern algorithm to Arabic numerals and the old-fashioned abbe assist with his counting board meanwhile arithmetic books to promote the hindu-arabic numerals were published by a number of people Fibonacci you may have come across Luca Pacioli and Robert record and on the right is a drawing from patios summer of 14 90 for that shows you how to do your how to calculate on your fingers let's now turn to our third number pi which arises in two ways as the ratio of those cuts the Conference of a circle to its diameter pi is C over D so C is PI D or 2 PI R where R is the radius and the point is that this ratio pi is the same for circles of any size from a pizzas of the moon but it's also the ratio of the area of a circle to the square of its radius pi is a over R squared or a is PI R squared and this ratio is also the same for all circles as usually approved in the 3rd century BC we can never write down PI exactly it's decimal expansion goes on forever but if the six figures I gave you earlier weren't enough for you here are a few more and if you happen to live in the car plus area of Vienna and you happen to have forgotten these digits don't worry your local metro stop has most of them written down but that's not enough for you here a few more but the point is we can never write out PI in fall although pi has actually been memorized to over a hundred thousand decimal places what a way to spend a life and calculated to only twenty twenty trillion places even that's only a beginning there's still a long long way to go but here are some ways of remembering the first few digits in the request may I have a large container of coffee the number of letters in each words spell out the first eight digits three one four one five nine two six if you want more than from the second sentence we get fourteen decimal places how I need a drink alcoholic of course after all these lectures informing Gresham audiences and below there's even one in Greek which gives us 22 decimal places when did people start to measure the circles several early civilizations needed estimates for the circumference or area and although they had no conception of pi as a number their results do yield approximations to its value a Mesopotamian clay tablet which survives relates the perimeter of a regular hexagon to the circumference of the surrounding circle and this ratio is a sexagesimal number zero 57 36 now if the radius of is R then each of the side of the hexagon it's also our six equilateral triangles here and so this ratio of six are over two pi r or three over pi is 57 over 60 plus 36 over 3,600 that's what that number means and after some calculation this gives us pi is 3 and 1/8 or 3.125 in decimals a lower estimate but it's within one percent of its true value and around the same time an Egyptian papyrus asked the following question given a round field of diameter 9 yet that's the universe length what is its area and the answer is given the steps take away 1/9 of the diameter which is 1 the remainder is 8 multiplied 8 times 8 it makes 64 therefore it contains 64 seat out of land so the area is 64 so to find the area they reduced the diameter by 1/9 and squared the result this method has probably found by experience in terms of the ratio the area turns out to be 256 over 81 R squared so that pie is about 256 over 81 which is about three point one six zero an upper estimate was also within 1% of the true value so there are some good values known even four thousand years ago a very convenient but much less accurate value appeared about a thousand years later in the Old Testament in 1 Kings and 2 Corinthians we learned that theorem a worker in bronze made a molten see with diameter 10 cubits in circumference 30 cubits giving Pike was 3 but a better better method of finding pi was introduced by the Greeks and would be used for almost 2,000 years often attributed to to Archimedes it actually dates back to the 5th century BC when the Greeks office antiphon and Bryson approximated a circle by regular polygons and tried to get better and better better better and better estimates by repeatedly doubling the number of size until the polygons eventually became the circle so antiphon began with the square inside a unit circle and found its area to be 2 so pi is greater than 2 he then doubled the number of sides to an octagon with area 2 root 2 or 2 point 8 to 8 Bryson's approached similar except that he also considered polygons outside the circle giving up a balance of four for the square and 3.3 tuned for the octagon not very good but now let's double the number of size so Archimedes adopted the same idea 200 years later but unlike antiphon and Bryson he worked with perimeters rather than areas starting with hexagons inside and outside the circle he then doubled the number of size of the polygons to 12 24 48 and 96 getting his balance with polygons with 96 size as three and ten seventy first and three and one seventh and 22 over 7 and this gives pie to two to two decimal places what was happening elsewhere in China around the year 263 Leo hue also used polygons to approximate pie starting with hexagons and dodecagon z-- he found simple methods for calculating the successive areas and pyramids whenever he doubled the number of sides and for polygons with 192 size of twice 96 he obtains bounds of about three point one four four more doublings led to polygons with 3072 sides and two pi is 3.14159 even more impressively around the Year 500 souchong G and his son doubled the number of size three more times two polygons with over 24,000 sides and obtained pi to six decimal places they also improved Archimedes fractional value of 22 over 7 to the more accurate 355 over 1 1 3 which also gives pi to six decimal places and this latter approximation wasn't rediscovered in Europe for another thousand years after this everyone got in on the game it's the number of size continued to double with kin with corresponding increases in accuracy leading eventually to the remarkable Dutchman Rudolph and : whose polygons had over 500 billion sides giving pie to 20 decimal places and his upper and lower estimates appear here just below his portrait but not content with this he then used polygons with 2 to the power 62 sides to find pie 235 decimal places proud of his results he asked for this latter value to appear on his tombstone in Leiden and for many years pie was known in Germany as the Nadal feein number a new and highly productive method for estimating pie which was used extensively throughout the 18th and 19th centuries involved the inverse tangent function usually written as 10 to the minus 1 x or arctan x this nick spec this next bit gets a little technical but hang in there it won't last long in particular I'll use Radian measure for angles where pi represents 180 degrees well if Y is 10 is 10 X then X is 10 to the minus 1 Y for example tan of pi so tan of theta from the triangle is a over B theta is 10 to the minus 1 a over B so in particular 10 of pi by 4 that's 10 45 degrees is 1 so 10 to the minus 1 of 1 is PI over 4 and tan of PI over 6 that's time 30 is 1 over 3 1 1 over root 3 so 10 to the minus 1 of 1 over root 3 is PI over 6 you're just turning things around well we can combine different values of 10 to the minus 1 for example when you add 10 to the minus 1/2 10 to the minus 1 or third you get PI over 4 that's 45 degrees and you can see this from the picture on the left and you can also prove it by simple geometry and in general we can combine any two inverse tangents by using the formula down below now some of you will know many functions can be written as infinite series for example we can write 10 to the minus 1 X as the infinite series shown here with only odd powers of X X X cubed s the fifth and with odd numbers as denominators 1 3 5 7 this result was already known in 15th century India but it's usually named after the Scotsman James Gregory shown here who rediscovered it 300 years later if we now let X equals 1 we get a serious expression for pi over 4 a result also known in India but usually credited to Leibniz this result pi over 4 is 1 minus 1/3 plus 1/5 minus 1/7 and so on this is one of the most remarkable results in the whole of mathematics by simply adding and subtracting numbers of the form 1 over n we get a result involving the circle number pi what of circles got to do with it unfortunately this series converges so slowly that we cannot use it to find pi in practice for example if you compute the first 300 terms of the series and get pi to only 2 decimal places and if you want 5 places you have to calculate the first half a million terms but we can use Gregory's series to estimate pi if we substitute values either other than 1 do you remember that 10 to the minus 1/2 and central minus one-third add up to pi over 4 so we can substitute X as 1/2 and X is 1/3 into the series for 10x giving the two series below and because of the increasing powers of 2 and 3 and the denominators needs to converge much faster giving good estimates for pi indeed in 1861 a gentle from a gentleman from Potsdam use these very series to find pi to 261 decimal places other other times the minus 1 results where the series converge even faster in 1706 the Englishman John Machin later incidentally aggression professor aggression professor of astronomy for many years used the addition formula to prove that pi is 16 times 10 to the minus 1/5 minus 4 times 10 to the minus 1 1 over 2 3 9 don't worry how about how he got it but he just used that formula over and over again and then he wrote out the 210 2-1 series shown here now these series converge rapidly because of the powers of 5 and 239 in the denominators for example if you only take three terms you already get the value 3.14 another advantage is that 5 is an easy number to divide by him so that Mason was able to calculate pi by hand to a hundred decimal places a great improvement on anything that went before 17:06 was a good year for pi as well as matrons result a Welsh math teacher caught william jones wrote a new introduction to the mathematics in which he introduced for the first time the symbol pi for measuring circles here are two act extracts from the book above you can see Mason's series and on the line below it is the first ever appearance of Pi that's the first time pi ever appeared and below in the lower frame is matrons value for pi in fall given them they're described as true two above a hundred places as computed by the accurate and ready pen of the truly ingenious mr. John Mason no wonder he became aggression professor such results can be used to obtain many different values for pi the most notorious of all is the one obtained by William shanks who in 1873 used Mason's formula to calculate PI to an impressive 707 decimal places these were later inscribed in a ceiling frieze in the PI room of the palace of discovery in Paris where they can still be seen unfortunately for him and for the palace it was later found that only the first 527 of these decimal places are correct let's now look at a very different way to find pi in 1777 the Court did perform describe an experiment for estimating it suppose you throw a large number of needles or matchsticks of length L onto a grid of parallel lines of the distance D apart and record the proportion of needles the cross a line it's not difficult to show that this proportion is 2 over pi times L over D from which we can calculate a value for pi for example here L over D is 4 over 5 and exactly 5 over 10 needles cross lines if you do the calculations you'll find that this cos pi is 3 over 2 3.2 which isn't too bad for only 10 needles incidentally in 1901 an Italian mathematician called Maria Lazzarini carried out such an experiment in which L over D was 5 over 6 performing 3408 trials and claiming 1800 and 8 crossings this gave pi is 355 over 113 which as we've seen is correct to 6 decimal places he was lucky if just one needle had landed differently his result would have been correct to only 2 decimal places in 1897 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 House bill attempted to legislate an incorrect value for pie provided by a local physician who would then allow the state to use the state to use this 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 often as a contribution to education to be used only in the state of Indiana free of cost by paying any royalties whatever on the same according to the proposed of the ratio of the diameter and circumference is 5/4 to 4 which gives PI's 3.2 for some reason the bill was then passed on to the House Committee on swamplands who in turn passes on to the committee of education it has been found as a circular area is to the quadrant of the circumference as the area of the equilateral rectangle is some square on one side well this makes little sense but even so it proceeded to the Committee on temperance who recommended its passage not knowing any better fortunately a mathematician from Purdue University happened to be visit the Statehouse on the very day 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 it's still with the Committee on temperance the 20th century saw a number of discoveries about Piatt many of them completely bizarre here are three of them are just mentioned briefly in 1940 in the indian mathematician Ramanujan found several remarkable exact formulas for one over pi including this one an infinite series in which strange numbers such as 11 103 and 26,000 390 seem to appear from nowhere such series converge extremely rapidly and form the basis of some of today's fastest algorithms for calculating pi and many years later in 1989 David and Gregory Chernov ski of New York produced a similar but even more complicated results with even larger numbers as you can see my third example is simpler but it caused much surprise its importance is that if we work in base 16 rather than in base 10 we can calculate each digit of pi one at a time without having to recalculate all the priests at preceding digits first I'd like to end my discussion of PI with a simple puzzle that appeared in 1702 in a book on Euclid's elements by the Cambridge mathematician William Weston if you haven't seen it before you may find its answer surprising the circumference of the earth is about 25,000 miles 132 million feet assuming there has to be a perfect sphere supposing we tie a piece of string of this great length tightly around it don't do this at home we then extend this string by about just six feet to 52 PI feet and prop it up equally all around the earth how high above the ground is the string most people think the resulting gap must be extremely small perhaps a tiny fraction of an inch but the answer is 1 foot and in fact you get the same answer whether you tie the string around the earth or tennis ball or any other sphere through the sphere has radius R feet then the original string has length 2 pi R when we extend it by 2 pi feet the new circumference is 2 PI R + 2 pi which is 2 pi times R plus 1 so the new radiuses are R + 1 1 foot more than before isn't that nice let's now move on to our next number the exponential number e 2.71828 etc 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 we know first that like pi the decimal expansion of e goes on forever the letter e first appeared in print in 1736 in in my book mechanika a book on the mathematics of motion and here it is in the penultimate line where e denotes the number whose hyperbolic logarithm is 1 that's the first appearance of e to indicate what we mean by exponential growth here's a story about the invention of the game of chess the wealthy king of a certain country was so impressed by this new game that he offered the wise man who invented it any reward he wished to which the wise man replied my prizes few to give me one grain of wheat for the first square of the chessboard 2 grains of the second square four grains 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 asked for such a tiny reward also he believed until his treasures calculated the total number of grains of wheat you can see however the number is growing quite fast and the total number of grains of wheat works out a 2 to the power 64 minus 1 that's enough wheat to form a pile size of Mount Everest placed end to end the grains had reached the nearest star Alpha Centauri and tour I and back again let's see how quickly other sequences can grow a very simple form of growth is linear growth as in the counting numbers 1 2 3 4 5 somewhat somewhat quicker is the way the perfect squares n Square to grow 1 4 9 16 25 and even faster is that of the queue in cute and these are all examples of polynomial growth because they involve evolved powers of n but alternatively we could look at powers of two or any other number as we saw in our chessboard story the sequence two to the N of powers of two starts off fairly slowly but soon gathers pace because each successive term is twice the previous one and the sequence 3 to the N 3 to the N of powers of three grows even more quickly these are examples of exponential growth where n appears as exponent to compare these types of growth let's calculate the running times of some polynomials and Exponential's when n is 10 30 and 50 for a computer performing a million operations per second for polynomial growth such as n cubed such a computer would take about one eighth of a second when n is 50 but exponential growth as such as 2 3 n 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 billions of years for 3 to the N so in the long run exponential growth tends to exceed polynomial growth often by a huge margin algorithms that run in polynomial time are generally thought to be efficient whereas those that run in exponential time normally take much longer to implement and the regarded as inefficient returning to e what exactly is this number and how did it arise 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 how does it grow and 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 simple let's see what happens if we invest one pound at the unlikely and you're 100% after one year our pound has doubles the two pounds but if we calculate the interest twice a year that is 50% each time then after six months our one pounders be multiplied by one and a half to give one one pound fifty and after a year that amount is multiplied by another one and a half to give two pounds twenty-five which is a bit more than before now let's calculate the interest every three months then there are four periods and after each one the amounts multiplied by one and a quarter first - one pound 25 and - about 1 pound 56 in 195 and then by the end of the year - about 2.2 pounds 44 which is 1 pound times 1 and 1/4 to the power 4 notice that these final amounts are increasing so as the periods get shorter what happens to them - they increase without bound or do they settle down to a limiting value well the results are shown in this table here to 5 decimal places and to find them notice of if the year is divided into n periods then after each period the amount is multiplied by 1 plus 1 over n so the amount at the end of the year is 1 plus 1 over N to the N we also see from this table the bottom row that has n increases these numbers tend to a limiting value that corresponds to when the interest is calculated continuously and this limiting amount is 2.71828 is our exponential number e now the greatest advances in understanding Exponential's were made in the early 18th century after Bernoulli the main figure in the story was myself who investigated the number E and the related exponential function e to the X and in 1748 my introduction to the analysis of infinite if I may say so one of the most important mathematics books ever written brought together many of my results from earlier here are some of my main findings we've just seen that e is the limit of the numbers one plus one over N to the N as n increases indefinitely and similarly you can show that e to the X is the limit of 1 plus X over N to the N for any number X but as Isaac Newton had already discovered the number is also the sum of the infinite series shown here where the denominators the factorials one factor is 1 2 factorial is 2 times 1 3 factorial 3 times 2 times 1 or 6 4 factorial is 24 and so on and more generally there's a similar series for e to the X which converges for all values of X and in fact these series converges to C extremely quickly because the factorials on the bottom increased so rapidly for example just the first 10 terms of the series for e already gives the correct value to 5 decimal places well I discovered all of these things and put them into my introduction and also on the right I've got the interesting fact here you can see the graph of y equals e to the X and one of its most important features is that at each point X the slope of the graph the steepness of the graph is also e to the X so the slope at any point is the Y value so the curve becomes steeper and steeper as x increases and that's what we mean by exponential growth so we'll end our discussion of Exponential's by returning to exponential growth and in 1798 Thomas Malthus wrote his essay on population week where he contrasted the steady linear growth of food supplies with the exponential growth in population he concluded that however one may make hope in the short-term exponential growth would win in the long term and there be severe food shortages a conclusion that was borne out in practice so how fast as the population grow it's a bit of calculus calculus needed here but it's quite quick so if n of T is the size of a population of time T and if the population grows at a fixed rate K proportional to its size then we have the differential equation DN / DT that's the slope as K times n which can be solved to give n is n 0 times e to the KT where n 0 is the initial population so you know the initial population and then it rises according to that formula an example of exponential growth 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 we come now to the last of our constants before we try to combine them together this is the imaginary square root of -1 which arose back in the 16th century when Cardno one of the Italian mathematicians who first solve cubic equations was trying to solve a number puzzle divide 10 into 2 parts whose product is 40 to solve this he let the two numbers be X + 10 - X he didn't actually have algebraic notation but essentially this is what he did he let the numbers be X + 10 - X so they add up to 10 and also x times 10 minus X is 40 solving this quadratic equation he found the solutions to be 5 plus the square root of minus 15 and 5 minus the square root of minus 15 which seemed meaningless commenting that nevertheless we will operate putting aside the mental tortures involved he checked that the answers worked and he but he was led to complain that so progresses arithmetic subtlety the end of which is as refined as it is useless is it useless let's see well trying to take the square root of a negative number doesn't seem to make sense for 1 times 1 is 1 and minus 1 times minus 1 is 1 and as the Victorian August's Tim Walker and mark 300 years later we have shown the same squirts of - a to be void of meaning or rather self contradictory and absurd while his contemporary the astronomer George Airy commented I have not the smallest confidence in any results which is essentially obtained by the use of imaginary symbols however Leibniz was more encouraging claiming that the imaginary numbers are a wonderful flight of God's Spirit they are almost an amphibian between being and not being to my shame even I who came to use them so effectively criticized them well for many purposes our ordinary numbers are enough and suppose we now agree to allow this mysterious object called the square root of minus 1 or I as I named it we can then form many more numbers such as 1 plus 3i and 2 plus I ignoring it for the moment what these actually mean we can carry out simple calculations with them adding is straightforward we just add the bits without the I and a plus C and the bits with I separately and so it's multiplying as long as we remember to replace I squared wherever it appears by minus 1 we can also represent these numbers geometrically this was first done by Casper vessel of Norway and later by Gauss and by jean-robert argon or Argand it's often called the Gaussian plane or Argand diagram but neither name is historically correct one should really call it the complex plane so we represent each complex number a + bi by the point with coordinates a B so the first picture here shows four points such as 1 plus 2i + 3 + I so represented we can then add two complex numbers together by using the parallelogram rule on the right and so as before 1 plus 3i plus 2 plus I is the other end of the parallelogram 3 plus 4i so multiplied by I we simply rotate through 90 degrees and the telephone operator said to me the other day the number you have dialed is purely imaginary please rotate your phone through 90 degrees and try again doing this again when multiplying by I squared or minus 1 that's just a rotation through 180 degrees as we've seen there's much suspicion in Victorian times about these imaginary numbers the Irish mathematician and astronomer William Rowan Hamilton largely ended a suspicion by defining the complex numbers as pairs a B of real numbers pairs which combined together according to the particular rules shown here now complet we've just seen that complex numbers can be represented in the plane so a natural question is can we extend the idea to three dimensions with the numbers of the form a + bi + CJ where I squared + J squared are both minus 1 and it turns out that addition is still okay but multiplication isn't because some of the terms have I times J and it's not it's all clear what that should be Hamilton tried all sorts of poles of possibilities and nothing worked 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 where - I was always obliged to reply with a sad shake of the head no I can only Adam subtract them they got some dealing with their cornflakes well after struggling with these triplets for many years Hamilton had his moment of glory on the 16th of October 1843 during a walk with his wife along Dublin's royal canal when as he said and the electric current seemed to close and a spark splashed forth I pulled out on the spot a pocketbook and made an entry there and then nor can I resist the impulse impulse to cut with a knife on a stone on broom bridge as we passed it the fundamental formula with the symbols I J and K name the ice I squared is minus 1 as this J squared and K squared and as is IJ K and these are the quaternions of four terms a + bi + CJ + DK here I squared J squared and K squared all minus 1 but multiplication is no longer commutative now our multiplication is commutative you can do it either way around 3 times 4 is the same as 4 times 3 but here that doesn't work I times J is minus J times I JK is minus KJ ki is minus I K it's a completely different number system anyway you can now multiply any two quaternions as long as you stick to these rules well now we've got our five constants let's return to my equation and my identity and for the last few minutes let's talk about that so recall that my identity connects the exponential function which goes shooting off to infinity with the functions cosine and sine which oscillate between 1 and minus 1 as you can see at the top to find this connection as some of you will know these functions can all be expanded as infinite series which are valid for all values of X so here you can see the series for e to the X cos x and sine x what happens if we now allow ourselves to introduce the complex number I the square root of -1 as I did in 1737 so you take the above series for e to the X and replace X everywhere by IX as shown here so e to the X is 1 plus IX over 1 factorial plus IX all squared over 2 factorial plus IX or CUDA over 3 factorial and but I squared is minus one so I cubed is minus I I to the forces is 1 and so on and if you sort it all out and collect terms you what you get it's a series for cosine plus I times the series for sine X that is e to the i-x is cos x plus I sine X which is my identity one of the most remarkable equations in the whole of mathematics beautifully connecting these seemingly unrelated functions all things to complex numbers I actually gave more than one proof of my identity here's part of a different approach this is from the introduction that I showed earlier and here I made use of so-called infinitesimal x' so this is the proof that appeared in the introductory in 1748 my identity appears in the penultimate line is e to the V times the square root of -1 that's e IV it's cos v plus square root of minus 1 times sine V this is the first appearance of my identity as I myself commented at the time from these equations we can understand how complex Exponential's can be expressed by real sines and cosines from my identity e to the i-x is cos x plus i sine x we can deduce my equation we simply let X is PI which is the Radian form of 180 degrees well cos of Pi cos of 180 is minus 1 sine of pi sine 180 is 0 and so we just get e to the I pi is minus 1 or e to the I pi plus 1 equals 0 and although I certainly made this is this just this deduction for some reason I never wrote it down explicitly in any of my published works I don't know why can we illustrate my equation pictorial in 1959 the English schoolteacher L WH holler showed how to do so take the power series for e to the X and put X's I pi giving one plus I PI over 1 factorial plus I PI squared over 2 factorial and so on which simplifies to 1 plus I PI minus PI squared over 2 minus I PI cubed over 6 and so on he then traced these terms on the complex plane as you can see on the right he starts at the point 1 on the right add I PI that means go upwards then you subtract 1/2 pi PI squared and subtract PI over 6 PI cubed add of 24th PI to the fourth and so on and this produces a spiral path which converges eventually to the sum of the series which is just e to the I pi which is minus 1 well although we call this result Oilers equation it was nearly discovered a few years earlier by Johann Bernoulli in 1702 Bernoulli was investigating the area of a sector of a circle of radius a that's the shaded area on the right between the x-axis and the line from the origin to the point XY and he found its area it found this area to be the expression you've got there a squared over 4 I times log of X plus I Y over X minus I Y well leaving but quite aside what's meant by the logarithm of a complex number I later observed that when X is zero this formula simplifies to a squared over 4 for I times the log of minus 1 that log thing that log term becomes log of minus 1 but the sector as you can see from the picture is now a quarter circle with area PI a squared over 4 and so you have the a squared over 4 I times log of minus 1 is PI a squared over 4 and if you do some canceling you get log of -1 is I times pi that's how you find the log of a negative number well although I wrote down this last result explicitly I didn't take exponential so to deduce my equation that form e to the I pi is minus 1 indeed I often credited Bernoulli with with discovering the spelling for log of minus 1 another near-miss arose from the work of the Cambridge mathematician Roger Coates who worked closely with Isaac Newton on the second edition of the Principia Mathematica and he was the one that's credited with introducing Radian measure for angles around 1712 he was investigating the surface area of an ellipsoid obtained by rotating an ellipse around an axis the details are complicated but he managed to find two mathematical expressions for the area one involving logarithms and the other involving trigonometry and both involving an angle Phi he first proved that the surface area is a certain multiple of log of cos Phi plus I sine Phi and in a different way he proved it to be the same multiple of I I Phi equating these results gives the results you have here which connects logarithms with the trigonometric no metric functions if he'd taken exponential he'd have found my identity in the form e to the I Phi is cos Phi plus I sine Phi but he didn't another near miss so to end with what shall recall the equation e to the I PI plus 1 equals 0 we've just seen how it follows the results of Johann Bernoulli and Roger Cotes but the neither of them seems to have done so even I never wrote it down explicitly that I certainly realized that it follows immediately from my identity e to the i-x is cos x plus I sine X in fact we don't know who first stated the equation explicitly though there's an early appearance in a French Journal of 1813 214 but almost everybody nowadays attributes the results of me so we're surely justified in naming into Euler's equation - one of the achievements of myself who's been called a truly great mathematical pioneer a word that if I same honestly describes me so well and which appropriately includes among its letters five constants pi I 0 1 and E thank you very much [Applause]

Channel: Gresham College

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Id: 1VL2fl1_wB8

Length: 60min 41sec (3641 seconds)

Published: Mon Feb 26 2018