A Tribute to Euler - William Dunham

so without further ado I'd like to introduce bill Dunham who will speak about boiler well thank you Jim pleasure to be here this evening to talk about Euler my favorite mathematician someone who's on everybody's shortlist of one of the greatest mathematicians of all time so here he is that is Euler the official title of this portrait is the haand Minh portrait after the artist who did it the unofficial title is Euler emerging from the shower you know if the portrait painter were coming over I don't think I'd wear this but that's what he decided to wear here's what I want to do tonight break this talk into three parts give a brief biography so you have some dates to hang his life on give a survey a broad survey of half a dozen or so of his great achievements but without the mathematical detail but just to give you a sense of the breadth and the depth of his legacy and third I actually want to show you an oil Arian proof with all the mathematical detail involved so we can look over his shoulder and watch him work now before you get scared the prerequisites to this are just high school math you don't have to be a Harvard graduate student to understand the mathematics but I predict that people are going to leave here impressed with what he could do when he got going so that's the goal so it's a busy agenda let's start with the biography and here it is in a nutshell Euler was born in 1707 in Basel Switzerland last year was his tercentenary people were celebrating all over the world in 1720 this very bright young lad was off to study with Johann Bernoulli a name you might know one of the members of the great Bernoulli family Bernoulli was in Basel and Johann would go I mean Oyler would go and study with him barbar newly gave him direction gave him guidance gave him mentorship in 1722 Euler graduated from the University of Basel and if you get out your calculator you'll see he was 15 at the time pretty good in 1727 he's off to st. Petersburg to the Academy at st. Petersburg and remember in those days the great courts of Europe were surrounded by these academies there was the Royal Society in London the Paris Academy the berlin Academy and Russia wanted its own so there was this new Academy in st. Petersburg and Euler went there to help give lustre to the to the court to the monarch to the nation he stays there until 1741 when he goes to the Berlin Academy which was then being run by Frederick the Great and he plopped down there it was like a free-agent went to another team stayed there until 1766 when he's backed the st. Petersburg and stayed in st. Petersburg until his death in 1783 Euler was buried in st. Petersburg subsequently he was buried in Leningrad and today he's buried in st. Petersburg so now these dates make him almost the exact contemporary of Benjamin Franklin so if you want to sort of compare him to someone from this side of the ocean think Ben Franklin is there almost exactly the same lifespan on the personal side he was married he and wife Katharina had 13 children but unfortunately with the child mortality being what it was in the 18th century only five of these children made it to adolescence so it would have been a great deal of heartache in the euler family he had by all accounts a phenomenal memory euler could memorize anything books plays tables you know most people have to look up logarithms but if you can memorize and that makes it a lot quicker and this memory would serve him well when physical ailments hit in 1730's he lost vision in his right eye apparently an infection got loose something we could probably cure with just an antibiotic but it not only cost him his destroyed his eyeball his hole I was was deformed and so he just had no use of his right eye so he his productivity goes down right Ron doesn't bother him he keeps going and continues until 1771 when he lost vision in his left eye this was a cataract 1771 they tried a surgery and you don't even want to think about eye surgery in 1771 it was painful and it failed and so Euler is essentially blind so his productivity goes down right Ron it goes up he actually gets more productive when he can't see what he would do is he would go into a room there'd be a table of scribes and he would just start spouting his mathematical papers and they'd be writing furiously Euler could think up mathematics faster than most people can write it if a new paper came in and he wanted to know what was going on with lagron then read it to him he would hear it with this incredible memory he could see it in his mind's eye and absorb it very well thus is the great inspirational story from the history of mathematics he's someone who is confronted with a disability and doesn't let it beat him he is the Beethoven of mathematics you know Beethoven composed music he never heard Oyler was producing mathematics he couldn't see and if you don't believe me that he was productive even though he lost his vision let me just indicate that in 1775 he produced over 50 papers and he was blind so Euler was incredibly productive even with the blindness this idea of producing a lot of papers feeds into the next slide his mathematics is distinguished by its quality and its quantity really distinguished both of these are superior beyond imagining let me address the quantity first nobody ever did more mathematics than Euler his output was phenomenal they tried to publish it all in 1911 the Swiss Academy of Sciences says we're going to publish oilers collected works the first volume is this big big heavy book right is heavy to carry around well in 1911 that came out another volume in 1912 1913 they're still coming they're not done it's 97 years later and Euler's work is still coming out at the moment the work called the opera omnia has 75 volumes this guy has 74 cousins all this mass and 25,000 pages of mathematics and nobody's quite sure when they're going to finish this I mean it's going to be deep into the 21st century the grandchildren of the original editors of this are old and and it's still coming out so the quantity of his work is breathtaking here's one other indication of that after he died there was a there was a backlog you know there were papers still in the chain paper still in his desk it took decades to clear the backlog and after he was dead he published 228 papers now trust me that's more than most living people could publish Euler published more mathematics dead than anybody else so the quantity of his work top-notch but the quality of his work was just as good you know if he done a lot of stuff and it was just drivel it wouldn't we wouldn't be remembering him here tonight the quant the quality of his work is extraordinary and although that's a little harder to measure maybe so what I did was this I went to the online dictionary called math world and I type in Euler and then any term in mathematics that carries his name came back as a hip and you know if it's in the dictionary it's really important something that's not just an insignificant little result but something so great that it's got his name on it it's in the dictionary so I typed in Euler and I got 96 entries 96 things in mathematics are named after him from the Euler line to the Euler identity to the Euler product some formula most branches of mathematics have something named after him and just for comparison's sake gauss 7ko she was a 33 and some people fare even less well so you know this guy was incredible he's so incredible he even has his own comic book there's an euler comic book that the birkhäuser put out a graphic novel about his life so how many I mean how cool is that right okay well so that's oil or now I want to start showing you some of the things he did that made him famous here's one that really is important the number II 1748 oil and here it is there's the document in which II appears and you see it there in the middle of the page he says for the sake of brevity we will use the letter e to stand for the number two point seven one eight two eight one eight two eight four five nine Euler loved calculating long strings of decimals so there it is he says which therefore denotes the base of the natural or hyperbolic logarithm so he knows he is important he knows it's the base of the natural logarithm and you want to calculate it at home there's how you do it down there at the bottom one plus one over one plus one over one times 2 plus 1 over one times two times three that converges very fast and you know if you know the series expansion of e to the X and you put in 1 sure enough that's it so oiler gave us e if he had done nothing else that would warrant a sentence in the hip math history books but but he did so much else here's one same year Euler's identity Wow hope everyone's seen this e to the IX equals cosine X plus I sine X a very amazing formula strange what connection between the exponential function and the trigonometric functions between real numbers and complex numbers all in one fabulous formula here is how he published it it's from his work e to the he wasn't using I so he wrote e to the plus V times the square root of -1 is equal to cos period V that's because this was an abbreviation for cosine was so you had to put a period there in those days plus the square root of minus 1 times the sine of e so there it is in 1740 now anytime anybody shows you boil errs identity they do the next thing you kind of have to math teachers are required you let in which case you get e to the I pi equals cosine PI plus I sine PI sine of PI 0 cosine PI is negative 1 e to the I PI is negative 1 bring the one over and you get this amazing equation e to the I pi plus 1 equals 0 falls right out of Euler's identity now math teachers at this point will always say the following they will say this is an incredible link among the most important numbers in mathematics if you are going to have a party and you wanted to invite the 5 most important numbers to your party you know who would you invite we invite 0 right the additive identity you'd invite 1 the multiplicative identity if you want to do calculus you invite e if you want to do geometry you invite PI if you want to do complex numbers you invite i-10 EEI pi the dream team of numbers right there and they're all in that one equation it's amazing it's amazing that it's amazing that there is an equation that links them actually and it's amazing that it's such a simple consequence of Oilers identity back in 1988 the journal mathematical Intelligencer did a survey of the world mathematical community asking people to vote for the most beautiful result in mathematics and you send in your you know your postcard what's the most beautiful result it was American Idol for math formulas of all the mathematics ever done this one is the most beautiful result ever and it's from Euler it's so beautiful that it inspired someone to write a poem which I shall now inflict upon you somebody wrote e to the I pi plus 1 equals 0 made the mathematician Euler a hero from real to complex with our brains in great flex he led us with zest but no fear so okay so that's something else Euler did how about this Oilers polyhedral formula 17:52 he tells us that V plus F equals Z plus 2 we're we're talking about polyhedra you know solid figures with polygons as faces like a cube and where V is the number of vertices F the number of faces a the number of edges and this amazing relationship holds like for the my cube V the number of vertices is 8 right four corners top and bottom F faces is six think of a dice or Die 8 and 6 is 14 number of edges for around the top for around the bottom for vertical 12 plus 2 bingo it works but when Euler realizes is it works not just for cubes but it works for icosahedron Akasa Hedra dodecahedra and a wide wide collection of solid bodies it's this amazing link in that math intelligencer survey the most beautiful results in history this was number two so Euler has the top two slots you know Otto himself and he himself was a little more circumspect about this he wrote I find it surprising that these general results in solid geometry have not previously been noticed by anyone so far as I am aware you know kind of surprised something this simple had escaped notice but of course there haven't been annoy lurvy for to notice it so that's pretty good what else well there's the Basel problem so the years is 1734 when Euler solves this great challenge and here's what it is back in 1689 Jakob Bernoulli who was Johann Bernoulli's brother Johann being euler's mentor challenged the mathematical community to find the exact sum woopsy got to go ever hear of this infinite series and he issued the challenge from Basel where Jakob was so it became known as the Basel problem what's the exact answer what is 1 plus 1/4 plus 1/9 plus 1/16 see what we're doing we're taking the whole numbers squaring them inverting them adding forever infinite series what's it come out to be well Jakob couldn't figure it out johann bernoulli couldn't figure it out Leibniz couldn't figure it out it was hard nobody could figure it out they knew it converged they knew it converge to something less than two but they wanted the exact answer and the first person to get that exact answer would be famous and that person was Euler and the answer you know the answer pi squared over 6 that is a strange answer right that is in fact correct and Euler gave at least four different proofs in his career that it really is PI squared over 6 one way to fill up 75 volumes is to keep doing the same thing multiple ways amazing result in the math and telogen sir pull of the most beautiful formulas of all time this was number 5 so oiler has topped the charts like the Beatles used to do when I was a kid you know he had 1 2 and 5 all right what else well the bridges of königsberg probably you note this one this was sort of famous this is a picture from one of euler's papers showing the town of canings burg with the river flowing through dividing around that island that a and then splitting and going around the landmass at D and you can see the little bridges connecting various pieces of land now the story is that the good burgers of canings burg on Sunday afternoon would take a stroll and the goal was to walk around here and cross each bridge once and only once what we now call an Euler path but they couldn't every time they tried it they either missed a bridge or they found themselves crossing a bridge I'd already crossed so they're perplexed you know what's going on here and so they asked the mayor I don't know why but the mayor of kenigsberg and he didn't know of course so he wrote to Euler and he says can you explain this you know is this possible and Euler proves that it isn't this particular configuration you cannot walk around here and cross each bridge once and only once and then Euler shows when you could do it he drew some other neat little pictures with complicated bridge arrangements which would work and nowadays we look at this and see this as the first instance of what's now called graph theory when you have vertices or nodes connected with edges the landmasses or the vertices the edges of the bridges and you know a whole subject is getting born here as Euler approaches this problem now he himself a little more circumspect he said this solution bears little relationship to mathematics and I do not understand why to expect the mathematician to produce it rather than anyone else for the solution is based on logic alone he saw it as like a puzzle and he didn't think it was math he didn't need integrals or anything now don't tell your graph theory friends that he said this ok actually if he were to come back today and see what has happened to graph theory I'm sure he would recognize it as mathematics all right anything else well yeah how about geometry plane geometry now you would think that that had all been discovered you know by the time wheeler got here Euclid had done all his geometry Archimedes was there anything left to discover about triangles well actually for big fat volumes of Euler's collective works or geometry he discovered lots of stuff and one of the things he does is the Euler line and you may or may not have ever seen this but if you have a triangle any triangle it could be isosceles it could be right but it needn't be anything special you can consider the intersection of the altitudes remember the altitudes go from each vertex perpendicular to the opposite side and they all go through a point which is called the orthocenter you can consider the intersection of the medians now the median goes from each vertex to the middle of the opposite side and those are concurrent they all go through a single point called the centroid and you can consider the intersection of the perpendicular bisectors so if you take each side take the bisector and drop a perpendicular those meeting the point which is called the circumcenter it's the center of the inscribed circle these points were known to the Greeks these aren't special or anything they've been around a long time now I'm going to try to do this with my powerpoint let's hope this works there's a triangle I want to look at the intersection of the altitudes from each vertex perpendicular to the opposite side so there's my altitudes kind of perpendicular and they all meet in that point that you are the center okay so we'll leave that there and get rid of that stuff okay now I want to get the intersection of the medians from each vertex to the middle of the opposite side so we want to do the medians and here they come from the vertex to the middle of the opposite side and they meet the point yeah get rid of that and then I want the intersection of the perpendicular bisectors so here comes there we go perfectly clear bisectors split each side put up the perpendicular they meet at a point called the circumcenter and what do you see they line up no one had seen that Euler proves that no matter what the triangle these three points will fall on a line which is now in his honor called the Euler line it was an amazing piece of geometry that had escaped all the great geometers in the past and Euler shared something else the ratio of those two segments is always one the two exactly that the centroid is half as far from the circumcenter as it is from the orthocenter it's a neat piece of geometry in the subsequent century lots of interest is devoted to plane geometry and I think part of it is because Euler thought this was important enough and he found interesting things and so you see a renaissance of geometry in the 19th century okay what else well how about number theory now Euler was one of the great number theorist for big fat volumes of number theory I want to show you something I like this is not the most important thing he did in number theory the you know what number theory is it's the study of whole numbers primes composites that sort of thing this isn't the most important thing he did but it's just kind of neat definition that had been around that said two whole numbers are amicable friendly if each is the sum of the proper whole number divisors of the other and needs an example okay so how about an example how about 220 and 284 if you ever see this this will be the example you see I promise you so what I want to do is look at the proper divisors of 220 by which I mean whole numbers that divide evenly into 220 but aren't 220 that's what the proper means it's less so if you collect them all there they are 1 2 4 5 10 11 20 22 44 55 and 110 that's all the proper divisors the 220 has and you add them up and trust me you get to 84 hmm now you take the proper divisors of 284 all the whole numbers that divide into 284 1 to 471 142 284 does but we don't count that it's my proper add those up and you get 220 each of these is the sum of the proper divisors of the other it's totally useless but but it's intriguing right if you're if you're a number theorist you will be intrigued by this strange reciprocity I have a friend this is the truth when he married the love of his life he gave her a keychain with a number 220 on it and his has to 84 to represent their amicability their eternal friendship it's really sweet nerdy but sweet right ok now here comes a brief history of amicable numbers it's a short history the Greeks knew that pair 220 and 284 they knew it somehow and remember the Greeks of the Pythagorean philosophy that underlay Greek mathematics sort of exalted the role of whole numbers sort of these metaphysical entities so the Greeks thought this was really spectacular and they sought another pair and they could find no others they couldn't find any more that's all nobody finds anymore until the ninth century Islamic mathematician tobot even Kura who finds a rule that generates two more pairs so there was the Greek pair and he found two more however this rule apparently didn't make it to Europe after the Renaissance so the folks in Europe didn't know that tobot had been down his path and so they thought there was just one pair and in 1636 the French mathematician Fermo finds another pair it's those two seventeen thousand two hundred ninety six and eighteen thousand four hundred sixteen the sum of the proper divisors of seventeen thousand two hundred ninety six is eighteen thousand four hundred sixteen and vice versa exercise for the reader you know you can check is it guess what that was one of table it's numbers actually thermo had gone down the same path he didn't know it in 17th century France there was a great mathematical rivalry Fermo had an opponent Descartes they hated each other and here is Fermo finding a pair so now the cart has to find a pair to prove his worth and he goes to work on it and in 1638 he finds that pair if you if you really loved your wife you'd give her a keychain with that number guess what that's the other one of table it's numbers these three are the low-hanging fruit of amicable numbers and so they the card in Fairmont we're just going down a previously trodden path so now it's 17 in 1638 three pairs you're known a hundred years later three pairs are known that's it these are very hard to come by when Euler's alive only those three we're known and so he thought well maybe I should give this some thought and he worked on it and he found 58 in a 1750 paper he finds 58 pairs so the world supply went from 3 to 61 with one person now this is what our would do to a problem he'd just blow it out of the water he increased the world supply of amicable numbers by a factor of 20 how'd he do it he saw a pattern that no one had seen and it allowed him to just sort of generate these things willy-nilly great insight um oiler also did Applied Mathematics in fact more than half of his 75 volumes are in things like mechanics optics acoustics I'm not talking about that tonight but let me just show you this cool little picture I found in one of his papers where he's talking about a pump that he has invented um so he did applied math and then while we're looking at pictures how about this never seen those before Venn diagram that's a van Venn diagram right we all know the little circles to connect ideas no that's not a Venn diagram then lived in the 19th century I took this out of Oilers work in the 18th century and not a Venn diagram we should call this an oiler diver however if we do this then is gone right what else did then do the so you know oiler doesn't need this you know for his credibility so I think we'll let then keep keep his diagram and then before I get to my proof one more thing I just find this intriguing this is not important particularly if it's a curiosity we'll say back in the 18th century that the issue of factoring polynomials was important and the question was could all polynomials be shattered into first and second degree pieces I'm talk about real polynomials and real factors so you know if I gave you X to the fourth minus one-fourth degree you could break it into x squared plus one x squared minus one two quadratics and then further break the x squared plus one X I mean the x squared minus 1 into X plus one X minus one no complex numbers here just real numbers so there's a fourth degree that can be broken into real first and second-degree pieces the question was is this at least theoretically possible for all real polynomials if you know the fundamental theorem of algebra you can see that's what this is in its real incarnation but people didn't know the answer it was up in the air if it if it's true such a decomposition is possible you had a supply proof but it's if it's false you just need one counterexample that can't be factored and guess what Nicolas Bernoulli who was johann son and jakob z' nephew wrote to Euler and said I found a fourth-degree that cannot be factored so that's answers the question this fourth degree polynomial X to the fourth minus four X cubed plus 2x squared plus four X plus four is irreducible it cannot be factored down case closed well wait a minute now Nicolas Bernoulli couldn't factor it but that doesn't mean it can't be factored oilers saw how to do it he broke it into two quadratics this one and this one okay now two issues here first of all that can't be right right look at this these quadratics with square roots embedded within square roots are you telling me those two things multiplied together to give that simple looking fourth degree on the top with just integer coefficients well I confess that one rainy afternoon I multiplied these back together and all the square roots cancel it really works so the one fact that's amazing is this is right this is true the factorization problem from hell right second question how did he do it you know is this finally proof that Euler is from the planet cooze Bane you know that he's not human well no because when he tells you how he did it you see it oh yeah I see that was very clever I could have thought of that you know you say but Euler was Euler so so there you go all right well now we've done our little survey we got time to do a proof and Alerian proof um I think if I don't show you an actual proof with the detail I'm kind of cheating you you know I could stand up here and tell you van Gogh was a great painter but at some point you want to see a picture right I tell you boiler was a great mathematician but at some point I got to show you boiler in action and so here comes the issue at hand is the partitioning of numbers partitioning and let me first of all introduce a little bit of notation here D of n is the number of ways of writing n as the sum of distinct whole numbers D of n is how many ways to split in into the sum of distinct whole numbers well we need an example right every time you get one of these definitions you want to see an example so how about five let's take a look at five so the question is how many ways can I split five up into different sized pieces one of them I'll just take five all by itself we'll call that a decomposition sort of the unitary decomposition how about four and one that's five and they're different how about three and two that's five and they're different anymore we could do two and three but hey that's the same as 3 into we won't count down you could say 6 plus negative 1 is 5 but you don't want negative sir these all have to be whole number so we can't count that one and a half and three and a half is five and they're different but they're not whole numbers I think that's it okay so how many ways can you write five is the sum of distinct pieces I think that's where we were three three ways to do it 5 4 and 1 3 and 2 so we're going to say D of 5 is 3 the number of decompositions in two distinct pieces I'm sorry - now it doesn't be two distinct pieces just distinct pieces ok so there's that now let Oliv n be the number of ways of writing n is the sum of not necessarily distinct odd pieces so the same issue decompose but this time the pieces all have to be odd even if they're repeated all right let's try that example five I could write 5 is 5 all by itself there's one way I could do now these all have to be odd how about 3 + 1 + 1 that's 5 and they're all odd now I repeated the 1 but that's ok for this game and one more what else 1 1 1 1 1 yeah and that's all so how many ways to do this d of 5 is 3 there's 3 ways to do it assume the o of Phi history another example I think we need another example especially since the screen run out here so let's let's try 8 alright so first issue distinct summands break it up into distinct piece 8 well there's 8 all by itself 7 and 1 that's 8 and they're different 6 and 2 bingo 5 and 3 bingo 4 and 4 now because they're not distinct they're different they're the same so I can't do that but I could do 5 and 2 and 1 those are all different that adds up to 8 and I can do 4 and 3 and 1 those are all different and that adds up to 8 and that's it so 1 2 3 4 5 6 ways break eight up so D of 8 is 6 let's do o of 8 let's break it into odd pieces 8 how about 7 and 1 odd pieces how about 5 and 3 yeah that's 8 they're odd how about 5 and 3 ones yeah let's see what else two threes and two ones that's eight one three and five ones yep and eight ones right so there's all the decompositions of eight in the odd pieces how many one two three four five six hmm hmm so what we know is the d of v and o v we're both three they were the same v of eight and all of eight were both six those are the same you know is there something going on here well in 1740 philippe nod a french mathematician wrote to Euler and asked him about partitioning by breaking numbers up like this and within days Euler has sent back a proof that D of n is always o of n for all N and he apologized for the delay caused by the bad eyesight I have been suffering and so this was a turn around almost instantaneous turn around given the mail in the 18th century you know that was that's kind of instantaneous so I want to show you his proof I don't know if you knew this the number of ways of writing a whole number in terms of distinct summands is always the same as the number of writing it in terms of odd some hands and oiler proves this for all integers at once it's one proof fits all so let me show you the proof so here's the theorem for all whole numbers in the event equals of them now his proof is built into three little pieces so we just have to get through these three pieces and we see it first piece he says I'm going to introduce that P of x equals 1 plus x times 1 plus x squared times 1 plus X cubed 1 plus 6 for 1 plus X to the fifth and infinite product of binomials why is he doing this hang on but this is what he wants to do ok so that's P of X now what do you do with this he says let's multiply it out but there's infinitely many terms that's alright he doesn't ever bought it here so we're gonna multiply this out now you know how multiplying works with binomials each binomial contributes one of the two terms to the product what's the constant term going to be here if I multiply this out one right each term there'll be a 1 times a 1 times a 1 so we're going to start with a 1 how about X is how many X's can come out of this well look that first term has an X in it and it can hit all the other ones and there's no other source of a 1 of an X so you're just going to get an X x squared that x squared can hit all the other ones and you'll just get one x squared this is looking kind of boring here but things get a little richer now for X cube there's two ways to do that the X cubed and the third term can hit all the other ones or I can get the x squared times the X to give me a second X cube and I'm going to write it this way just to see where these are coming from the X cubed plus the other guy is coming from the x squared times the X X to the fourth you're going to get two of those the X to the fourth all by itself and the X cubed times the X and one more X to the fifth three of those the X to the five all by itself the four and one the three and two look at those exponents seen them before 5 4 & 1 3 & 2 those are exactly the decompositions of five in two distinct pieces there were three such decompositions there's three X to the fifth and you can see where they're coming from and you see why they have to be distinct because look at P there it has all different powers in the different terms you'll never get a repetition so by thinking through this Euler realize that P of X if we want to write it out as a series is 1 up front plus the sum as n goes from 1 to infinity and how many X to the ends are there going to be D event precisely as many as there are ways of writing n is the sum of this thing piece so like how many X to the eights are there going to be here there's going to be six X to the 8th and they're going to correspond to the six decompositions of eight in the distinct summands so there's the first formula first part of the three that's P of X now I'm going to remove this but it'll be back in a minute but it's going the next piece requires one little prerequisite so this is the one thing I'm not going to prove but I hope you all remember this the sum of an infinite geometric series 1 plus a plus a squared plus a cubed if this goes on is an infinite series the sum is 1 over 1 minus a and I hope people remember that and there are conditions here to worry about them he didn't worry them if you don't like this if you want to know where this is coming from do long division on the right you'll get the series or a better yet cross multiply and you'll see it works so I'm going to need this infinite geometric series formula in fact I'll put it up here at the top and now second part of the proof we introduce Q of X to be that 1 over 1 minus x times 1 over 1 minus X cubed times 1 over 1 minus X fifth and so on why hang on now what a whaler says to do with this this is a bunch of reciprocals with the odd powers he wants to write them without denominators so you see that first guy 1 over 1 minus X if I use the geometric series formula I can write that as a series you know usually we go the other way with that top formula we have the series and we want the sum but there's no reason why we couldn't start with the sum and turn it back into the series so 1 over 1 minus X I just let a be X and I get 1 plus X plus x squared plus X cubed the next term also fits the pattern 1 over 1 minus X cube you let the a be X cubed and you'll get 1 plus X cubed plus X cube squared plus X cubed cubed so there it is 1 plus X cubed x 6 6 9 the next guy 1 over 1 minus X to the fifth let a be X to the fifth and you'll get 5 10 15 20 s of exponents so that's what Q is ok one more thing I want to do with this instead of writing that is that X I'm going to write X to the first instead of the x squared I'm going to write 1 plus 1 instead of X cube 1 plus 1 plus 1 I'm going to break these up into equal pieces and the second guy 3 3 plus 3 3 plus 3 plus 3 5 5 plus 5 it said so if I do that then Q will come out to be remember the first one was 1 X x squared X cubed I just put it the ones in the next one 5 10 3 6 9 12 becomes 3 3 plus 3 3 plus 3 plus 3 5 5 plus 5 so that's what Q is now I have infinitely many infinite series this time this is worse than P but multiply it out so we're going to multiply it out all right so here goes multiply it out what's the constant term 1 every piece will contribute a wine we'll start with a 1 how about X is the only X I'm ever going to get out of this is when that X to the first hits all the other ones there's no other source of an X so you'll just get an X to the first X Squared's well that X to the 1 plus 1 can hit all the other ones but there's no other way to get an x squared so that's how we start but there's more than one way to get an X cubed the X to the 3 in the second term can hit all the other ones or there's an X to the 1 plus 1 plus 1 they can hit all the other ones so there comes to X cubes and I'll show you where they come from by writing the exponents in that form X to the fourth I think there's two of those there's a three plus one and there's a one and one and one and one and one more X to the fifth where is this coming from well there's going to be an X to the 5 an X to the 3 plus 1 plus 1 and an X to the 1 plus 1 plus 1 plus 1 plus 1 look at those exponents 5 3 plus 1 plus 1 1 plus what are those those are the decompositions of 5 into odd sum and these have to be odd because look at the top line everything up there was odd you're putting together odd pieces they might repeat but they're odd every such decomposition will give me another X to the fifth everything that coming out here on X to the fifth was built out of odd summands how many X to the 8th sar they're going to be in this six based on the six decompositions of a in the odd some ian's and so what our has realized that Q of X which is where this started is one plus the sum n goes from one to infinity and how many X to the ends you're going to get o of n the number of ways of breaking in into odd pieces so there is the second part of the proof okay so now third and final part and let me just review here we'll take a breath okay P of X was 1 plus X times 1 plus x squared times 1 plus X cubed then he saw that that was 1 plus the sum of the d of n X to the Q of X was this different thing 1 over 1 minus x times 1 over 1 minus x cubed 1 over 1 minus x to the fifth and he showed that that was 1 plus the sum of the o of N X ax T now it's been so long since we started this theorem I forget what we're trying to prove we're trying to prove that D of n is always equal to o of N and what other said was if P and Q were the same which they're not but if they were the same then when you wrote out the series all the coefficients would have to be the same D of 1 would be o of 1 D of 2 would be O of 2 these would have to match up perfectly and you'd be done if proof would be over if only P and Q were the same which they're not so let me write that up there to prove the D of n equals o of n for all N we need only show that P equals Q everybody see how that would do it if P and Q are the same the series are the same the coefficients are the same we can go home but they're different they're different or are they ah oiler says they're the same they're the same if I can show you that proofs over so why are these in the same they look different to me here's what earlier says remember P of X it was 1 plus X times 1 plus x squared times 1 plus X cubed etc and there they are and I sort of moved them apart any little elbow room that goes on forever turn this into a fraction that ah okay in the numerator I'm going to multiply by a 1 minus X well if I do that in the numerator I must do the same in the denominator so they're still still P of X so I haven't changed anything I could cancel this back out up there at the top I'm going to put a 1 minus x squared down below 1 minus x squared to keep it equal up on top 1 minus X cubed down below 1 minus X cubed 1 minus X for that so on so on so on so you do this forever all those could be cancelled just still P all right now look at the first two terms in the numerator 1 plus X times 1 minus X what's that 1 minus x squared the first two terms up there whoops here we go 1 plus x times 1 minus X take out 1 minus x squared the next 2 1 plus x squared times 1 minus x squared that's 1 minus X to the fourth they take that 1 plus X cube 1 minus X cubed takes the 1 minus X to the sixth away you go what's left in the numerator 1 what's left in the denominator all the odd ones and so what you get is 1 over 1 plus 1 minus x times 1 minus 6 cube times 1 minus X fix and that's Q P and Q are the same and hence D of N equals L of n for all N QED pretty good I think I will pass your applause to Professor Euler who deserves it and he does all this in a few days and his eyes are hurting you know this is where the whole theory of partitioning of numbers starts right here and it's now a vast branch of mathematics with really great work going on it started right there so I will leave you with two quotations which I think are fitting one is from the mathematician Frobenius and he looked at things like this and he said Euler lacked only one thing to make him a perfect genius he failed to be incomprehensible and one of the things I liked as I read through Euler is that you can really understand what he's doing it's not so complicated he's a very good expose etre he tries to make it clear and he is never incomprehensible the other quotation I don't have a source for but I like it I think it's fitting somebody said the talent talent is doing easily what others find difficult genius is doing easily what others find impossible and I think by that definition Euler is a genius he could do the seemingly impossible and he did it throughout his long and illustrious life certainly a good reason for us to give a tribute to him and I'll leave but with our shout out way to go uncle Oh bill that was a really fine talk I think we can probably take just a couple of questions before doing so I'd like to express my thanks to Harvard University for allowing us to host this talk here so maybe a couple of questions for bill and then we'll be out of here yes question is what was ours influence on the calculus he wrote a gigantic text on calculus his differential calculus book was this big and his integral calculus book was three volumes so you know if you think your calculus book is big today you know you should have seen oilers calculus for this sort of first year calculus yes we sort of model our textbooks on his except they're not that big but he never did the epsilon-delta stuff the analysis that you hit beyond calculus that sort of limit based calculus limits weren't around when Euler was working he worked with infinitely small quantities in a kind of philosophical metaphysical world that we now either don't like or we put it over in non-standard analysis but the the epsilon-delta stuff that you know if you've taken analysis comes with Koshi in the 19th century so in that sense his calculus book isn't a modern one but the basic topics are pretty much accurate oh that's okay you can go out now yes yes right the big yet the question though is the soiler supply a lot of detail about his thought processes and the answer is yes the two models are oiler versus gals oiler will write and tell you what he's thinking I tried this it didn't work he'll tell you oh and then I tried this well that didn't work either and then I tried this and wow it worked and you really can see him thinking now he could do this because at the st. Petersburg Academy in the Berlin Academy he had the right to publish anything he want wanted without an editor so he could just be as expansive as he wanted and nobody was going to cut so that doesn't work anymore the other side of the coin is Gauss Gauss would take out everything all the intermediate thought and just leave the basic beautiful structures which makes Gauss much harder to figure out you know he doesn't help you along people criticize Gauss they said Gauss was like the Fox that walked through the stand and dragged his tail behind him to erase the footprints you know you didn't you didn't see how he did it you didn't see how he passed this way Gauss responded he says yes but the architect of the great Cathedral doesn't leave the scaffolding up you take that down and you just leave the gem of the idea so if you want to see somebody that helps you understand oiler is your guy and that's we should all be grateful for that yes he made mistakes yes he had conjectures that turned out not to be true he offered a proof of the fundamental theorem of algebra that thing about breaking all real polynomials in the first and second degree pieces that was flawed and gauss caught the error so he was human he didn't make too many though I mean in 25,000 pages everybody can have a couple you know but he was pretty good and very often when he was working on what we would almost call intuition you know he didn't have the rigorous foundations that we have yet he got it right time and again and you know you get this sense that he sort of knew more than he was letting on as to how fabulous was his insight yes German princess yes no no they weren't poetry yes his most popular work the one that has sold the most copies is called letters to a German princess and what these are are essays on popular science not so much math but supposedly he was teaching this young girl about the world so you know why does the Sun rise in the east and set in the West why is the sky blue and hid right these little letters you know that went to her supposedly but in fact they got published and in German and French Russian and NACA and there is an english translation of that and they're considered very high quality popular science and you know not every great genius can write popular science with equals success and in fact Euler could do that so you can go find out in the library if you want it's it's neat stuff you you

Channel: PoincareDuality

Views: 277,740

Rating: 4.9283371 out of 5

Keywords: Leonhard Euler, William, Dunham, harvard, math, Harvard University, mathematics, clay, institute, public, lecture

Id: fEWj93XjON0

Length: 55min 7sec (3307 seconds)

Published: Tue Nov 22 2011

Reddit Comments

[u/jevonbiggums2]

Anyone with a brain says top 3 or 4 in history. To say he's the greatest ever is just asking for an argument (although I agree with you).

[u/angryWinds]

Really glad I had an hour free to watch that. Quite cool. Thanks for posting.

[u/[deleted]]

The Prince of Mathematicians would like a word.

[u/rhlewis]

Euler was not "the greatest mathematician."

[u/[deleted]]

Dunham as in "Journey through Genius" Dunham? He's a cool dude.

[u/[deleted]]

Really enjoyed that. Thanks for sharing.

[u/[deleted]]

the greatest?

[u/BendoHendo]

On this topic, could someone explain to me why the consensus seems to be that Newton, Gauss, and Archimedes are regarded as greater mathematicians than Euler? What exactly did those three do to surpass Euler?

[u/Banach-Tarski]

Gauss > Euler