An Evening with Euler: A math club guest talk by Bill Dunham

as a research associate Tamar he's a nice Orion who specializes in the work so well I could wait before so that I got my marks one time one time yes sorry so it's a place for me here study her to talk about my favorite mathematician boiler here's what I plan to do break this into three parts first a general biography quick quick biography planner so you have some dates the second I'm going to do a broad survey of some of his contributions to mathematics very disappearance subject and then finally I'll actually give you approve a variant proof from the 18th century that you will understand this is Euler this is the famous portrait when you see this you realize why he was never an international icon of fashion but he sprained his legs aware and they're gonna come and paint your picture that's what he wore he was born in Basel Switzerland in 1707 and was brilliant so that at the age of 13 he's studying johann bernoulli now y'all have really probably run into the bernoulli man in some of your classes he was at that time probably the greatest living mathematician work and he was in but and so this little kid I learned was sent to study the Greek were new and initially it's terrified being the same room really it doesn't take long before we really recognizes what a special cases is where I graduated from the University of Basel in 1722 he was 15 and when he was 20 he went off to the st. Petersburg Academy in Russia now in those days in Europe the different courts were surrounded by academies which were meant to enhance the the nation in the monitor so there was a Royal Society in London and Paris Academy and Berlin Academy in Russia wanted one so they set up this st. Petersburg Academy and whether it was one of the early one of the first people there he stays there until 1741 when he goes to the Berlin air trading or something team he stay there until 72 1766 when he was back the st. Petersburg and stays for the rest of his life and the interesting thing happens to him in 1782 he's made a foreign member of the American Academy of Arts and Sciences now the United States hadn't quite finished its Revolutionary War the peace treaty was 1783 since with a brand-new country but they wanted to have Academy of Arts and Sciences and you know Benjamin Franklin was anatomists Jefferson people like that but they wanted some foreign members and who was the most distinguished position well so they wrote a letter to the Roger pen said you know I would like to make you a member in the American Academy of Arts and Sciences and he wrote back and said thank you very pleased I once said the membership but I probably won't be able to make it to a needs is old but it's kind of neat and they still have that letter in the US and the American Academy lots of other people are since been conducted at the point I joined it the next year our last dock and here just tuned in st. Petersburg Leonardo or level is it citizen it's lacking high score I like the slide because it gives you a context book or a comparison if you know American history over here Ben Franklin you know things character of scientists politician and even almost exactly the same years that they were almost exactly temporaries - towering figures of the Enlightenment and Franklin was not to be outdone by corner because he's wearing it did in the hall around his neck fashion statement it would have been needed to knit but they were too pretty on the personal side Euler married the ham wife Catherine have had 13 children Bobby this was the 18th century and child mortality was rare and of those 13 only five survived to adolescence our concept phenomenal memory apparently you could memorize anything and where other people have to look something up in the law table I know it's much easier to memorize a lot of that and this becomes important in the 1730's when he loses his vision in his right eye this vision is gone and what happens is the aimer actually true but it was the origami eyeball okay it's some sort of infection and we think and it destroyed and so his eye was worthless so he gives her right you know all he keeps going he still has good on myself in his output does not diminish and in 1771 he loses vision and the other left eye and this is a camera and nowadays that's easy fix but not in the 18th century and in fact they tried surgery and you don't even want to think about eye surgery in 17 it's horrible it was painful and it failed so now he's why can't see so his productivity falls off right wrong he keeps going he now will commit and he would stand there and dictate mathematics that people furiously writing it down that he could see in his mind's eye with his phenomenal memory his productivity did not diminish even when he went blind and if you don't believe me there were 17 75° 1775 he produces 50 papers as a blind man so this makes oiler kind of be an inspirational figure in mathematics he was someone that would not be by this physical malady and now this man is distinguished by its quantity and its quality he's off the charts on both of those axes and let me mention the quantity first and then for the quality I'll show you some terms of quantity nobody ever did more mathematics I'm sorry not a page a page if what I did have a lot with you here let me let me to expect the value of 1783 but just being dead does not stop over the next was that 47 years he published they had found these things is death away is 228 papers want dead no other dead mathematician has ever or will ever published 228 papers but where did that it's just unbelievable in 1910 somebody that good stock management decided you catalog all of Oilers works you know compliment cult cut them all right little summaries up put this up keep on 866 broke some papers it's like and in that regard errors might have a higher number than that but Oilers paint was a paper might be 100 pages that counts as one paper a book might be in this calculus books were 1500 pages long that was a kind so in terms of pages east you know wait wait off the charts ah the catalog that in astre brooked giving a description of each of mileage results was 380 page one month works so the next year the Swiss Academy Sciences decided they would republish all boilers works nice organized series they called the over Aquaman and he 1911 the first ball game came out and these volumes are big and think of an encyclopedia they're not little books are giant tunnels so I cannot next year volume came out a few years later they're still coming they're not done they've been at this for well over a century and there's still that last count there were 75 any evidence slide the whole might be 77 belongings of language collected works 25,000 pages from one person if you're by time productivity was 250 pages multiplied by ten twenty five hundred pictures multiplied by 10 again so that was the quant not the quality right so let me show you some of the things he did in his career one was in his great work the introductory Oh Annelise in infinite Oram of 70-48 he said the proper focus of geology and geography the trig and calculus was not the curve which is what people had been studying prior to this curve lines geometrically he said huh that's not what you want to study it's a function and he made function central to analysis and if you study the analysis function why because prior to that curves were the object of study and they said I gotta give you the functions you need and token so he gives you the polynomial function he introduces a logarithmic functions exponential trig function inverse trig functions they're all in this great work boiling and there's still functions that are in your toolkit this is what has been called the greatest math textbook of modern times in terms of what it did so he gave us functions that's pretty good he gave us the number of e and here's where it sure this is this is his birth this was a 1748 he said let us place for the sake of brevity that's this number to be for the sake of brevity people replace this number two seven one eight two eight cetera by that way e let's use that which he says therefore denotes the base of the natural or hyperbolic logarithm so he knew what it was and he says if you want to calculate it at home there's how you do it 1 plus 1 over 1 plus 1 over 1 times 2 plus 1 over 1 times 2 times 3 if you know the Taylor series for e to the X you put in 1 of course that's and it converges very fast and it converges and that's how I got this is very accurate although he loved this kind of stuff it could multiply ten digit numbers in his head you know so doing long this was kind of fun for sidelight little exercise yeah well maybe these papers were laughing about this is weird if he was in Russia he wrote a plan if he was in Berlin Germany he wrote in French because the Academy in Berlin they insisted that everyone speak French not Germans Frederick the Great the German the pressure bleeder hated Germans and he loved the French so so if you see it at a women paper and it's written in French you know it's coming from Berlin and it is written in Latin it's coming from Russia but of course in those days all scholars in Europe knew laughs so this was the common language it makes makes it difficult to read this stuff unless you know Latin and French call this room okay he is the owner called poly Negro for the E Plus F equals E Plus de panamá through sickness in your courses but this involves this is about polyhedra which are three my and three-dimensional bodies whose faces are polygons and vide is the number of vertices number of corners and that was the number of faces and E is the number of edges on these things and what over there says it is B plus F equals D plus 2 not for my Q there that's cubes are perfectly good apology drew what's me how many vertices right for the top four on the bottom how many faces sex and think of a duck and how many edges what you count them you see one two three four five six seven eight nine ten 11 12 and 8 plus 6 is 12 let's do it so it works for a cute easy but where it shows is it works for a wide array of other calling like prisms and pyramids and icosahedron like this and lots of other things and this general property is quite significant to this day now we think highly of this but back in the day wherever it said surprise these general results solid geometry had my previously but others fly anyone so far as I am or he said what's the big deal maybe you've seen this philosopher 11734 and there is a challenge afoot in Europe Yakov Bernoulli who is the brother of Johan Johan de Villiers mentor young challenge the mathematical community you'd find the exact sum of that infinity plus 1/4 plus 1/8 plus 1/16 the sum of the reciprocals of the square and this was these big I'm soft brother nobody could do it we really could do it yes there's a lot of challenges that went on back in the day and sometimes you would have issued the challenge until you did it yourself and they can issue the challenge then but nobody else been doing because I see how smart but then this one Yahoo couldn't do it either he didn't do this to discredit he showed this convert in our language to something less - it was a comparison test essentially the very girl comparison test I'll talk about this tomorrow speaking about but he knew it converge to something but he wanted to know exact and he couldn't figure out and boilers for one with us and do you know the answer you seen this weird answer this is strange strange strange result looks like can't be right but it is of course and what I'm going to publish was this time expands this is called the result of a fantasy it's still pretty young and after this everyone here knew there's this superstar at this time he was it would you know if we would like today and this right there it would have been on the front page of the New York Times there's somebody played insisted what I've given a million dollars or something gr he worked in GRE you think how much geometry could be left you know the Greeks did geometry over the past years before surely all the geometry was done not for big fat bodies of Oilers collective work took geometry he went back to the sort of old subject and thought new things supposed to intron anything doesn't happen to be equal at all or you know isosceles moved to be any and you find the the point where the three altitudes meet you know the altitude is what it goes from each vertex perpendicular to the opposite side the three altitudes go through a point called the earthers the three medians median point now median remember this it goes from each vertex to the middle of the opposite side and those be and then you take the perpendicular bisectors should you take each side bisect that put up the perfect thing though does making point which is the center of the circumscribed circle is called a certainty okay so these points combine any triangle the Greeks knew about these places that's not but let me show you a picture so here's a triangle so let's do the intersection of the altitude the earthís so each one be perpendicular perpendicular perpendicular and there is the now we'll do the intersection of the medians so they go from each vertex to the middle of the opposite side and they all mean the same and it will do the intersection of the perpendicular bisectors so you take each side bisect put up the perpendicular and the airspeed circumcise what did we see that no one had ever seen they always like for any trend these are going to form a line which we not call the oiler because he's the one who discovered it and proved that these have to be aligned and he proves something more but the distance between the centroid and the circumcenter is always exactly half the distance between the centroid and the orthocenter that line segment is always split in the ratio of 2 to 1 no matter what triangle you sir so there was still some geometry to be done in the world ok here's the famous paper is a solution pertaining to a problem of the geometry of position Gale metrion zinthos the geometry position this was the Bridget's of Kenneth and you probably had a kind of quest this summer and there your life this is a map of kenigsberg but the river Fraggle I guess comes in he goes around this island that there were these bridges and the story you have to tell off you tell me was is that supposedly in kenigsberg the burghers wanted to take a stroll on Sundays and they block here and the block around I'm going to go over the bridges look at the target on this and try to do this so that they cross each bridge once and only once and they could every time they travel to miss the bridge or they King we're back across the bridge they already cross so they were you know computers so they asked the mayor of Canonsburg can this be done and he said I know I'm the mayor kidding you asking me for ask that guy down you that's so they write a letter go in there and whatever analyzes it and shows that in fact you can't even for this computer that's impossible but if I that configuration is possible and he explains why it gives a proof and it turns out to be the origins of what we now call grafting the theory of graphic vertices in those edges and it's a fast fast subject today but it's often true origins traced back to that we're impressed well there wasn't so impressed he said man this solution bears little relationship to mathematics and I do not understand why you expected mathematician to produce it rather than anyone else for the solution is based on logic alone so he said this isn't even mathematics that cosines derivatives and this is just recently constructed combinatory Harkins so he said the mayor probably sure whatever but yeah and in fact if we're going to come back today and see what happened certainly recognize in action all right one of the things he did it was really good was never here he's among the five greatest number theorist ever probably not worth Erie the study of the whole yeah products and composites and all that stuff it sounds like it should be pretty easy poll numbers one two three and what's so hard about that you know irrational are those are actually really fire or other did four fat volumes of number theory and let me just pick out one thing he did this definition had been kicking around M&N are called amicable numbers which means friendly if each is the song the proper whole number devices of the other so that's the definition of an example as an order and here's the example you'll see 220 and 284 amicable knowledge here's what we mean like that what you want to do is look at all the proper divisors of 220 with by which I mean whole numbers to divide evenly into 220 but proper means not 220 it'll come back it's not a small so there they are the numbers that divide into 220 exactly or one two four five 5510 admah 284 now you take the proper divisors with 284 all the numbers will divide evenly in the 284 there there are two 471 142 are not 20 to 80 for this problem add those up to 20 each of these is the sum of the property lancers at the end it's totally anything but this is what number theorist whoa these kind of strange weird properties of the whole um so I'm sorry this way I always call story which was a true story I had a friend it was bad he was getting married and he gave his wife to be a bracelet with the number 284 and he has one he wears with the number 222 show their turn internal friendship it's very nerdy finding a life partner just cos that as a surgeon it's very recent oh yeah here comes the brief history of amicable dollars the Greeks knew that pair wait they had discovered the 220 and 284 had the strange reciprocity each was the son of the property advisors in the other and they wanted to find more and they couldn't find villa nobody finds anymore until the ninth century when the Islamic scholar Tabata in Korea discovers the rule that he yields two more parents but apparently this rule does not make it back to Europe in the sense that European mathematicians were nowhere nervous so they think there's just this there in 1716 36 Fairmont lost their fame finds another pair mr. seventeen thousand two hundred ninety six eighteen thousand four hundred sixteen to take all the proper divisors of seventeen to 96 and half of them you get an 18 for sixty and vice versa so that was a pair but that was one of top it's not top it they'd already discovered that pair but pheromone didn't know it's a thermal is padding so back maybe improperly but he didn't independent okay so not fair upon his phone now in France in the in the 17th century there was a terrible mathematical rivalry fair law and they cart the hey teacher and if everyone just found an animal care so now if a car has to find an amicable pair of cruisers word so he goes to work and so there guess what that's the other one these are the easy ones this is the law hanging fruit of amicable numbers at what comet had discovered was a rule that would have yielded those if fair market card had known hey if my friend really loved his wife gonna put that eight digits of love that's where the situation's good in 1638 and that's where it stood when okay no one in the next century found any more and then in 1750 or letter looked at this problem we're okay and I try to do because well this here it is but I like to tighten the Newbery's Amiga billions on amicable numbers but I must be same word nobility this is the paper here's the 284 the 220 he starts off with that example and then goes on so what's significant about this paper in this paper he finds 58 more pairs in 2000 years mathematic competition set by the three pairs in one paper for the clients vehicle it's one more that we're doing he would did he saw something that nobody else had seen he had afford another just too much I would come on the carts no but I would kind of so many so hey how're things before I get to my fruit he worked in applied math he did affect happened his volume use this collector works are applications physics mechanics acoustics he designed a pump this is his little picture for - books where he's showing up and then there's this these little circles intersecting all right everybody knows these as band diagrams see no that was in the 19th century I took this out of Oilers work in the 80s boiler was drawing a little diagrams intersections to explain logical principles so we should call this and what about if we were to be blocks however if we were to do that then this is gone what else - so if we did this or other we get one more thing invented the obscure doesn't and then two other things and I promise I'll get to my proof here was a challenge given to a boiler near the end of this life find for whole numbers the sum of any two of which is a perfect square now this is easy I will give you an example here come for 9 8 8 8 10 8 okay it's different thank you sir now is bodies and if you just try to do this but see your pants young just took a trial and error you know you think of a number one find the number to add to it to make a perfect square how about three so there you go now you got to have something add the tree to be a perfect square 6 so 3 is 9 but what happens when you do that the six of the 1 don't work anymore oh you have a back and change this took 10 so now 10 + 6 is 16 which is a perfect square then the 10 to the 3 don't work so it's like you're trying to juggle you can't get all the balls in the air and that's just three numbers any heated floor so if you try this you know just really Nellie it will not work I promise you you need an assistant so whatever thinks about it or puts it through system he comes up with these four not those four numbers any two of them act together and you get a perfect square but you know this is done centuries before there were computers or calculators and you just look at this and you say how in the world the somebody in the 18th century come of those foreigners and the answer so it was real smart this is my car two more times and then back in high school you factor polynomials right he's spending gobs of top of factory things x squared minus one the quadratic breaks into two liters and X to the fourth minus one can break in the x squared plus one x squared minus one and you could further factor although this four plus ones irreducible if you're just so a conjecture was floating around in Europe that said any real polynomial no matter its degree can't be factors at least theoretically into first and second degree factors like this linear and and if you've seen the fundamental theorem of algebra where it's recast in terms of complex numbers and you can further factor this and then the statement is that any an three polynomial can be broken into and linear pieces but they weren't talking about complex polynomials in but so they buy even when the question was can you break any polynomial in the first one second degree fact nobody knew nobody could prove it but a lot times Nicolas Bernoulli yet another of these four duties and he said you can't do it you cannot factor any polynomial the first and second degree pieces and I it was really have a congressman and he says that the plane X to the fourth - more x cubed plus 2x squared plus 4x plus 4 fourth degree he says you cannot break it into two quadratics or later it's introduced he can't well Nicolas Bernoulli couldn't factor it but that's not the same as saying it can't be done well it says obviously yes the two quadratics this one and now two questions is this right it doesn't one way you know opposed to me this is nuts because there's square roots within square roots here you know are you telling me this quadratic times this quadratic turns into something as simple as that for tabletop but if you want to spend half an hour doing this or something wall climb back in the world works this is ring so that's correct and Bernoulli's counter example wasn't it kind of presentable at all but then the other question is how it's brilliant when you're done when you read everything you say just like in the air ok the figures but no you okay so now fine I'm gonna get to right there and my theorem this is quite significant one whatever is identity from 1748 this e to the i-x equals cosine X plus I sine X and before I proceed hi they've seen this at some point and if you've seen it prove almost certain things done the series right you do the series for cosine the series for sine a series for e and put it together that's probably one of the ways he did like to return do a problem in Hamlet there I get a different group that's how you thought 75 so I'm gonna show you with very clip here's how these steak notice he is using the key but he had to be had adopted the I so he was still writing the square root minus 1 here and here and then this is curious he what's this one right the whole function so during cos period Coast period because it's an abbreviation for cosines and si Han period is an abbreviation for silence so he actually felt compelled to put in the periods we adopted his notation okay so now how in the world do you prove this strange result well he's going to prove it with integral calculus this is going to become something that's so we're gonna prove this theorem as he did and he begins well I'd say he doesn't actually like I swear - well I'm gonna let I'd be this occasional convention and he integrates this he says this is the trick start with the internal these the over the square root of 1 plus C square now all right the calculus students here at Stony Brook can do this I guess if you give this to the man if they remember trig substitution everybody's favorite trick you can do this with trigonometric substitution or you can punch it in this is the anti-griddle log of Z plus the square root of 1 plus Z squared remember out of a trig substitution you can almost differentiate it backwards you just see it works so this is going to start if you ever lost a point from one of your professors for forgetting the plus C take that paper back so this is how we started to prove this substitution I'm gonna let Z be my Y or eyes then BC he says will be ID 1 the eye comes out the differential of Z will be I times the difference he's going to make that substitution I believe that if Z is final then DZ is 5v 1 the constant 5 comes out of the differential but this isn't this is a complex number are you sure it comes up of the difference can you pull out eyes and whether responded to that kind of criticism of this state he says if you deny the validity of pulling up complex classes you will shatter the foundation of all analysis which consists consists principally in the generality of the rules and operations Jardine's true whatever the nature which ones supposes for the quantities to which they are applied so you say you can pull up a real constant out of the differential you can pull a complex constant out of a differential paper real cost amount of the different from any other kind of constant otherwise you will shatter the foundation well analysis so yeah these are the entities so there okay stick with obviously and what happens here is when the DZ is we got an ID Y square root of one plus I Y squared because it's a Z is over here log Y plus 1 plus on X square now what's I'll I swear I swear it Y squared what was I square minus 1 yeah so from this we can get instead of the square root 1 plus I Y squared it's a square R 1 minus 1 3 is I squared is minus 1 likewise here in hand I guess I turn these around psych them to square root first so you get 1 minus y squared and he pulls up that I outside the internet taken area numbers outside of intervals yes let's apply these sine X and dy is cosine X DX and we substitute again so now we have I up here dy is cosine X DX square root of 1 minus y squared 1 minus sine squared a lot of the square root of 1 minus sine squared plus I sine X now what's 1 minus sine cosine and what's the square root of it yeah and so what's where one minus sine squared becomes coastal in both spots here cancel the cosines and he's got I times the integral of DX on the left and the log of cosine X plus sine is from the integral DX that's X so IX is the log of cosine X plus I sine X and you're saying he's not gonna do what I think he's been yes yes he's got an exponentiating both sides and on the left you get e to the i-x and what's either the log cosine X plus I sine X cosine X plus i silence and that is how I was I did it by integral calculus I think that's amazing now you can argue there's all sorts of questions that he just sort of swept under the rug and pulling up these constants you know but for the time this is from this equation you are identity if you're teaching this you have to you have it's in the contract mathematicians have you let X equal PI and you get e to the I PI cosine PI plus I sine PI but cosine PI is negative 1 sine PI 0 and so e to the I PI is negative 1 follows immediately from this and then what they usually do is put the minus 1 over the other side and so you get e to the I PI plus 1 is equal bat Falls and this has been called the most beautiful formula in mathematics they actually did a poll of mathematicians this was some years ago in fact was so long ago whenever if you wanted to participate in the employee had sent a postcard you couldn't do it online cuz there weren't computers but people did this you know what's the most do before we solve down Pythagorean theorem this one e to the I plus 1 that was regarded as the greatest because well because it connects the five greatest constant so if you're gonna have a party than what I did invite the greatest constant unity by zero and one additive the multiplicative identity given by E and one of the do calculus pie if you wanted to do geometry I into the realm of complex numbers e I Phi 0 one dream team and they're all connected by this point there's no reason they should be why should the five greatest numbers all be connected letters and it's just brilliant that it just falls right now so it becomes the number one greatest of all time so people that someone wrote a poem and even though it is almost 8 o'clock mr. e to the I pi plus 1 equals 0 made the mathematician Euler a hero from the real complex with our brains and great flex he let us with zest the no beer okay okay one last thing here's something else overdose he says I want to figure out what is the square root of minus 1 to the power square root of minus 1 it doesn't accept a complex of imaginary number to an imaginary puck or we can write it I to the eye and he's trying to figure out what this it might be does because he goes back to why was I then e to the i-x equals cosine X plus I sine X this time unless X Q PI over 2 so e to the I PI over 2 is code this cosine PI over 2 plus I sine PI over 2 cosine PI over 2 0 sine PI over 2 is 1 and so this comes out to be so e to the I PI over 2 it turns out to be hi now if you've got a complex analysis or anything that's obvious but it wasn't obvious Batman this is kind of amazing that itself I could be really bad but now what he says is if I wanted I to the up leave the exponent alone that this I am the base can be written as e to the I PI over 2 so hi to the I is easy the I PI over 2 to the I and now he said you know the rules you multiply the exponents if you a Q squared and multiply to 6 but you say wait a minute oh these are hot integers up here these are complex numbers and you know what he was saying that we don't want to shatter that's not exponents work people on applies them I squared minus one city to the minus PI over two and so out of this emerges to some basic fact but I to the N on the square root of minus one to the power square root of minus one is that iya to minus PI over 2 which is 1 over square root of e to the PI it's a real one square root of minus 1 to the square minus 1 is a real number you touch another country and nobody had ever seen anything like that it's really quite miraculous and there's a story that mathematician Benjamin purse had been teaching this problem at Harvard this is bank first taught this expression that showed that either the ogresses wonderful square root of e to the PI and then he was close with the Tully's class we have no idea what this equation means but we may be sure it means something ok so this is or this is what is giving us let me leave you with a quotation I don't have a source but it's a good one I think it's after somebody said the talent is doing easily others difficult genius is doing easily and satisfies so I believe with a shot way to go oh yeah that's one of the questions in each of zv+ name was a boiler actually probably not he was a very modest person he's very nice person some of these great mathematicians if you get to them as people necessarily very nice scallop Gauss was kind of icy and Newton he was absolutely but whether it was a nice guy and you just don't think he would try to do something brand Yzerman use the letter of its last name it was for exponential he chose for exponential there's a story about him that when he went to st. Petersburg where all the scholars were from somewhere else they weren't rushing they are imported to France to Germany in Switzerland they didn't learn Russian because Russia was the language of the peasants people to clean your house here in Russia because he wanted to talk to the grocer this is good but the e is for expert in his achievement is something that we know his children were not of course this was having one of these sons was named Johan Albrecht Euler and he became a mathematician brain man now he collaborated with his father was father was still alive and and there's this story there's something or something called the Paris Prize it'd be a challenge my challenge is the challenge and people have here is submit a solution and whoever got the best one would win a lot of money thousands and thousands of francs but in modern bucks and a world are 113 times or something this last but the one here is sensible y'all wanna open it one great so y'all have gotten some power surprised but this dad stole a lot after that when his dad died and Johan Albirex submitted a solution never won but otherwise I don't know how to do any of the other children when in the sciences and beyond that generation I don't know at all but it's not like like the Bernoulli's all these Bernoulli's and their sons and cousins and nephews were all scientists or the box or musicians race car drivers kids become racecar drivers but Oilers family to repeat

Channel: Math Club at Stony Brook University - SBU Math Club

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Published: Wed Oct 23 2019