'How to Solve Equations' - Dr Vicky Neale

thank you very much it's very nice to be here can you all hear me okay okay great well it's a great pleasure to see you all here it's a sort of pleasure for me to come back for the day and see Cambridge again so about a hundred years ago when I was your age I was preparing for step and I just kind of wanted to reassure you that it's possible to do things like step and mat and come out the other side I found the questions I was doing really hard there wasn't anybody in my sixth form college who'd done step recently so the teachers hadn't sort of done the questions and so on I kind of do it by myself they were really hard questions I really enjoyed the challenge where I practiced I got better at doing them so that's my tip for you I'm supposed tell you about how to solve equations I kind of wish I knew how to solve equations solving equations is quite hard but let's see how we go so what I thought I'd do is give you some equations and I don't want you to try to solve them I just want you to look at them and get a sense of how many solutions you think they have and you know talk to your neighbor spend a minute so this isn't get your pens and paper out and I want you to spend lots of time solving equations just use your your instincts which is a kind of way of saying your experience your your intuition your your feeling for how many solutions each of these equations should have okay so if you don't know the person next to you well just be brave and talk to them anyway okay and we'll have a vote in a moment so here are the equations okay and this is this isn't supposed to be easy right I don't do easy problems and it's multiple choice okay so you can tell me there are no solutions one solution two solutions three or more solutions but a finite number or infinitely many solutions okay so take a minute touch your neighbor how many solutions what a fantastic buzz of mathematical chatter okay so I'm really going to interrupt you I don't mind if you don't definitely know the answer okay this isn't about definitely known the answer this is just about what's your what's your feeling for it what's your intuition so let's do a vote but you've had a chance to talk to your neighbor I hopefully that means voting is a little bit less scary and I promise there are no consequences okay I'm not going to pick on you whatever you vote so you can be brave so the first one x squared plus two equals zero who votes for no solutions a few people be bold okay be bold great he waits for one solution okay two solutions interesting three or more but finitely many infinitely many okay so map systems of democracy right I mean you vote something that doesn't change that doesn't make it whatever it is but that's really interesting okay what about the next one two over X plus three over y equals one over six who votes for no solutions one solution okay so there's a there's some hands up this is good two solutions slightly more three or more solutions infinitely many solutions okay so there's quite a lot of undecideds that X to the 17 plus y to the 17 equals e to the 17 no solutions few people one solution to three or more in Philly many okay this is good we're at the last one y squared is X cube minus 2 who waits for no solutions one solution two solutions few hands up three or more quite a lot infinitely many okay so what I'm learning from this is that you do have a feeling for some of these problems kind of what it might be but actually they're quite hard equations so some of them you're looking at and you don't automatically so I think people felt much happier the first one maybe than the later ones that that's my impression um there's a bit immune thinks this is kind of an unfair question because it matters quite a lot what kind of solutions we're looking for right so there's lots of people thinking yeah yeah that's what I thought why didn't you say that before well yeah in a parallel universe there's a version in this slide that asks for integer solutions for example hole number my lady's word integer integer just means hole number positive zero negative they're all integers so so some of you might think oh well if she said integer I'd have changed my answer some of you might have been thinking about integers all along so yeah we have to be precise I guess so basically I'm just sort of planning to tell you a bit about each of these four equations and I'm particularly interested in integer solutions so Julia mentioned I wanted to question number theory number theory is all about properties of whole numbers it's a part of maths that I find particularly interesting but along the way I'll mention some of the other kinds of places we might look for solutions as well so X square plus 2 equals 0 so I guess that has no integer solutions because if I take any whole number and square it I get something that's greater than or equal to 0 if I add 2 I definitely going don't get 0 in fact there are no rational solutions am I allowed to talk about rational numbers rational numbers means fractions so a ratio of one whole number divided by another just not dividing by zero why is they have no rational solutions where exactly the same reason it had no integer solutions right if it's got no real solutions as some of you said well this this equations got no solutions and you might have been thinking about introduced you might be thinking about real numbers but if we extend the pool of numbers that we're working with if we allow ourselves to use complex numbers as well then we find that it's got two solutions and maybe I can deduce how many of you in the room have started thinking about complex numbers by how many of you said oh it's got two solutions so yeah the answers depend crucially on where we're working and this sort of mirrors the kind of historical sort of yes as we've mathematicians have evolved our study of equations we've wanted to introduce new kinds of numbers to allow ourselves to solve more and more equation so complex numbers became crucial when we realized we just needed to be able to solve more kinds of equations that we were currently able to deal with so good let's talk about this equation which started to cause a few more difficulties I think so one thing we might try to do I guess it's often quite helpful when we think about solutions is to draw a graph yeah so I hope you're all good at sketching graphs I really recommend being good at sketching graphs but I'm not very good at sketching graphs on my computer so I got a website to draw it for me so what this is showing is that there are lots of pairs x and y that satisfy this equation but the graph is just think about real numbers that satisfy and I'm interested in whole numbers that satisfy it's very hard to tell looking at a graph like this about whole number equate solutions to this equation so let's do some substance or different so there are lots and lots of pairs XY there satisfy this but it's not quite sort of obvious to me when I look at this equation how am I supposed to know how many integers solutions it has how many whole numbers it's kind of odd looking equation well let's make it look less odd I'm interested in whole numbers and this is I mean one of the things I find difficult about this has got all these kind of denominators and x and y on the bottom of stuff is sort of horrible so let's multiply through and that's really nerve-wracking having like algebra on my slides because the chances of my having made a mistake are quite high but I think I've multiplied everything in sight by 6 X Y to clear the denominators and what I like about this equation is there are no kind of fraction e type things about everything in slices a whole number I feel much happier in the world of whole numbers and I can strip everything across to one side because it's nice to have zero on the right hand side there's something there's a bit of me the sort of things I should have given you half an hour to think about this problem but then I wouldn't have been able to tell you so much stuff so I'm just going to keep pressing on so looking at this equation think well how are we going to figure out which x and y satisfy this equation I don't know about you but I look at this left-hand side and I sort of want to try to factorize its fact writing good way of solving equations right so things are equal to zero and sort of factories and quadratics and things so we could try to factorize the left-hand side and what I've done here is I just kind of guess what the brackets ought to be this is really powerful no guessing is a really good thing to do so I've got XY here this is gonna be my minus 12y it's giving me my minus 18x so this is all good but I get another term from these brackets as well right I get a plus 12 times 18 so all I get to do is write down on the right hand side whatever I need to write down to make this a true equation which is 12 times 18 and I really love number theory and I really don't love arithmetic so I couldn't be bothered to work out with 12 times 18 is so I'll just leave it's 12 times 18 because actually I'm kind of thinking ahead to the next step how is this going to help me because normally you're I factorize and then I know one of the factors must be 0 because the product 0 that kind of thing I'm sort of looking in case any of you go to like nods as though you know what I'm talking about yeah there's some nuts thank you but I haven't got 0 on the right-hand side here so I feel really careful but here's where it really matters that we're looking for x and y to be whole numbers right because it means that X minus 12 and Y minus 18 have to be whole numbers so they have to be factors of the thing on the right and that's kind of why I didn't bother to work out what 12 times 18 is because if I'm just thinking about factors I may as well leave it like that so what we discover is that X minus 12 has to be a factor of 12 times 18 and then Y minus 18 has to be whatever the factor is to you know complete that product okay I sort of feel like I've morally solved the problem now this is a classic mathematician I don't care what the solutions I don't actually care how many of them are I feel completely confident that if you locked me in a room for five minutes and said you're not coming out until you've written a list of the solutions that I'd be able to do it right because I'd write a list of the factors of 12 x 18 I carefully remember there are some negative factors of 12 x 18 and then I just work out what the corresponding X's and Y's are so I sort of feel like I've basically finished at this point so let's not write a long list of factors because nobody cares but I guess the message here is they'll go through quite a lot of solutions but even though we've got two variables in just one equation we've got this extra piece of information that we're looking for whole number solutions so normally we'd have got infinitely many because we saw that real graph but the extra information that we're only looking for whole number solutions narrows that down I find that kind of intrigue I quite like this kind of argument where we really exploit the fact that we're looking for whole number solutions you know you could sort of stare at the equation and maybe yes happen to stumble across get lucky find some solutions but actually being clear that we've got all of them I really like this if you like this kind of problem I thought we'd have a little advert break okay this is the time for the adverb break so don't leave the room to make a cup of tea don't get your popcorn out here's the adverb break so maybe some of you've heard of the UK mass trust who organized maths challenges and team mass challenges and that kind of thing and you KMT organizes a mathematical Olympiad for girls the UK mug in 2016 it's going to be on that date and if you look at this website you can find some past papers and see the kinds of things so the way this works is that you KMT sends some teams to international mass competitions each year including European girls math Olympia which in fact you KMT was involved in setting up a few years ago and this is the beginning of the Talent Search to sort of encourage girls to get involved join in with the mentoring program we have sort of training practice problem-solving and then potentially be picked for a team to go and represent the UK so this year's UK team who have just got back from the European girls Math Olympiad in Romania and have come back with a gold medal a silver medal and two bronze medals which is the best ever team performance so we were very excited and if you like solving really hard maths problems then this is the thing for you right let's go back to solving our maths problems here's an equation this is quite a hard equation I mean where do you even start and I just picked a certain number seventeen because I like it basically so let's have a warm-up problem it's a really good thing to do when you were stuck on a problem pick and easy a problem and do that in statins a really good thing to do so what professional mathematicians do all the time so I thought we might think about this problem instead x squared plus y squared equals Z squared anyone recognize this equation few people yeah Pythagoras so if I've got a right-angled triangle besides x and y the hypothesis new Zed they satisfy this relationship and maybe you know some integer right-angled triangles cuz math textbooks like the math teachers like them right they're sort of learning about Pythagoras is there it's really nice to be able to practice applying it and find some troy girls that have whole number answers so we have a name for these these are called Pythagorean triple so if you've got x y&z satisfy this question it's called Pythagorean triples so maybe you've seen three four fives very famous so there's a right-angled triangle decides three four or five and then you can see three squared I guess that's 9 plus 4 squared 16 is 25 which is 5 squared so it's it satisfies this equation 5 12 13 is another favorite of the textbooks there yeah you sort of play around you'll find examples of these things so for me one of these stands out as being much less interesting than the others so I thought I'd give you 20 seconds to talk to your neighbor to discuss whether you think one of these is less interesting or whether they're all as interesting is each other talk to your neighbors see what you think so I don't think I'm cheating here I think these are all genuinely Pythagorean triples so it's it's not that but who thinks they're all as interesting as each other who thinks one of them is less interesting few people okay which one do you think is less interesting who think is a top one it's least interesting second one third one fourth one interesting so I think the greatest number of hands was so the third one is so many brave enough to say why they think the third one is the least interesting yep the same difference between the numbers so you mean this difference is six and this is six yeah interesting yep is it six times the first one so you mean this 18 is 6 times 3 24 is 6 times 4 30 is 6 times 5 which is in fact related to your you've noticed the same thing in a different way yes that this one is somehow 6 times this one if you imagine that triangle that's like measuring your triangle with a ruler with different units or something you're blowing up your trial it's the same sort of proportions of triangles do and the reason that's relevant so I'm really pleased do you spotted that that's great the reason that's relevant is if I gave you this multiple choice question about how many integer solutions there are to this equation well the answer is definitely 3 or more because here are 4 but there's a silly reason why the answer is infinitely many we says I could just pick that 3 4 5 1 and just multiply it up scale it up by whatever I like so there are infinitely many solutions for a sort of rubbish reason so a more interesting question is are there infinitely many genuinely different solutions like genuinely different shaped triangles not just the same thing scaled up over and over again now how much do I want to give away here because you can think about this problem for yourselves right this is a problem that is within your reach if you got a bus journey back this evening you can think about this so maybe I will tell you that there are infinitely many genuinely different solutions and there's a really beautiful way of describing them also you sort of give a kind of formula a recipe for constructing them also if you think really carefully about this this is really lovely this goes right back to the ancient Greek so as a chap called Diophantus who wrote a number theory textbook called the arithmetic ax and he describes how to find all of these infinitely many genuinely different Pythagorean triples and a lot of centuries later my precise grasp of counting decades century is there somebody else came along and said well this is very interesting you know I'm reading diophantus this book what if I change the power what if instead of thinking about X square plus y square it because that's what if I just you know forget about right-angled triangles but what if I make it X cubed plus y cubed equals their cubed or X to the power 17 plus y to the power 17 equals Z to the power 17 how many solutions that and presumably he went away and did some doodling on the back of an envelope or something we don't know Lucian's he somehow convinced himself positive whole numbers so for powers greater than or equal to three there are no so I have to say positive integers here because there are sort of silly solutions where one of the variables is zero if I have three cubes and minus three cubes they had to give zero cube that kind of thing so rule like that the uninteresting solutions there are interesting solutions with positive whole numbers so firma was this French mathematician his job positive mathematician he was a civil son he was a lawyer he did maths in his spare time as you do and he was reading this ancient Greek textbook and he said God this is really cool and it was just his textbook he was just doodling in the margin and he wrote in the margin there are no solutions to this equation and I have a truly marvelous proof of this which the margin is too small to contain don't write this in your maths homework he goes really down really badly with math teachers so he was just writing this for his own benefit but after he died his son published this textbook with all of his father's kind of scribbling sin he'd written all sorts of things ferma was really interested in understanding the number theory that the Greeks had done but then taking it further seeing what else he could do what equations can I solve what can I understand better than the Greeks can I add to them what they did Oh let's try again I love blackboards they never do that T where was I Furman so yes so he had all these these various assertions of firmware was really shocking at writing down his work he just basically didn't bother so after his death his son published his book and other people had to come along and try to check it was fair my right or not and sometimes he was and other people were able to fill in the proofs and sometimes he was wrong he made mistakes this is called Fermat's Last Theorem here I can explain 2/3 of the name ok I can explain firmer because firm I wrote it down in his textbook I can explain last because it was the last one of these claims that firm are made for the mathematicians hadn't decided one way or the other I can't explain the theorem because he didn't have a proof so it really shouldn't have been called a theorem claiming to have a proof it's no good you actually have to have one and people spent a long time trying to figure out is this a true statement or not can we prove this could we show that this is really the case so firma himself in fact when the rare occasions when he did show his working showed this result for the case n is 4 so he showed there are no positive whole number solutions to the equation X to the power 4 plus y to the power 4 equals Z to the power 4 he did genuinely do that which is part of the reason why I think maybe he even realized he made a mistake when he claimed to have done this because if you claim to have proved it for all of these why would you go back and do that special case later olá dealt with the case of the cubes a few years later the first person to deal with a whole bunch of cases all at once was Sofia's Gemma I'm sorry this is not a great picture the quality of the digital cameras and the period she was alive didn't have very high resolution so if he's known as really interesting person so she lived in Paris an incredibly exciting time they were building this fantastic University just down the road from her a really exciting moment except women weren't allowed to study there this is just completely shocking but you know it was a long time ago Sophie Scherman was a very determined individual she really wanted to study her parents hadn't been so keen on this so she used to lie there under the bedclothes with a candle doing maths please don't do this at home so she somehow managed to get hold of the lecture notes from this fantastic University just down the road and she was reading about what people were doing and she was studying and like Farabaugh she wanted to take it further she wasn't satisfied with reading what other people had done she wanted to contribute her own work to it and she worked on various things so she worked toward trying to develop a mathematical theory of elasticity which was one of the kind of really exciting scientific problems at the time she did a lot of work on number theory so she read this book by Gauss one of the greatest mathematicians of all times fantastic number theory book and she really understood what this was about and she was adding her own ideas and she was thinking about flowers equation and she wanted to write to Gauss and say we'll look mr. Gauss yeah this is what I'm working on what do you think she was worried that Gauss would not take her seriously if she wrote a Sophie Zuma so she pretended to be a bloke she signed her letter Monsieur Leblanc she had this pseudonym and they correspond it and gasp thought what a great mathematician Monsieur LeBlanc was he did eventually discover that mr. LeBlanc was in fact Sophie's young man to his credit he thought that was great he didn't think she's a woman so Sophie's amount was somebody who was able to deal with a whole bunch of cases of Fermat's Last Theorem all at once so firma had done horsepowers Euler had done cubes that's great but if you just pick them off one at a time it's going to take a long time because there are infinitely many cases to deal with so if he's young I was able to deal with a whole family of them which was really exciting and then others came along after that building on her work and it was finally solved in 1990s it took a long time so this is Andrew Wiles kind of as he is now he had a little bit more hair in 1995 so he was then in Princeton he's now in Oxford so the building I work in the equivalence of the sense of mathematical sciences here is called the Andrew Wiles building in Oxford and his office is there mine I haven't quite plucked up the courage to ask him how he feels about getting post addressed to Professor Sir Andrew Wiles the Andrew Wiles building it's one of the most celebrated mathematical achievements of all time is such a famous problem but the way he did it is just utterly beyond anything that farmer could have begun to imagine so what ad you did was he proved a particular case of a result called the tiny Amish Amuro conjecture and he had some help from his graduate student then graduate student form graduate student Richard Taylor and they proved this particular result but somebody else had shown well if that's results true then Fermat's Last Theorem would also be true so that was a really big motivation for Andrew Wiles to be working on this problem he spent seven years working on this problem just I just want you to put in context you know maybe you were working on some hard problems today maybe you spent like half an hour being stuck or something maybe if you have a particularly kind teacher they might sometimes give you a really hard problem and you might go away and be stuck on it for a week or something I spent four years proving a result for my PhD and that was like a tiny result not like this kind of result and you are spent seven years proving Fermat's Last Theorem it's pretty amazing achievement most people kind of talk to each other about that what they're doing so lots of mathematicians will collaborate and share ideas and so on Andrew was really focused on I'm just going to prove Fermat's Last Theorem it's a really extraordinary so there's this fantastic BBC horizon documentary if you haven't watched it you are obliged to go and watch it so you google iPlayer Fermat's Last Theorem so the documentary was made at the time which is a long time ago by your standards it's quite a long time ago by my standards but it's online it's on the iPlayer and the best-ever horizon documentaries or something it's fantastic it is less than an hour of your life and it's superb good so what this means is that if I'm looking for positive integer solutions the equation I gave you has no solutions there are some boring solutions if you allow zero um okay good let's go to the force equation I gave you y squared is X cubed minus 2 yeah so I guess this is 30 or undergraduate material you won't mind if I carry on with this excellent I like a bit of ambition so if we just switch it round it's like so I've deliberately witnesses y squared equals x q minus 2 and I'll try to remember to tell you why in a minute but if we switch it around it's just slightly easier to think about so just think about cases so if Y is even then Y squared is even so Y square plus 2 is even so X cubed is even so X is even right exactly if we think about this if X is even then X cubed is a multiple of 8 2 times something times 2 2 have some things I pursuits have some multiple of 8 but Y square plus 2 is not a multiple of 8 y squared is a multiple of 4 I had to I don't get a multiple of 8 don't worry if you don't getting all the details here I just want to give you a kind of taste to it so what we've said is if Y is even then we kind of discover the X and y that they don't satisfy this equation so if we've got us an equation sort a solution to this equation then Y has to be odd that's great so we started out with infinitely many possible values of Y and now we've got half of them so that's still infinitely many possible values of Y great yeah that's sort of exhausted that line of thought so the equations is really hard and sometimes you need lots of ideas and sometimes you try this idea and it doesn't seem so helpful and then you have to have another idea you just kind of have to be fearless about trying it I mean if you sit there and think oh gosh I've got no idea at all how to solve this equation you're guaranteed not to solve it whereas if you think well there's half a percent chance that this idea might work well that's already a lot of progress right you might as well try it and see happens okay here's another idea so if I look at this X cubed is y squared plus 2 so the right hand side is a difference of two squares so I can factor eyes that it is just possible that you are not looking at this and thinking is y squared plus 2 that's a difference of two squares partly because it's not a difference and partly because they're not squares but I say be more open-minded because if I thought about this is y squared minus minus 2 and I thought the minus 2 was a square it would be a difference of two squares right I mean what's not to like this is how you make progress in mathematics by just kind of thinking well what if I'll give it a go so here we go I factorize it it's a difference of two squares so I've written this thing called square root of minus 2 and my estimate from thinking about X square plus 2 equals 0 earlier honest there may be a third if you haven't learnt about complex numbers yet and I think that's illegal well that's ok it'll be fine ok so the square root of minus 2 here's what you need to know about the square root of minus 2 okay if you square it you get minus 2 I mean if I give you the number square root of 2 what do you need to know about the square root of 2 you need to know that if you square it you get 2 I don't see why the square root of 2 is somehow more superior than the square root of minus 2 just because I can put it on some number line somewhere no big deal I'm just going to write the square root of minus 2 so if you know about complex numbers you might be thinking oh that's I times the square root of 2 that would be fine they're sort of convention in this area of number theory is not to write I is to write the square root of minus 2 so what i'm doing here is kind of conventional but you could write it in terms of i these are just complex numbers and what i'm going to do is allow myself to work in this extended pool of numbers where instead of just having whole numbers I'm allowed whole number plus whole number times the square root of minus 2 it's a little bit like real and imaginary parts for those who come across complex numbers if you haven't come across complex numbers don't panic it'll be fine just every time you seem - 2 squared we'll replace it by minus 2 so good we factorize this equation now I'm going to not fill in the details of a step for you because it's just a little bit of fiddly and I don't spend time on that but you can think about this so I claim that the two factors on the right have highest common factor 1 so if you have something that divides into both of them it has to be like 1 or minus 1 and I'll let you think about that what's really powerful about that that incidentally uses the fact that Y is odd so that little bit of working we did on the previous slide where we pointlessly ruled out half the cases actually turns out to be really helpful because knowing that Y is odd is useful in this stage of the argument so now what we've got is two numbers they kind of have no factors in common apart from one and minus one and their product is a cube what that means is that each of these numbers individually must be a cube ok this is hard right I told you it's 30 or undergraduate stuff but no I figured you'd like a challenge so each of these as a cube so for example I can write Y plus the square root of 2 as the cube of some number in this extended pool of numbers I'm using so a and B a whole numbers here this is just an arbitrary number in this pool of numbers I'm using and Y plus 4 minus 2 is equal to the cube of it and so I guess you're probably you're better at the binomial theorem than I am so we can expand this thing out there are so it looks a bit messy oh look here here's a square root of minus 2 squared so you can place that by -2 and there's one in here we can simplify as well so if I tidy that up I've got these two terms which are just whole numbers so that's this term and this term and then this part which is something times the square to minus 2 where is all this guy don't know I mean basically this was sort of doing this because I couldn't think of any other idea so you might as well try it right but actually this is quite interesting because what we know is that the coefficient of root minus 2 is 1 so this lot must be equal to 1 so for those of you who know about these things this is like comparing real and imaginary parts that's idea and I've sneakily factored out this B here and now we can recycle an idea we had earlier mathematicians do this all the time right it's good for the environment it saves having new ideas if you just recycle them so earlier on we have that X minus 12 and y minus 18 or whatever it was is equal to something simple that means X minus 12 had to be a factor of 12 times 18 we're here B times some other whole number has to be one and there are very few ways to get one as a product of two whole numbers right so B has to be plus or minus one and then you can solve this has to be plus or minus one and you just figure out an a has to be plus or minus one and now we're pretty much done so why here is this if you like the real part we can substitute in a and B and we get what Y is and then we can go back and figure out what X is so you might have spotted from this equation that 25 is a square number is 27 cube number minus 2 which would give us these two solutions 3 into plus or minus 5 what's great about this is it shows that the only solutions along the way we've done some fairly sophisticated algebraic number theory but you know it's good to have a challenge this equation y squared is X cube minus 2 is an example of an elliptic curve so the reason that I wrote it in that format is that we traditionally write elliptic curves as y squared equals cubic in X so it's cubic because the highest power of X there is a 3 so I've drawn this nice picture of it I didn't feel like that was totally representative of what elliptic curves might look like if we draw a picture so I drew some more for you if you haven't thought about graphs of the form y squared equals something you might like to go and play with that idea a little bit if yourself is kind of interesting so you're kind of maybe used to thinking about y equals something what about y squared equals something it's quite a interesting you might think about what features of the graphs here are a consequence of that so these are thinking about the the real points here whereas before we would think about the whole number solutions that's where we they're only two there's quite a lot to be discovered by thinking about complex solutions so if you allow an x and y to be complex numbers you end up with some exciting kind of donut torus type shape because there are some extra dimensions because each of the complex numbers is kind of two dimensions and then you pick out a surface within it it's all very exciting it ties in with kind of topology so this arithmetic geometry which is all about these kind of rational points integer points on these equations and so on elliptic curves are utterly crucial in Andrew Wiles his proof of Fermat's Last Theorem so the the taniyama-shimura conjecture that he proved was for a particular type of particular family of elliptic curves and he proved that they all had certain properties so I'm a long way off explain to you how his proof goes because it's really hard but these elliptic curves are fundamental objects but they're also kind of interesting in themselves there's a lots of structure to explore they're they link up with algebra geometry number theory they all kind of come together here okay um let me tell you why this is all really hard so David Hilbert was a leading German mathematician end of the 19th beginning of the 20th century and every four years mathematical community gets together what's called the International Congress of mathematicians and everybody gets together and there are sort of talks by fantastic mathematicians to say well this is what I've been doing my research recently and people exchange ideas and they present prizes and there's all sorts of things the International Congress in 1900 but announced these problems he he made his list of problems that he they're kind of like a challenge to the mathematical community these are the problems I think are really important here's 23 of them and the one that's relevant for us here is Hilbert's tenth problem so some of Hilbert's problems have been solved some of them have not yet been solved so yeah when you've sorted out the other little loose ends I've left you for think about on the bus back or maybe that'll take you to a tomorrow you go look up Hilbert's 23 problems and pick one to work on that'll keep you busy for a while and when you've got a solution I will be the first to congratulate you because that would be terrific especially if there's Riemann hypothesis I'm very interested in the Riemann hypothesis coefficients the numbers I hold them as and you want whole number solution so they're exactly the kinds of things that we've been talking about today okay so that's just deciphering that so any number of variables we want whole number solutions everything in sight is a whole number and what Hilbert wanted in his problem was to come up with a kind of algorithm a process that a computer could run he wasn't saying that in 1900 but that's how we might now think about it that would decide whether this equation has whole number solutions so not even as difficult as the problem I gave you right at the beginning of you know how many are there just are there any solutions or no solutions and this algorithm has to you know you feed in your equation the computer whirs and clunks a bit and crucially after a finite amount of time comes back and says yes it's got introduced solutions or no it hasn't okay so that's what held what wanted Hilbert very much had this agenda that that we would find out the answers to this that you can answer all mathematical problems by the 1930s it was becoming clear that that's just not true I don't mean not true in some sort of vague philosophical sense I mean not true in a provable mathematical sense so there's this extraordinary mathematician girdle who came along and turned it all upside down said well hang on a minute so people just started to think well is it even possible to come up with this algorithm this process that Hilbert wants and one of the first people to make progress in this was Julia Robinson who's a really interesting mathematician and not not enormous ly famous but should be more famous I think it's a really interesting story so things like she married a mathematician and because he had a job in the department she wasn't allowed a job as well because there were rules then they couldn't both be employed after she'd kind of solved this world famous problem they did eventually decide they give her a job so in 1952 she made a breakthrough that wasn't immediately related to Hilbert's tenth problem but wrote this paper that would turn out to be enormous ly relevant so she was thinking about something slightly different and then in 1960 she got this draft of a paper by two other mathematicians Davis and Putnam and she realized that if they've combined her work with there were they'd have made a lot of progress on this problem of showing that in fact there isn't such a process and then they kind of so they wrote up this paper in 1961 they said well you know his where we're at in 1970 it's very young Russian mathematician Matias a managed to fill in the gap they were missing and having seen how he did that Julia Robinson realized that in fact she sort of been quite close 20 years earlier but you have to look at the problem the right way you have to have the right ideas in the right order so it's this kind of collaboration show there is no computer program that will is guaranteed to decide in a finite amount of time yes this dyfan tiny equation has integer solutions no it doesn't now depending on your perspective that's devastating or fantastic I mean yo it means the different equations are really hard right you can't just use a computer to decide yes there's a solution no there isn't so that's great because it means there are lots of different equations out there to study so the tools that Andrew Wiles used for studying Fermat's Last Theorem that equation people are using developing to tackle more equations there are other equations out there that we still just don't have the tools to begin to understand and you know it would be great if some of you got involved in sorting that out so my question for you is which tie finds phone equations will you tackle next thank you very much you you

Channel: Millennium Mathematics Project - maths.org

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Rating: 4.8557115 out of 5

Keywords: Mathematics, Pure mathematics, Number theory, Mathematics education, Solving equations, Vicky Neale, maths.org, nrich.maths.org, University of Cambridge, University of Oxford, Mathematical thinking, Mathematical problem solving, Maths, Pure maths, Developing mathematical thinking, STEP, Pythagorean triples, Fermat's last theorem, Women in mathematics, Equations, Math

Id: lfXej2qAd8c

Length: 43min 49sec (2629 seconds)

Published: Tue May 17 2016