{"version":"1.0.0","segments":[{"startTime":5.145,"endTime":10.401,"body":"So in this then my final Gresham lecture, I want to talk about one of"},{"startTime":10.401,"endTime":13.205,"body":"the defining characteristics of mathematics and that's proof."},{"startTime":14.265,"endTime":19.262,"body":"So if we imagine ourselves 2000 years ago in ancient Greece, as we we're all"},{"startTime":19.262,"endTime":24.259,"body":"well educated people, we all know that there are four elements that make up everything"},{"startTime":24.259,"endTime":26.925,"body":"in the universe, earth, air, fire, and water."},{"startTime":27.585,"endTime":30.995,"body":"And we also all know that if you have a right angled triangle, the square"},{"startTime":30.995,"endTime":34.405,"body":"on the hypo news is the sum of the squares on the other two sides."},{"startTime":35.185,"endTime":40.566,"body":"Now, today we still believe one of those things, but probably we don't believe the"},{"startTime":40.566,"endTime":40.925,"body":"other."},{"startTime":41.465,"endTime":43.285,"body":"And the difference is proof."},{"startTime":44.425,"endTime":47.285,"body":"We have proved pyrosis theorem mathematically."},{"startTime":48.345,"endTime":52.325,"body":"And when you have a mathematical proof, as long as it's correct, then what you've"},{"startTime":52.325,"endTime":55.245,"body":"proved is true and it's true now and it'll stay true."},{"startTime":56.625,"endTime":61.845,"body":"So today I'm gonna show you some of my all time favorite mathematical proofs."},{"startTime":62.495,"endTime":67.285,"body":"We're gonna talk along the way about what makes a great mathematical proof."},{"startTime":67.905,"endTime":72.621,"body":"But everything I'll show you today has something in common and that is a flash"},{"startTime":72.621,"endTime":75.765,"body":"of pure genius that makes these things even more delightful."},{"startTime":76.065,"endTime":77.165,"body":"So I hope you'll enjoy them."},{"startTime":77.165,"endTime":81.823,"body":"And I thought we should start with, since we've already mentioned one theorem proving pyres"},{"startTime":81.823,"endTime":82.445,"body":"is theorem."},{"startTime":82.705,"endTime":84.485,"body":"Now there are lots and lots of different proofs."},{"startTime":84.485,"endTime":89.635,"body":"This theorem has been discovered and rediscovered in lots of different cultures over many different"},{"startTime":89.635,"endTime":92.725,"body":"times, many proofs, hundreds of them are out there."},{"startTime":93.225,"endTime":96.045,"body":"So it's hard to pick the favorite."},{"startTime":96.385,"endTime":99.765,"body":"Uh, but I've got two to show you 'cause I couldn't pick."},{"startTime":99.985,"endTime":104.554,"body":"Uh, let's just remind ourselves, we all all know rissy theum, but just to be"},{"startTime":104.554,"endTime":109.124,"body":"sure, if you take any old RightAngle triangle and it sides are AB and hypotonus"},{"startTime":109.124,"endTime":113.085,"body":"C, then we know that A squared plus B squared equals C squared."},{"startTime":113.355,"endTime":113.645,"body":"Okay?"},{"startTime":113.905,"endTime":114.885,"body":"Iconic equation."},{"startTime":115.785,"endTime":120.856,"body":"So we're gonna take this triangle and we are gonna look at two diagrams and"},{"startTime":120.856,"endTime":125.928,"body":"you almost don't need any words this point, these two diagrams almost just tell their"},{"startTime":125.928,"endTime":126.605,"body":"own story."},{"startTime":127.225,"endTime":130.645,"body":"So both of those show a big square."},{"startTime":131.585,"endTime":133.565,"body":"Uh, its side is A plus B."},{"startTime":134.145,"endTime":138.962,"body":"And inside that square we've got four copies of our triangle and then some space"},{"startTime":138.962,"endTime":139.605,"body":"that's left."},{"startTime":140.145,"endTime":144.245,"body":"Now on the left hand side, the space that's left is a square of side"},{"startTime":144.245,"endTime":145.885,"body":"C in the right hand side."},{"startTime":146.735,"endTime":150.926,"body":"Apart, once you've taken account of those four triangles, the space that's left is divided"},{"startTime":150.926,"endTime":151.765,"body":"into two squares."},{"startTime":151.985,"endTime":154.525,"body":"One has side A and one has side B."},{"startTime":155.185,"endTime":160.805,"body":"So because you've got the same area in each picture, those areas are the same."},{"startTime":160.945,"endTime":165.517,"body":"So we can straightaway say that C squared plus four times the area of that"},{"startTime":165.517,"endTime":170.09,"body":"build triangle is equal to A squared plus B squared plus four times the area"},{"startTime":170.09,"endTime":171.005,"body":"of that triangle."},{"startTime":171.505,"endTime":173.765,"body":"And so those things are equal."},{"startTime":173.785,"endTime":175.925,"body":"So we get A squared plus B squared equals C squared."},{"startTime":176.245,"endTime":177.965,"body":"I think it's a very pretty little proof."},{"startTime":179.085,"endTime":180.445,"body":"I wanna show you another proof though."},{"startTime":180.545,"endTime":183.085,"body":"And this is not because we need to prove something twice."},{"startTime":183.545,"endTime":186.805,"body":"If our first proof is correct, that's fine, that's all we need."},{"startTime":187.345,"endTime":191.722,"body":"But often we see that if you look at it things from a different viewpoint"},{"startTime":191.722,"endTime":193.765,"body":"and prove things in a different way."},{"startTime":194.525,"endTime":196.725,"body":"A, it's fun, there's no problem with that."},{"startTime":196.865,"endTime":202.633,"body":"But also it can show you different approaches, maybe ways you might generalize what you're"},{"startTime":202.633,"endTime":205.325,"body":"talking about, techniques you might use elsewhere."},{"startTime":206.145,"endTime":211.366,"body":"And the other proof of phys serum, I'm gonna show you I really like it"},{"startTime":211.366,"endTime":216.588,"body":"because to me it feels surprising that the outcome tells you something about the areas"},{"startTime":216.588,"endTime":217.285,"body":"of squares."},{"startTime":217.885,"endTime":220.165,"body":"'cause it doesn't seem to involve any of those things along the way."},{"startTime":220.825,"endTime":221.645,"body":"So let's see it."},{"startTime":221.785,"endTime":225.165,"body":"Here's our triangle, our right angle, triangle sides, A, B, and C as before."},{"startTime":225.825,"endTime":229.965,"body":"And we're gonna drop just a perpendicular from that top vertex like that."},{"startTime":230.195,"endTime":233.085,"body":"That divides our triangle into two smaller triangles."},{"startTime":233.705,"endTime":238.9,"body":"Now each of those smaller triangles is similar to the one we started with, similar"},{"startTime":238.9,"endTime":241.325,"body":"as in all the angles match up."},{"startTime":242.065,"endTime":245.365,"body":"And just make sure we believe that the one on the left, that green one,"},{"startTime":245.365,"endTime":248.665,"body":"we can see the right angle there, we can see it's got that yellow angle"},{"startTime":248.665,"endTime":250.205,"body":"the same as the, the big triangle."},{"startTime":250.555,"endTime":253.78,"body":"Well, whatever's left, we know yet the yellow and red angles add up to 90"},{"startTime":253.78,"endTime":255.285,"body":"degrees because they're in the big triangle."},{"startTime":255.745,"endTime":258.045,"body":"So the remaining angle must be the same as this red one."},{"startTime":258.625,"endTime":261.872,"body":"And in the the blue triangle on the right, same thing we get, we get"},{"startTime":261.872,"endTime":263.605,"body":"now the red and yellow angles as before."},{"startTime":264.065,"endTime":267.005,"body":"So those two triangles are similar to the big one."},{"startTime":267.545,"endTime":274.351,"body":"And we know about similar triangles that uh, ratios of lengths of corresponding sides are"},{"startTime":274.351,"endTime":274.805,"body":"equal."},{"startTime":275.265,"endTime":280.49,"body":"So if we look at that big triangle, if we take the hypotonus length C"},{"startTime":280.49,"endTime":285.715,"body":"and the second longest side, a C divided by A will equal that corresponding ratio"},{"startTime":285.715,"endTime":287.805,"body":"in, in the left hand triangle."},{"startTime":287.905,"endTime":291.381,"body":"So the, the hypo news of that one is A, and the next longest side"},{"startTime":291.381,"endTime":291.845,"body":"is X."},{"startTime":292.065,"endTime":294.885,"body":"So we get C over A equals A over X."},{"startTime":295.545,"endTime":299.302,"body":"And we can do the same thing in the right hand triangle this time comparing"},{"startTime":299.302,"endTime":300.805,"body":"the hypotonus to the shortest side."},{"startTime":301.265,"endTime":305.196,"body":"So in the big triangle, C over B must be equal to in the small"},{"startTime":305.196,"endTime":306.245,"body":"triangle, B over Y."},{"startTime":306.755,"endTime":308.645,"body":"Okay, so we get these two expressions."},{"startTime":309.665,"endTime":313.07,"body":"Now all we're gonna do is tidy those up a bit, get rid of the"},{"startTime":313.07,"endTime":314.205,"body":"fractions, rearrange a little bit."},{"startTime":314.465,"endTime":318.685,"body":"And we see from these we find an expression for A squared it equals cx."},{"startTime":318.745,"endTime":321.285,"body":"And for B squared on the right that equals CY."},{"startTime":321.585,"endTime":323.685,"body":"Now we want to know what's A squared plus B squared."},{"startTime":324.705,"endTime":326.285,"body":"So we can just add those up."},{"startTime":326.525,"endTime":330.845,"body":"A squared is cx, B squared is CY, bring out that factor of C."},{"startTime":331.385,"endTime":332.525,"body":"Now what's x plus Y?"},{"startTime":332.705,"endTime":336.71,"body":"Oh it's just CX and Y is how we divided up that that long bottom"},{"startTime":336.71,"endTime":337.245,"body":"side C."},{"startTime":337.865,"endTime":341.354,"body":"So again, we get this, it just falls out pys Ethereum A squared plus B"},{"startTime":341.354,"endTime":342.285,"body":"squared equals C squared."},{"startTime":342.685,"endTime":346.165,"body":"I really love this 'cause yeah, it's just a surprise to me."},{"startTime":346.165,"endTime":347.925,"body":"Nothing about areas really in here."},{"startTime":348.195,"endTime":352.631,"body":"Some we just do a couple of little calculations with similar triangles and outcomes Ethereum"},{"startTime":352.631,"endTime":355.885,"body":"in a completely different way from how we saw it before."},{"startTime":356.585,"endTime":360.125,"body":"So that's just beautiful and there are hundreds of proofs of pyros the out there."},{"startTime":360.145,"endTime":364.42,"body":"So I encourage you to go and have a look and try and find your"},{"startTime":364.42,"endTime":365.845,"body":"favorite proof of pyrosis theorem."},{"startTime":367.115,"endTime":370.605,"body":"Well, let's think then about what comprises of proof."},{"startTime":372.225,"endTime":376.157,"body":"Before you can do anything, before you can even write the word proof and begin"},{"startTime":376.157,"endTime":379.565,"body":"what you're going to do, you need something that you want to prove."},{"startTime":380.025,"endTime":385.743,"body":"So Pyrosis theorem tells us something about right angle triangles, which had to first be"},{"startTime":385.743,"endTime":386.125,"body":"noticed."},{"startTime":386.825,"endTime":390.584,"body":"So you look at lots of different RightAngle triangles and after a while you notice"},{"startTime":390.584,"endTime":394.343,"body":"this relationship between the length of the sides, you spot a pattern and you make"},{"startTime":394.343,"endTime":394.845,"body":"a conjecture."},{"startTime":395.265,"endTime":397.605,"body":"And then, and only then are you gonna try and prove it."},{"startTime":398.505,"endTime":401.52,"body":"So in order to do any kind of proofs at all, we have to be"},{"startTime":401.52,"endTime":402.325,"body":"good at making conjectures."},{"startTime":402.625,"endTime":404.445,"body":"And that's all about spotting patterns."},{"startTime":404.705,"endTime":407.365,"body":"So we're gonna have a practice pattern spotting."},{"startTime":407.365,"endTime":409.485,"body":"Now, uh, so here's a question."},{"startTime":409.705,"endTime":412.405,"body":"Um, we've gotta work out what the pattern is here."},{"startTime":412.745,"endTime":417.766,"body":"So you draw a circle, um, you draw some points on the circumference end points,"},{"startTime":417.766,"endTime":420.445,"body":"let's say n equals three to begin with."},{"startTime":420.785,"endTime":424.205,"body":"Uh, and we are gonna join those points with lines like this."},{"startTime":424.505,"endTime":427.085,"body":"So n equals three, join the points with lines."},{"startTime":427.945,"endTime":433.026,"body":"And the question is into how many regions can we divide the circle by doing"},{"startTime":433.026,"endTime":433.365,"body":"this?"},{"startTime":433.475,"endTime":437.802,"body":"Alright, so with three points you can instantly see I've got four regions that I've"},{"startTime":437.802,"endTime":439.245,"body":"divided that circle up into."},{"startTime":439.785,"endTime":443.715,"body":"So a bit early perhaps to make a conjecture and equals three gives answer four,"},{"startTime":443.715,"endTime":447.645,"body":"we don't, you know, we haven't got enough information, so let's get some more data."},{"startTime":448.865,"endTime":449.845,"body":"Here's some more data."},{"startTime":450.155,"endTime":454.627,"body":"Okay, so the numbers here are how many regions we've managed to divide the circle"},{"startTime":454.627,"endTime":458.205,"body":"into with one point, just the circle itself, one region, two points."},{"startTime":458.295,"endTime":460.085,"body":"We've divided that region up into two."},{"startTime":460.105,"endTime":463.965,"body":"So we've got two regions, three points gives you four, four points."},{"startTime":464.025,"endTime":466.605,"body":"You can eyeball that and see it's eight five points."},{"startTime":466.665,"endTime":473.011,"body":"You might not necessarily instantly count that in your head, but let's count it 1,"},{"startTime":473.011,"endTime":479.357,"body":"2, 3, 4, 5, uh, 6, 7, 8, let's see, 9, 10, 11, and then"},{"startTime":479.357,"endTime":483.165,"body":"that last bit, 12, 13, 14, 15 and 16."},{"startTime":483.315,"endTime":486.325,"body":"Okay, so look at that pattern."},{"startTime":486.825,"endTime":488.725,"body":"1, 2, 4, 8, 16."},{"startTime":490.025,"endTime":492.125,"body":"We are in a position to make a conjecture, right?"},{"startTime":492.125,"endTime":496.351,"body":"We, we, this is like oh, powers of two, but if we want to prove"},{"startTime":496.351,"endTime":500.577,"body":"this, uh, you sort of think, well how, okay, the number of regions seems to"},{"startTime":500.577,"endTime":504.803,"body":"be doubling whenever I add a point and all of the lines that go with"},{"startTime":504.803,"endTime":505.085,"body":"it."},{"startTime":505.425,"endTime":509.605,"body":"So somehow doing this is gonna perhaps divide each region into two."},{"startTime":509.665,"endTime":514.044,"body":"That's how we might double, but it seems really hard to do this calculation."},{"startTime":514.345,"endTime":518.965,"body":"And the way I counted those off there was a bit haphazard."},{"startTime":519.155,"endTime":521.885,"body":"It's hard to spot what's, what's going on the structure of what's going on."},{"startTime":522.265,"endTime":525.492,"body":"So if we don't know how to do our proof, perhaps we, we do the"},{"startTime":525.492,"endTime":528.719,"body":"next one up and see if we can do it in a bit more of"},{"startTime":528.719,"endTime":529.365,"body":"an organized way."},{"startTime":529.385,"endTime":531.685,"body":"So let's, let's think what happens when we have six points."},{"startTime":532.305,"endTime":535.77,"body":"So I've sort of drawn them kind of haphazardly on this circle and let's count"},{"startTime":535.77,"endTime":536.925,"body":"in a more systematic way."},{"startTime":537.225,"endTime":541.089,"body":"So with six points here and we've got the regions on the outside, there's six"},{"startTime":541.089,"endTime":541.605,"body":"of those."},{"startTime":542.675,"endTime":544.765,"body":"Were the ones with, you know, curved parts to them."},{"startTime":545.225,"endTime":548.275,"body":"And then we can kind of go back in the triangles next to those, those"},{"startTime":548.275,"endTime":548.885,"body":"little regions there."},{"startTime":548.885,"endTime":549.565,"body":"There's six there."},{"startTime":549.625,"endTime":550.925,"body":"So six plus that went up to 12."},{"startTime":551.625,"endTime":556.203,"body":"And then we've got this kind of six pointed star and the points each comprised"},{"startTime":556.203,"endTime":558.645,"body":"two more regions, the ones I've colored yellow."},{"startTime":558.665,"endTime":563.005,"body":"So we've got another 12 regions there, so 24 so far."},{"startTime":563.545,"endTime":567.005,"body":"And then we've got another six regions that I'm colored in green here."},{"startTime":567.005,"endTime":567.685,"body":"So up to 30."},{"startTime":568.705,"endTime":575.365,"body":"And then there's only one region left, which rather messes up our lovely \u003claugh\u003e, our"},{"startTime":575.365,"endTime":579.805,"body":"lovely idea that we were gonna prove powers of two."},{"startTime":579.805,"endTime":583.367,"body":"Well, well what's happening here and I have to break it to you only gets"},{"startTime":583.367,"endTime":583.605,"body":"worse."},{"startTime":584.305,"endTime":588.165,"body":"The next two numbers in this pattern are 57 and 99, right?"},{"startTime":588.165,"endTime":591.045,"body":"So we're getting further and further away from the powers of two that we thought."},{"startTime":591.425,"endTime":595.592,"body":"So what I was really doing here is saying yes, we need to make conjectures"},{"startTime":595.592,"endTime":599.759,"body":"and spot patterns, but we really, really, really need proofs proof means never having to"},{"startTime":599.759,"endTime":603.927,"body":"say you are sorry \u003claugh\u003e, we are not gonna have to go back on our"},{"startTime":603.927,"endTime":604.205,"body":"word."},{"startTime":604.205,"endTime":607.765,"body":"If we could prove we can work out what this pattern is and prove, um,"},{"startTime":607.765,"endTime":611.326,"body":"prove it, get a formula and prove it, then we are going to know for"},{"startTime":611.326,"endTime":614.887,"body":"sure what happens rather than having to make a guess which might then later be"},{"startTime":614.887,"endTime":615.125,"body":"wrong."},{"startTime":615.705,"endTime":619.377,"body":"Um, so I'm gonna leave to you if you want to think about what on"},{"startTime":619.377,"endTime":622.805,"body":"earth this might be and what's going on in the transcript for the lecture."},{"startTime":623.085,"endTime":626.797,"body":"I have put what the formula is and I've given a proof of what it"},{"startTime":626.797,"endTime":627.045,"body":"is."},{"startTime":627.105,"endTime":629.961,"body":"So it's a bit long to long to do now, but if if you're interested"},{"startTime":629.961,"endTime":631.485,"body":"in that, you can look at that later."},{"startTime":631.825,"endTime":634.925,"body":"So this is, this is my, we need to prove things for sure."},{"startTime":636.585,"endTime":639.729,"body":"So then once we've made a conjecture and we think we know what the right"},{"startTime":639.729,"endTime":642.245,"body":"answer is, we want to prove it, we have to prove it."},{"startTime":643.635,"endTime":644.685,"body":"What is a good proof?"},{"startTime":644.685,"endTime":644.925,"body":"Then?"},{"startTime":645.025,"endTime":646.325,"body":"I'm talking about, I'm saying the proof."},{"startTime":646.485,"endTime":649.453,"body":"I'm showing you a a lovely and beautiful and I can say that because I"},{"startTime":649.453,"endTime":650.245,"body":"didn't think of them."},{"startTime":650.265,"endTime":654.599,"body":"You know, these are, these are wonderful mathematical edifices passed down, uh, through the ages"},{"startTime":654.599,"endTime":658.645,"body":"for us to all marvel at and be just, just enjoy for their loveliness."},{"startTime":659.395,"endTime":661.765,"body":"What is it that mathematicians rate about a proof?"},{"startTime":661.765,"endTime":663.165,"body":"What do we think makes a good proof?"},{"startTime":663.675,"endTime":667.006,"body":"Well, let me give you an example of something that will, first of all, not"},{"startTime":667.006,"endTime":670.338,"body":"a good proof and then we'll we'll see how to make it good and it's"},{"startTime":670.338,"endTime":671.005,"body":"another possible pattern."},{"startTime":671.505,"endTime":674.805,"body":"So this one is about, uh, an expression, I'll call it F of N."},{"startTime":674.805,"endTime":678.045,"body":"And you work out n squared plus n plus 41 A."},{"startTime":678.105,"endTime":680.285,"body":"And we're working this out for numbers N starting at zero."},{"startTime":680.345,"endTime":681.845,"body":"So whole, whole number integers."},{"startTime":682.225,"endTime":686.106,"body":"So if you put N equals zero into that, you just get the constant term"},{"startTime":686.106,"endTime":686.365,"body":"41."},{"startTime":686.905,"endTime":687.925,"body":"So that's a prime number."},{"startTime":688.185,"endTime":690.885,"body":"If you put n equals one, you get 43."},{"startTime":691.395,"endTime":697.104,"body":"Also a prime number N equals two, uh, 47, also a prime number N equals"},{"startTime":697.104,"endTime":697.485,"body":"three."},{"startTime":697.855,"endTime":698.365,"body":"Let's do it."},{"startTime":698.665,"endTime":700.525,"body":"Is it 53 also a prime number?"},{"startTime":700.905,"endTime":703.205,"body":"Oh gosh, lots of prime numbers coming out here."},{"startTime":703.585,"endTime":707.35,"body":"If you take this a bit further, um, the first few terms you can see"},{"startTime":707.35,"endTime":708.605,"body":"all of them are prime."},{"startTime":709.315,"endTime":709.765,"body":"It's prime."},{"startTime":709.765,"endTime":713.43,"body":"For the first 10 values, it's prime for the first 20, it's primed."},{"startTime":713.43,"endTime":716.478,"body":"For the first 30, you start thinking, oh my goodness, I've got a prime number"},{"startTime":716.478,"endTime":716.885,"body":"generator here."},{"startTime":717.835,"endTime":722.516,"body":"Well, since the incident of the circles, uh, perhaps we don't necessarily believe that all"},{"startTime":722.516,"endTime":725.325,"body":"of these numbers are going to be primed forever."},{"startTime":725.665,"endTime":727.125,"body":"And in fact we'd be right to be skeptical."},{"startTime":727.695,"endTime":729.005,"body":"These are not always prime."},{"startTime":729.745,"endTime":733.125,"body":"So how could we prove that once we've discovered it?"},{"startTime":734.065,"endTime":737.749,"body":"So what we might do, in fact the first time we see a question like"},{"startTime":737.749,"endTime":741.434,"body":"this, we might, and it's perfectly legitimate thing to do, just keep plugging in values"},{"startTime":741.434,"endTime":743.645,"body":"and see, just keep going and going and going."},{"startTime":744.065,"endTime":748.154,"body":"And eventually this is the bad proof after a very long time and I'm not"},{"startTime":748.154,"endTime":749.245,"body":"going to do it."},{"startTime":749.245,"endTime":750.725,"body":"But it wouldn't make for a great lecture."},{"startTime":751.385,"endTime":756.965,"body":"We find we get to the number 1,681 and we notice, oh, that's not prime."},{"startTime":757.025,"endTime":758.045,"body":"Oh, it's not prime, right?"},{"startTime":758.045,"endTime":758.565,"body":"We can stop."},{"startTime":758.565,"endTime":761.005,"body":"We found a value event, which that is not prime."},{"startTime":761.465,"endTime":766.4,"body":"Uh, and so you, you can, this is, it would be a correct proof, it's"},{"startTime":766.4,"endTime":768.045,"body":"just not a great proof."},{"startTime":768.265,"endTime":769.525,"body":"It doesn't really tell you anything."},{"startTime":769.525,"endTime":770.365,"body":"It doesn't have insight."},{"startTime":770.365,"endTime":772.525,"body":"You've just plugged away at it."},{"startTime":772.665,"endTime":775.005,"body":"So you can call this a proof by exhaustion if you like."},{"startTime":776.225,"endTime":781.085,"body":"But a great proof of this is to notice something about what's going on."},{"startTime":781.435,"endTime":787.899,"body":"That 41, if we put n equals 41 into that expression, then every term 41"},{"startTime":787.899,"endTime":790.485,"body":"is a constant at the end."},{"startTime":790.575,"endTime":795.133,"body":"Everything with an n in, if you put n is 41, um, will now be"},{"startTime":795.133,"endTime":796.045,"body":"divisible by 41."},{"startTime":796.145,"endTime":798.645,"body":"So every term will be divisible by 41."},{"startTime":799.345,"endTime":802.701,"body":"So we've got a, a multiple of 41 will come out a bigger multiple than"},{"startTime":802.701,"endTime":802.925,"body":"one."},{"startTime":803.865,"endTime":806.645,"body":"And so that can't be a prime number."},{"startTime":807.145,"endTime":812.016,"body":"So that observation, once you've noticed that means you don't have to do any calculations"},{"startTime":812.016,"endTime":813.965,"body":"at all, which is always good."},{"startTime":814.105,"endTime":816.005,"body":"And you certainly don't have to do loads and loads of them."},{"startTime":816.425,"endTime":818.725,"body":"You just notice of course this can never be prime."},{"startTime":819.985,"endTime":825.601,"body":"And another great thing about this observation, which makes it a good, a good proof,"},{"startTime":825.601,"endTime":831.217,"body":"is that you can generalize it pretty much instantly because there was nothing special about"},{"startTime":831.217,"endTime":836.085,"body":"the number 41 there really, uh, that enabled us to do this trick."},{"startTime":837.305,"endTime":841.761,"body":"So we can instantly generalize this and make our own little theorem that says actually"},{"startTime":841.761,"endTime":846.217,"body":"no expression like this, but of this form, something like a n squared plus BN"},{"startTime":846.217,"endTime":850.673,"body":"plus C, no expression like that will always give you prime numbers for n greater"},{"startTime":850.673,"endTime":851.565,"body":"new to zero."},{"startTime":852.465,"endTime":854.725,"body":"And we can, that proof will go something like this."},{"startTime":855.065,"endTime":859.512,"body":"Um, if the number C the constant term is not a prime number, then you"},{"startTime":859.512,"endTime":861.885,"body":"are already happy at when N equals zero."},{"startTime":862.445,"endTime":865.725,"body":"'cause when n equals zero, the output is C not prime done."},{"startTime":866.185,"endTime":868.685,"body":"So we only need to worry about what happens when C is prime."},{"startTime":868.745,"endTime":872.957,"body":"But when C is prime the trick I've just noticed instantly works F of C"},{"startTime":872.957,"endTime":875.765,"body":"will give you something that isn't not a prime number."},{"startTime":876.005,"endTime":877.645,"body":"'cause we will have a factor of that prime."},{"startTime":877.945,"endTime":878.725,"body":"So we are done."},{"startTime":879.105,"endTime":884.055,"body":"So it's, it's an observation that one makes your proof extremely short, but two, we"},{"startTime":884.055,"endTime":887.685,"body":"can generalize it and get a much more, um, wider result."},{"startTime":888.145,"endTime":891.35,"body":"So that for me is, is the essence of what makes a good, good proof."},{"startTime":891.395,"endTime":898.78,"body":"It's elegant, it's short, it's ingenious, and, and it tells you something about what the"},{"startTime":898.78,"endTime":899.765,"body":"situation, okay?"},{"startTime":899.765,"endTime":905.894,"body":"So I'm gonna show you now, uh, three different proofs that use three ideas or"},{"startTime":905.894,"endTime":912.024,"body":"concepts that are really ingenious and they sort of lead us to techniques that we"},{"startTime":912.024,"endTime":914.885,"body":"might be able to use more widely."},{"startTime":915.145,"endTime":917.405,"body":"And each of them, i i I love for their different reasons."},{"startTime":918.025,"endTime":923.507,"body":"So the first one is about this question, how many different sums add up to"},{"startTime":923.507,"endTime":928.99,"body":"n So n you know, positive whole number like three and what I'm talking about,"},{"startTime":928.99,"endTime":932.645,"body":"everything's a whole number here and I'm counting as different."},{"startTime":932.645,"endTime":935.605,"body":"You'll notice two plus one I'm saying is different from one plus two."},{"startTime":935.625,"endTime":939.497,"body":"So I I'm, the order matters to me if, if you, if that doesn't matter"},{"startTime":939.497,"endTime":943.369,"body":"to you, it's a different question with a different answer, um, that you might think"},{"startTime":943.369,"endTime":947.241,"body":"about separately for in this case, just genuinely what are the things in what order"},{"startTime":947.241,"endTime":949.565,"body":"add up to three and there are four possibilities."},{"startTime":950.135,"endTime":952.685,"body":"Three, two plus one, one plus two, one plus one plus one."},{"startTime":953.195,"endTime":953.485,"body":"Okay?"},{"startTime":953.485,"endTime":957.466,"body":"So there's four sums that make three and there are eight sums that make four"},{"startTime":957.466,"endTime":959.325,"body":"if you could check if you like."},{"startTime":959.865,"endTime":961.605,"body":"So is well is it powers of two?"},{"startTime":962.025,"endTime":966.928,"body":"Now we, again, we are still a bit suspicious after the, the circles in this"},{"startTime":966.928,"endTime":970.525,"body":"case though it does turn out that it's powers of two."},{"startTime":971.025,"endTime":973.605,"body":"But how could we, how on earth could we prove this?"},{"startTime":973.605,"endTime":977.415,"body":"We're somehow thinking it's gotta double, the number of sums has to double every time"},{"startTime":977.415,"endTime":978.685,"body":"you go up by one."},{"startTime":979.065,"endTime":983.581,"body":"So you might think, well is there a way to make those sums that that"},{"startTime":983.581,"endTime":984.485,"body":"give you three?"},{"startTime":985.285,"endTime":990.37,"body":"I can turn those into sums that give you four by just adding one somewhere,"},{"startTime":990.37,"endTime":992.405,"body":"but I've gotta do it systematically."},{"startTime":993.505,"endTime":996.165,"body":"Um, maybe I just increase the first number by one."},{"startTime":996.165,"endTime":997.205,"body":"That would work."},{"startTime":997.345,"endTime":1000.525,"body":"All of those are sums that give you 4, 4, 3 plus one."},{"startTime":1000.745,"endTime":1003.525,"body":"So the three becomes four, two plus one becomes three plus one."},{"startTime":1003.665,"endTime":1006.325,"body":"So on that obviously isn't giving us everything."},{"startTime":1006.505,"endTime":1008.005,"body":"So is there something else we could do?"},{"startTime":1008.475,"endTime":1012.027,"body":"Well, maybe we can add an extra term, we can add a one at the"},{"startTime":1012.027,"endTime":1015.58,"body":"end if we do that, then three becomes three plus one, two plus one becomes"},{"startTime":1015.58,"endTime":1016.765,"body":"two plus one plus one."},{"startTime":1016.825,"endTime":1017.165,"body":"So on."},{"startTime":1017.625,"endTime":1024.682,"body":"So now you've got eight things, but unfortunately, uh, we've got some duplicates and we've"},{"startTime":1024.682,"endTime":1026.565,"body":"got some missing subs."},{"startTime":1026.944,"endTime":1028.285,"body":"So this doesn't give us everything."},{"startTime":1028.285,"endTime":1029.685,"body":"Maybe there's more things we can do."},{"startTime":1029.944,"endTime":1034.48,"body":"But if you try and explore the possibilities here, it becomes so horrible and complicated"},{"startTime":1034.48,"endTime":1035.085,"body":"and messy."},{"startTime":1035.355,"endTime":1039.204,"body":"It's really, really difficult to make progress if you take this approach."},{"startTime":1039.983,"endTime":1041.483,"body":"And so you might feel a bit stuck."},{"startTime":1042.025,"endTime":1046.784,"body":"And then inspiration has a lovely, lovely argument that that settles this and shows that"},{"startTime":1046.784,"endTime":1051.543,"body":"indeed if you wanna add, make the thumbs, the add to n, there are two"},{"startTime":1051.543,"endTime":1053.765,"body":"to the N minus one of them."},{"startTime":1053.825,"endTime":1058.606,"body":"So for example, when n equals four, N minus one is three, two cubed is"},{"startTime":1058.606,"endTime":1058.925,"body":"eight."},{"startTime":1059.025,"endTime":1060.245,"body":"That's the correct answer."},{"startTime":1060.865,"endTime":1066.187,"body":"And the little bit of genius that allows this to happen is, is this, it's"},{"startTime":1066.187,"endTime":1071.51,"body":"turning this question into a different question with the same answer, which it's easier to"},{"startTime":1071.51,"endTime":1073.285,"body":"calculate in that other context."},{"startTime":1074.025,"endTime":1078.48,"body":"So what you do is you take a a, a, a string of ones n"},{"startTime":1078.48,"endTime":1082.045,"body":"of them, um, just in a line with little gaps between them."},{"startTime":1082.105,"endTime":1086.755,"body":"So if you've got N ones, how many gaps N minus one, okay, between in"},{"startTime":1086.755,"endTime":1091.405,"body":"each of those gaps we are gonna put a symbol, we've got a choice of"},{"startTime":1091.405,"endTime":1093.885,"body":"two symbols and we can do every possibility."},{"startTime":1094.545,"endTime":1097.525,"body":"So the first symbol we can use if we want is a plus sign."},{"startTime":1097.985,"endTime":1100.045,"body":"And the next one is a letter G."},{"startTime":1101.035,"endTime":1101.845,"body":"That seems a bit odd."},{"startTime":1102.125,"endTime":1103.325,"body":"G is gonna stand for glue."},{"startTime":1103.745,"endTime":1107.235,"body":"That's why, because if you've got two ones with it with a g between them,"},{"startTime":1107.235,"endTime":1110.725,"body":"you are going to glue them together to make a bigger number is an example."},{"startTime":1111.105,"endTime":1115.17,"body":"If you take five ones in a row, um, and where you have a g,"},{"startTime":1115.17,"endTime":1116.525,"body":"you're gonna glue those together."},{"startTime":1116.625,"endTime":1120.276,"body":"So you've got one glue to one that makes two, then a plus sign, that"},{"startTime":1120.276,"endTime":1123.685,"body":"means plus, uh, then one glue to one glue to one that makes three."},{"startTime":1124.305,"endTime":1130.091,"body":"So we've coalesced into two plus three and we can see that actually every possible"},{"startTime":1130.091,"endTime":1135.878,"body":"sum that makes in this case five in general n you, you can arrive at"},{"startTime":1135.878,"endTime":1138.965,"body":"by doing this process in exactly one way."},{"startTime":1139.235,"endTime":1142.205,"body":"It's really kind of well-defined, uh, setup."},{"startTime":1143.065,"endTime":1148.688,"body":"So this is allows us to translate that question about thumbs that make a number"},{"startTime":1148.688,"endTime":1154.311,"body":"into saying how many ways are there to put in n minus one spaces, um,"},{"startTime":1154.311,"endTime":1157.685,"body":"symbols where you've got two possibilities for each one."},{"startTime":1158.225,"endTime":1159.925,"body":"So plus or G plus or G."},{"startTime":1160.065,"endTime":1163.629,"body":"So there's two ways to fill the first space times two for the second space"},{"startTime":1163.629,"endTime":1166.005,"body":"times two times two times two N minus one spaces."},{"startTime":1166.385,"endTime":1167.285,"body":"That's our answer."},{"startTime":1167.825,"endTime":1169.845,"body":"That's the number of ways to assign those symbols."},{"startTime":1169.865,"endTime":1171.005,"body":"Two to the n minus one."},{"startTime":1171.305,"endTime":1172.925,"body":"And therefore that's the number of sums."},{"startTime":1173.185,"endTime":1178.099,"body":"So that little flash of insight there that allows you to translate this question to"},{"startTime":1178.099,"endTime":1183.014,"body":"something we can answer much more easily, that's the key to this beautifully simple argument,"},{"startTime":1183.014,"endTime":1184.325,"body":"which I really love."},{"startTime":1185.675,"endTime":1187.725,"body":"Okay, next one."},{"startTime":1188.385,"endTime":1194.121,"body":"The principle here is so kind of self-evident that when I tell you it, you're"},{"startTime":1194.121,"endTime":1199.857,"body":"gonna think that's ridiculous \u003claugh\u003e, that it's even got a name, but it's extremely useful,"},{"startTime":1199.857,"endTime":1201.005,"body":"extremely, extremely useful."},{"startTime":1201.545,"endTime":1207.029,"body":"And it's this, it's called the pigeonhole principle, uh, because the, the, you imagine some"},{"startTime":1207.029,"endTime":1210.685,"body":"pigeons and they're going to each go to a pigeonhole."},{"startTime":1211.105,"endTime":1214.742,"body":"Um, but if you've got more pigeons than pigeonhole, there's gonna be at least one"},{"startTime":1214.742,"endTime":1216.925,"body":"pigeonhole that has more than one pigeon in it."},{"startTime":1217.305,"endTime":1220.765,"body":"That's, that's literally as easy as as this is the whole principle, right?"},{"startTime":1221.185,"endTime":1224.871,"body":"Um, if you wanna be more formally, if you have K boxes and n objects"},{"startTime":1224.871,"endTime":1228.558,"body":"to put in them, if n the number of objects is bigger than K, the"},{"startTime":1228.558,"endTime":1232.244,"body":"number of boxes, then we are guaranteed that at least one box will contain more"},{"startTime":1232.244,"endTime":1233.965,"body":"than one object, however they arrange themselves."},{"startTime":1234.025,"endTime":1235.405,"body":"Or maybe they all go in one box."},{"startTime":1235.425,"endTime":1238.477,"body":"So maybe they spread themselves out as widely as they can, like people sitting on"},{"startTime":1238.477,"endTime":1238.885,"body":"a bus."},{"startTime":1238.905,"endTime":1242.815,"body":"But in this case, if there's more than more objects than boxes, we are guaranteed"},{"startTime":1242.815,"endTime":1246.725,"body":"to have a box with at least two more than one uh, object in it."},{"startTime":1246.955,"endTime":1247.245,"body":"Okay?"},{"startTime":1247.505,"endTime":1251.308,"body":"So this is used in so many proofs in mathematics and many are, you know,"},{"startTime":1251.308,"endTime":1255.111,"body":"very beautiful but would take a long time to, to uh, give you the background"},{"startTime":1255.111,"endTime":1255.365,"body":"behind."},{"startTime":1255.385,"endTime":1257.485,"body":"So I'm gonna give you kind of a toy example."},{"startTime":1258.075,"endTime":1260.925,"body":"It's not particularly profound, but it's kind of fun."},{"startTime":1261.745,"endTime":1266.725,"body":"Um, I picked a totally random number here, 42."},{"startTime":1266.945,"endTime":1269.285,"body":"You can, uh, this works for any number, uh, any whole number."},{"startTime":1269.625,"endTime":1274.525,"body":"Um, it says there's a multiple of 42, which consists just of zeros and ones."},{"startTime":1274.905,"endTime":1278.827,"body":"Now, okay, I'm not being clever, I'm not saying write it in binary \u003claugh\u003e, I"},{"startTime":1278.827,"endTime":1282.75,"body":"mean genuinely writing in base 10 in our normal number system, we can guarantee that"},{"startTime":1282.75,"endTime":1286.672,"body":"there will be a multiple of 42 or in fact any number as we'll see,"},{"startTime":1286.672,"endTime":1288.765,"body":"which just has ones and zeros in it."},{"startTime":1289.385,"endTime":1291.725,"body":"So lots of possible numbers out there of that form."},{"startTime":1291.945,"endTime":1294.902,"body":"But how do we go looking for these things and how do we prove that"},{"startTime":1294.902,"endTime":1296.085,"body":"this is always gonna be possible?"},{"startTime":1296.755,"endTime":1298.045,"body":"Well, the argument goes like this."},{"startTime":1298.775,"endTime":1303.525,"body":"Let's just think about numbers that only have ones 1, 1111 and salt."},{"startTime":1303.825,"endTime":1306.605,"body":"So obviously you can carry on that sequence as, as far as you like."},{"startTime":1308.095,"endTime":1310.205,"body":"Those are gonna be the, the things that are going in the boxes."},{"startTime":1310.205,"endTime":1312.725,"body":"They're, those are the objects that we're gonna put into boxes."},{"startTime":1313.105,"endTime":1318.605,"body":"The boxes are going to be remainders, take any number."},{"startTime":1319.425,"endTime":1323.645,"body":"If you divide it by 42, you get some quotient, don't care about that."},{"startTime":1323.785,"endTime":1327.165,"body":"And a remainder, the remainder is somewhere between zero and 41."},{"startTime":1327.535,"endTime":1328.685,"body":"Can't be 42 or more."},{"startTime":1328.845,"endTime":1330.245,"body":"'cause otherwise we could take off another multiple."},{"startTime":1330.665,"endTime":1335.447,"body":"So when you divide by 42, there are exactly 42 possible remainders from zero to"},{"startTime":1335.447,"endTime":1336.085,"body":"41 inclusive."},{"startTime":1337.225,"endTime":1342.677,"body":"So we're gonna take these numbers, we only need to go as far as the"},{"startTime":1342.677,"endTime":1343.405,"body":"43rd number."},{"startTime":1344.945,"endTime":1349.649,"body":"And at that point we know by the pigeonhole principle we're putting those numbers into"},{"startTime":1349.649,"endTime":1351.845,"body":"the boxes that go with the remainder."},{"startTime":1352.775,"endTime":1356.603,"body":"There must be at least two of these numbers somewhere in the sequence that have"},{"startTime":1356.603,"endTime":1358.645,"body":"the same remainder when you divide by 42."},{"startTime":1359.085,"endTime":1363.435,"body":"'cause there are 43 numbers going into 42 possible boxes, which we label with the"},{"startTime":1363.435,"endTime":1363.725,"body":"remainder."},{"startTime":1364.025,"endTime":1365.685,"body":"So that must, it just must happen."},{"startTime":1365.685,"endTime":1366.725,"body":"Pigeonhole principle."},{"startTime":1367.305,"endTime":1369.445,"body":"So we have these two numbers, at least two numbers."},{"startTime":1370.225,"endTime":1372.965,"body":"So we'll take the bigger one, there's the bigger one, there's the smaller one."},{"startTime":1372.995,"endTime":1374.645,"body":"They're both some multiple of 42."},{"startTime":1374.745,"endTime":1377.645,"body":"We don't care what plus the same remainder are."},{"startTime":1378.185,"endTime":1380.645,"body":"So all we need to do now is subtract the smaller from the bigger."},{"startTime":1381.145,"endTime":1384.826,"body":"If you do that, you get some string of ones followed by a string of"},{"startTime":1384.826,"endTime":1388.508,"body":"zeros and that is gonna be a multiple of 42 'cause we the remainders cancel"},{"startTime":1388.508,"endTime":1389.245,"body":"with each other."},{"startTime":1390.065,"endTime":1393.165,"body":"So this is just guaranteed to happen, which I think is very cool."},{"startTime":1393.185,"endTime":1398.005,"body":"And it, and it, the 40 twoness of this was not important at all."},{"startTime":1398.005,"endTime":1402.107,"body":"We could do this for any number n just take the first I guess n"},{"startTime":1402.107,"endTime":1406.21,"body":"plus one numbers that are a sequence of ones and then look at what happens"},{"startTime":1406.21,"endTime":1408.125,"body":"when you divide by n the remainder."},{"startTime":1408.125,"endTime":1411.866,"body":"There are n possible remainders n plus one numbers to put in them is that"},{"startTime":1411.866,"endTime":1412.365,"body":"same argument."},{"startTime":1413.465,"endTime":1418.04,"body":"So any positive integer n divides some number of this form, which I think is"},{"startTime":1418.04,"endTime":1419.565,"body":"kind of cool and unexpected."},{"startTime":1419.565,"endTime":1421.565,"body":"There's no reason to think that that would be guaranteed."},{"startTime":1422.505,"endTime":1423.525,"body":"So that's quite fun."},{"startTime":1423.705,"endTime":1426.805,"body":"And the pigeonholed principle, yeah, really, really useful technique."},{"startTime":1427.915,"endTime":1434.523,"body":"Okay, the next idea is something that geometry proofs just really, really often do this"},{"startTime":1434.523,"endTime":1435.845,"body":"kind of thing."},{"startTime":1436.225,"endTime":1440.539,"body":"So I'll show you a, a nice little theorem which is perhaps not as well"},{"startTime":1440.539,"endTime":1444.854,"body":"known as some of the other theorems in geometry that we may encounter, but it's"},{"startTime":1444.854,"endTime":1446.005,"body":"very pretty viviana's theorem."},{"startTime":1446.105,"endTime":1451.021,"body":"So take an equilateral triangle, put draw a point anywhere inside that triangle you like"},{"startTime":1451.021,"endTime":1452.005,"body":"anywhere at all."},{"startTime":1452.625,"endTime":1456.778,"body":"And if you work out the distances from that point to the, to the sides"},{"startTime":1456.778,"endTime":1460.931,"body":"of the triangle, the sum of those distances, it's always equal to the height of"},{"startTime":1460.931,"endTime":1461.485,"body":"the triangle."},{"startTime":1462.425,"endTime":1463.845,"body":"So I think it's a lovely little fact."},{"startTime":1464.625,"endTime":1466.365,"body":"So let's see a proof of this."},{"startTime":1467.425,"endTime":1471.128,"body":"The first thing to notice in, in this triangle, indeed in any triangle, the area"},{"startTime":1471.128,"endTime":1473.845,"body":"of a triangle is half the base time is the height."},{"startTime":1474.075,"endTime":1474.365,"body":"Okay?"},{"startTime":1474.665,"endTime":1477.933,"body":"So if the base, I'll just call the base B, the area is half base"},{"startTime":1477.933,"endTime":1478.805,"body":"time is height fine."},{"startTime":1479.425,"endTime":1483.596,"body":"Now here's the trick, uh, we're gonna join some extra lines that weren't there in"},{"startTime":1483.596,"endTime":1487.767,"body":"the question \u003claugh\u003e, this is, you know, it's when you're trying to do geometry proofs"},{"startTime":1487.767,"endTime":1491.938,"body":"yourself and you see one of these things maybe your first go, oh, why didn't"},{"startTime":1491.938,"endTime":1493.885,"body":"they just put that line in already?"},{"startTime":1494.065,"endTime":1498.298,"body":"But you know, \u003claugh\u003e, once it's okay, like we don't always, I may not have"},{"startTime":1498.298,"endTime":1502.531,"body":"thought of is proof and we may not have done but this we're just admiring"},{"startTime":1502.531,"endTime":1506.765,"body":"the beautiful masterpieces here, you know, without feeling annoyed that we didn't think of them."},{"startTime":1506.985,"endTime":1508.205,"body":"We just admire the loveliness."},{"startTime":1508.305,"endTime":1509.565,"body":"So let's admire the loveliness."},{"startTime":1509.615,"endTime":1510.845,"body":"Let's join some lines."},{"startTime":1511.015,"endTime":1514.836,"body":"We're gonna join this point to all the vertices of the triangle like that, that"},{"startTime":1514.836,"endTime":1516.365,"body":"divides this triangle up into three."},{"startTime":1517.545,"endTime":1523.028,"body":"Uh, so now we're gonna just unfold the diagram and we see we've now got"},{"startTime":1523.028,"endTime":1524.125,"body":"these three triangles."},{"startTime":1524.545,"endTime":1525.725,"body":"All of 'em have the same base."},{"startTime":1525.835,"endTime":1529.085,"body":"It's an equilateral, the original triangle is equilateral, same base B."},{"startTime":1529.665,"endTime":1532.645,"body":"And so that total area is the sum of the areas of these three."},{"startTime":1533.305,"endTime":1536.886,"body":"And of course what is all of their areas, half of the base times their"},{"startTime":1536.886,"endTime":1537.125,"body":"height."},{"startTime":1538.065,"endTime":1542.081,"body":"So we get this, we've got half BX plus a half B, uh, z plus"},{"startTime":1542.081,"endTime":1542.885,"body":"a half BY."},{"startTime":1543.705,"endTime":1547.59,"body":"And if you just bring out the common factor, you can see instantly x plus"},{"startTime":1547.59,"endTime":1548.885,"body":"Y plus Z equals h."},{"startTime":1548.985,"endTime":1552.765,"body":"So really, really nice little fact about equilateral triangles that I love."},{"startTime":1552.785,"endTime":1557.565,"body":"And the, the, the key to this was just drawing in those lines."},{"startTime":1557.585,"endTime":1560.656,"body":"So always look for lines that you can draw in if it's a circle through"},{"startTime":1560.656,"endTime":1561.885,"body":"drawing a tangent, something like that."},{"startTime":1562.515,"endTime":1567.75,"body":"Okay, so this of course this proof involved a diagram calls, it's a proof, it's"},{"startTime":1567.75,"endTime":1569.845,"body":"a result in geometry about triangles."},{"startTime":1569.845,"endTime":1576.784,"body":"You might expect a diagram, but diagrams and visual helps can be really useful in"},{"startTime":1576.784,"endTime":1580.485,"body":"proofs that are not necessarily coming from geometry."},{"startTime":1580.945,"endTime":1582.165,"body":"So I'm gonna show you a couple of those."},{"startTime":1582.785,"endTime":1586.125,"body":"And the first is probably better known than the second."},{"startTime":1586.825,"endTime":1589.005,"body":"Um, but they, they, the one leads to the other."},{"startTime":1591.185,"endTime":1594.725,"body":"So the first one is about we sometimes call these triangular numbers."},{"startTime":1595.355,"endTime":1598.019,"body":"It's basically when you add up the first few numbers, one plus two plus three,"},{"startTime":1598.019,"endTime":1599.085,"body":"and so on up to N."},{"startTime":1599.985,"endTime":1604.313,"body":"So you can think of this as making triangles and how many dots you need"},{"startTime":1604.313,"endTime":1606.045,"body":"to make triangles of each size."},{"startTime":1606.045,"endTime":1609.213,"body":"So for the first one it's like one dot and then a, a a a"},{"startTime":1609.213,"endTime":1611.325,"body":"two by two triangle if I can call it that."},{"startTime":1611.545,"endTime":1614.525,"body":"Um, you have to add another row which has two dots, dots in it."},{"startTime":1614.525,"endTime":1615.925,"body":"So one plus two gives you three."},{"startTime":1616.355,"endTime":1619.88,"body":"Then for the, the triangle of kind of size three, um, it's got, you have"},{"startTime":1619.88,"endTime":1621.525,"body":"to add three more in the bottom."},{"startTime":1621.585,"endTime":1622.925,"body":"So one plus two plus three six."},{"startTime":1622.985,"endTime":1626.567,"body":"And then for the fourth triangle number, I'm calling these T four, T five and"},{"startTime":1626.567,"endTime":1627.045,"body":"so on."},{"startTime":1627.425,"endTime":1630.63,"body":"We have one plus two plus three plus four, which is 10, you can go"},{"startTime":1630.63,"endTime":1631.485,"body":"on 15, 21, 28."},{"startTime":1631.585,"endTime":1635.085,"body":"So, so this is the number we're interested in this quantity."},{"startTime":1635.745,"endTime":1637.845,"body":"Is there a formula for this?"},{"startTime":1638.625,"endTime":1643.181,"body":"Adding the numbers from one to n uh, we'll call that TN the n triangular"},{"startTime":1643.181,"endTime":1643.485,"body":"number."},{"startTime":1643.905,"endTime":1646.245,"body":"Is there a formula that will save us some time?"},{"startTime":1646.465,"endTime":1651.259,"body":"So we're sort of instinctively a bit lazy, we don't wanna spend all day adding"},{"startTime":1651.259,"endTime":1656.054,"body":"up laboriously the numbers all the way from one up to have a big N"},{"startTime":1656.054,"endTime":1660.849,"body":"is a million, whatever, it's, well, okay, there is \u003claugh\u003e, spoiler alert, there is, and,"},{"startTime":1660.849,"endTime":1665.325,"body":"and the way to see what the formula is one way relies on pictures."},{"startTime":1666.145,"endTime":1668.125,"body":"So here is the nth triangular number."},{"startTime":1668.435,"endTime":1671.663,"body":"Well in this case n equals six, but you know, general, here we go, all"},{"startTime":1671.663,"endTime":1672.525,"body":"these dots piled up."},{"startTime":1672.585,"endTime":1676.626,"body":"So we've got, uh, kind of this is uh, n high and n dots along"},{"startTime":1676.626,"endTime":1677.165,"body":"the bottom."},{"startTime":1677.825,"endTime":1680.525,"body":"And all we're gonna do is just not change the number of dots at all."},{"startTime":1680.525,"endTime":1685.506,"body":"We're just gonna technical terms, scooch this along so that we've now created a RightAngle"},{"startTime":1685.506,"endTime":1690.488,"body":"triangle there and the number of dots, I've just put each of them in a"},{"startTime":1690.488,"endTime":1691.485,"body":"side little square."},{"startTime":1692.025,"endTime":1697.382,"body":"And what we're gonna do is we're gonna take that exact picture, copy it, turn"},{"startTime":1697.382,"endTime":1699.525,"body":"it over and join it on."},{"startTime":1699.525,"endTime":1700.925,"body":"So we're now gonna get two of these things."},{"startTime":1701.345,"endTime":1706.246,"body":"And now what we've got, um, is in, we've got a rectangle and the total"},{"startTime":1706.246,"endTime":1711.148,"body":"number of dots in that is the area of that rectangle, which is its base"},{"startTime":1711.148,"endTime":1716.05,"body":"times its height, and that is n plus one is but obscured along the bottom"},{"startTime":1716.05,"endTime":1719.645,"body":"we've got n from the original triangle plus that one extra."},{"startTime":1719.905,"endTime":1724.222,"body":"And the height of this thing is our n So the total area of this"},{"startTime":1724.222,"endTime":1726.525,"body":"rectangle is just n times n plus one."},{"startTime":1727.345,"endTime":1728.725,"body":"That's twice the number we need."},{"startTime":1729.085,"endTime":1730.485,"body":"'cause we use two copies of our triangle."},{"startTime":1731.305,"endTime":1735.093,"body":"So all we need to do to, to get the formula for the end trying"},{"startTime":1735.093,"endTime":1738.125,"body":"the number is just divide by two and this is the formula."},{"startTime":1738.905,"endTime":1741.085,"body":"So there's a lovely, uh, anecdote."},{"startTime":1741.515,"endTime":1745.085,"body":"It's unclear how much evidence there is that it really happened."},{"startTime":1745.345,"endTime":1749.657,"body":"But the great mathematician, uh, gauss, it said when he was a a small child"},{"startTime":1749.657,"endTime":1753.97,"body":"at school, the teacher at the end of probably a long day, I, I imagine"},{"startTime":1753.97,"endTime":1756.845,"body":"this teacher wanting just a half an hour of peace."},{"startTime":1756.845,"endTime":1760.415,"body":"So he, so he said to his pupils, okay guys, you are gonna sit down"},{"startTime":1760.415,"endTime":1763.985,"body":"and I want you to add up all the numbers from one to a hundred"},{"startTime":1763.985,"endTime":1766.365,"body":"while I just sit quietly over here, be my brow."},{"startTime":1766.825,"endTime":1772.246,"body":"Um, and then his lovely pupil g immediately, sir, it's 5,050 \u003claugh\u003e because he had"},{"startTime":1772.246,"endTime":1777.667,"body":"the legend, has it worked out this formula and realize in order to add all"},{"startTime":1777.667,"endTime":1782.005,"body":"the numbers from one to a hundred, that's n equals a hundred."},{"startTime":1782.305,"endTime":1785.365,"body":"You just have to do a half times, a hundred times 101."},{"startTime":1786.025,"endTime":1789.228,"body":"And so you can do that instantly, whereas adding them all up one by one"},{"startTime":1789.228,"endTime":1792.005,"body":"takes you an an awful long time and you probably make some mistakes."},{"startTime":1792.385,"endTime":1794.285,"body":"So I dunno if that's true, but it's a nice story."},{"startTime":1795.025,"endTime":1798.418,"body":"Um, okay, so this is the numbers adding up just the numbers from one to"},{"startTime":1798.418,"endTime":1798.645,"body":"end."},{"startTime":1798.655,"endTime":1800.085,"body":"Let's raise our game a little bit."},{"startTime":1800.815,"endTime":1805.125,"body":"Let's add the squares, the first n squares one plus four plus nine plus 16."},{"startTime":1805.125,"endTime":1809.245,"body":"And so on up to n squared, I'll call that sn for squares."},{"startTime":1809.745,"endTime":1810.365,"body":"What's that?"},{"startTime":1811.065,"endTime":1817.056,"body":"Now you may have seen, if you've seen that first proof with do with visual"},{"startTime":1817.056,"endTime":1823.048,"body":"proof for the triangular numbers, you may have seen, I've certainly seen proofs of this"},{"startTime":1823.048,"endTime":1825.445,"body":"that involve fitting pyramids into cuboids."},{"startTime":1825.865,"endTime":1829.933,"body":"Now for me, at that point it kind of switches and I find the algebraic"},{"startTime":1829.933,"endTime":1832.645,"body":"proof of whatever this formula's gonna turn out to be."},{"startTime":1832.805,"endTime":1840.405,"body":"I find the algebraic proof more easy and intuitive than fitting pyramids into cuboids."},{"startTime":1840.535,"endTime":1841.805,"body":"Maybe that's different for you."},{"startTime":1841.805,"endTime":1846.763,"body":"And of course everyone has their own tastes in these things, but there is a"},{"startTime":1846.763,"endTime":1851.722,"body":"visual proof that I find really helpful in seeing some of the connections with the,"},{"startTime":1851.722,"endTime":1853.045,"body":"the triangular number formula."},{"startTime":1853.865,"endTime":1856.965,"body":"I'm gonna show you that, um, I like it a lot."},{"startTime":1857.025,"endTime":1860.445,"body":"So here's a triangle, but instead of dots, we've now got numbers."},{"startTime":1860.545,"endTime":1863.765,"body":"So in the first row it's ones and then twos, threes, fours and so on."},{"startTime":1864.385,"endTime":1865.805,"body":"And so what, what have we got in here?"},{"startTime":1865.805,"endTime":1869.765,"body":"Well, the, the second row, two twos at four, the third row three threes is"},{"startTime":1869.765,"endTime":1871.085,"body":"nine, four fours is 16."},{"startTime":1871.585,"endTime":1876.023,"body":"So what we've got here is the sum of each row is that square number"},{"startTime":1876.023,"endTime":1880.461,"body":"and the sum of the numbers in the whole triangle is exactly the thing we're"},{"startTime":1880.461,"endTime":1883.125,"body":"interested in the sum of the first end squares."},{"startTime":1884.025,"endTime":1886.765,"body":"So if we could work out what the, the total of the numbers was in"},{"startTime":1886.765,"endTime":1888.045,"body":"this triangle, that's the thing we're after."},{"startTime":1888.705,"endTime":1892.417,"body":"So you take that triangle, we're just gonna knock it over to the right and"},{"startTime":1892.417,"endTime":1894.645,"body":"to the left, get two more copies of it."},{"startTime":1895.035,"endTime":1898.165,"body":"So's exactly the same numbers involved, we've just tipped them over."},{"startTime":1898.945,"endTime":1902.703,"body":"So each of these triangles, the sum of the entries is our SN that we're"},{"startTime":1902.703,"endTime":1903.205,"body":"interested in."},{"startTime":1903.785,"endTime":1908.005,"body":"Now what we're gonna do is superimpose these triangles on top of each other."},{"startTime":1908.665,"endTime":1913.111,"body":"So we'll now could of combine all of these three triangles and we'll do that"},{"startTime":1913.111,"endTime":1916.965,"body":"the numbers, we are just gonna add the numbers in each place together."},{"startTime":1918.945,"endTime":1921.713,"body":"And if you do that, now this might take a little bit of thinking, you"},{"startTime":1921.713,"endTime":1923.005,"body":"might have to convince yourself of this."},{"startTime":1923.415,"endTime":1927.565,"body":"Every entry in this combined triangle is two n plus one."},{"startTime":1927.745,"endTime":1930.463,"body":"If I just add another little row in there, just on each one we can"},{"startTime":1930.463,"endTime":1930.645,"body":"see."},{"startTime":1930.865,"endTime":1934.747,"body":"So if you look at the top corner there, n plus one plus n, that's"},{"startTime":1934.747,"endTime":1938.63,"body":"two n plus one, uh, go down, we get n plus one plus two plus"},{"startTime":1938.63,"endTime":1939.925,"body":"n two n plus one."},{"startTime":1941.235,"endTime":1946.185,"body":"Hopefully you can convince yourselves that every entry in this combined triangle is two n"},{"startTime":1946.185,"endTime":1946.845,"body":"plus one."},{"startTime":1947.425,"endTime":1949.605,"body":"So we can add all of those up."},{"startTime":1949.625,"endTime":1953.09,"body":"If we know how many entries there are, how many two n plus ones have"},{"startTime":1953.09,"endTime":1954.245,"body":"we got to add together?"},{"startTime":1955.065,"endTime":1959.423,"body":"And if we do that well how many entries are there in the each of"},{"startTime":1959.423,"endTime":1960.005,"body":"these triangles?"},{"startTime":1960.275,"endTime":1963.756,"body":"It's just the number of dots in a, in a ular size, n So it's"},{"startTime":1963.756,"endTime":1964.685,"body":"the n triangular number."},{"startTime":1965.745,"endTime":1969.365,"body":"And if we do that, we've got these three triangles combined."},{"startTime":1969.365,"endTime":1971.325,"body":"So we've got three times the sum we're interested in."},{"startTime":1971.905,"endTime":1976.507,"body":"So we find that three times our sn that we want is just equal to"},{"startTime":1976.507,"endTime":1981.11,"body":"two and plus one times the nth triangular number, which we just proved was N"},{"startTime":1981.11,"endTime":1982.645,"body":"half N and plus one."},{"startTime":1983.545,"endTime":1987.405,"body":"And so to get our sn we just divide everything by three, right?"},{"startTime":1987.865,"endTime":1989.165,"body":"And this is the formula we get."},{"startTime":1989.665,"endTime":1993.787,"body":"So if you've seen this formula before, you will, you know, it's, it's come out"},{"startTime":1993.787,"endTime":1997.909,"body":"correct for me, I saw this formula and the way I was, um, taught to"},{"startTime":1997.909,"endTime":2002.032,"body":"prove it originally uses a technique, an algebraic technique called proof by induction, which we're,"},{"startTime":2002.032,"endTime":2005.605,"body":"we're not gonna talk about today, but it's a very excellent proof technique."},{"startTime":2006.265,"endTime":2011.504,"body":"But it does not that just doing it purely algebraically does not really explain why"},{"startTime":2011.504,"endTime":2016.045,"body":"this formula is so clearly related to the formula for the triangular numbers."},{"startTime":2016.465,"endTime":2020.415,"body":"So I really love this proof because it gives an insight into what the connection"},{"startTime":2020.415,"endTime":2021.205,"body":"is between them."},{"startTime":2021.825,"endTime":2025.325,"body":"Um, so yeah, I top mark this proof for me, I really enjoy it."},{"startTime":2026.385,"endTime":2029.605,"body":"So we've seen these proofs, uh, these two visual proofs."},{"startTime":2029.905,"endTime":2034.192,"body":"I'm gonna go back to one that is, does not involve drawing a diagram, but"},{"startTime":2034.192,"endTime":2038.479,"body":"it's one of the first oldest proofs that we, that we have dates back to"},{"startTime":2038.479,"endTime":2038.765,"body":"Euclid."},{"startTime":2038.905,"endTime":2040.485,"body":"And it uses an amazing idea."},{"startTime":2041.785,"endTime":2045.359,"body":"Uh, we're gonna prove that there are infinitely many prime numbers and you may have"},{"startTime":2045.359,"endTime":2048.934,"body":"seen this proof before, but I have to include it in a lecture of the"},{"startTime":2048.934,"endTime":2050.364,"body":"best proofs of all time, right?"},{"startTime":2050.364,"endTime":2054.329,"body":"It's kind of, you know, you can't, I'd be thrown outta the mathematician's guild, which"},{"startTime":2054.329,"endTime":2056.445,"body":"doesn't exist if I didn't include this proof."},{"startTime":2056.995,"endTime":2061.202,"body":"It's so amazing because it really, if you think how on earth can you prove"},{"startTime":2061.202,"endTime":2062.605,"body":"that something goes on forever?"},{"startTime":2063.065,"endTime":2065.165,"body":"How, how is that we can't do, how can we do this?"},{"startTime":2065.784,"endTime":2068.165,"body":"And it involves doing a little fault experiment."},{"startTime":2068.784,"endTime":2069.405,"body":"So here we go."},{"startTime":2069.635,"endTime":2070.965,"body":"Imagine this wasn't the case."},{"startTime":2070.965,"endTime":2074.794,"body":"Imagine that actually there's only a finite number of primes, then you can imagine a"},{"startTime":2074.794,"endTime":2078.623,"body":"list of them all 2, 3, 5, 7, and so on up to whatever the"},{"startTime":2078.623,"endTime":2079.645,"body":"highest prime number is."},{"startTime":2079.645,"endTime":2084.022,"body":"Let's call it p Now we're gonna make a number, I'll call it n you"},{"startTime":2084.022,"endTime":2087.525,"body":"make it by multiplying all the prime numbers together and adding one."},{"startTime":2088.585,"endTime":2094.295,"body":"Now that number n let's think about it if we're gonna try and factorized it,"},{"startTime":2094.295,"endTime":2099.245,"body":"well, it's not divisible by any of these prime numbers on our list."},{"startTime":2099.355,"endTime":2100.805,"body":"It's one more than a multiple of two."},{"startTime":2100.825,"endTime":2103.405,"body":"So it's not divisible by two, it's one more than a multiple of three."},{"startTime":2103.405,"endTime":2107.396,"body":"It's not divisible by three, it's one more than a multiple of every single prime"},{"startTime":2107.396,"endTime":2109.525,"body":"on the list of primes that we have."},{"startTime":2110.145,"endTime":2114.365,"body":"So this number N is not divisible by any of the primes on our list."},{"startTime":2115.665,"endTime":2118.085,"body":"So it doesn't have any factors that are primes on our list."},{"startTime":2118.225,"endTime":2122.325,"body":"So that either means the n itself is prime, that's what is a prime number."},{"startTime":2122.325,"endTime":2123.445,"body":"It's only divisible by itself."},{"startTime":2123.445,"endTime":2129.245,"body":"And one, either it's prime itself or it's got a prime factor that we didn't"},{"startTime":2129.245,"endTime":2130.405,"body":"know about before."},{"startTime":2131.385,"endTime":2134.005,"body":"Either way, there's something gone wrong."},{"startTime":2134.105,"endTime":2137.525,"body":"We found a new prime that wasn't on our allegedly complete list of primes."},{"startTime":2138.465,"endTime":2141.805,"body":"So this, at this point we say magic word contradiction."},{"startTime":2142.515,"endTime":2144.805,"body":"This contradicts our, our assumption that we made."},{"startTime":2145.625,"endTime":2149.15,"body":"The, the assumption that we made can't be right 'cause we've ended up with something"},{"startTime":2149.15,"endTime":2150.325,"body":"that that doesn't make sense."},{"startTime":2151.045,"endTime":2152.965,"body":"I call this the Sherlock Holmes method of proof."},{"startTime":2153.355,"endTime":2158.445,"body":"Once you've eliminated the impossible, whatever remains, however improbable must be the truth."},{"startTime":2159.065,"endTime":2162.491,"body":"So in fact, our assumption that there's only a finite number of primes led us"},{"startTime":2162.491,"endTime":2165.005,"body":"down the garden path of something that ended up being impossible."},{"startTime":2165.005,"endTime":2168.038,"body":"We had a complete list of primes and now we found a prime that wasn't"},{"startTime":2168.038,"endTime":2168.645,"body":"on our list."},{"startTime":2169.545,"endTime":2172.565,"body":"So actually all along there must have been infinitely many prime numbers."},{"startTime":2173.305,"endTime":2175.805,"body":"So this is a fantastic idea."},{"startTime":2175.865,"endTime":2178.894,"body":"You know, you take your life in your hands and say, okay, I think this"},{"startTime":2178.894,"endTime":2181.923,"body":"thing is right, but I'm gonna just assume the opposite and see where it takes"},{"startTime":2181.923,"endTime":2182.125,"body":"me."},{"startTime":2182.125,"endTime":2185.583,"body":"Hopefully I find some, you know, I prove that one equals zero, which we know"},{"startTime":2185.583,"endTime":2186.045,"body":"isn't correct."},{"startTime":2186.105,"endTime":2189.725,"body":"So I, my initial assumption, uh, was wrong."},{"startTime":2189.725,"endTime":2192.285,"body":"And so my thing I really wanted to prove must be true."},{"startTime":2192.985,"endTime":2196.085,"body":"The great mathematician Gh Hardy, there he is looking very severe."},{"startTime":2196.585,"endTime":2202.256,"body":"Uh, he says, this is one of the great techniques, one of the finest weapons"},{"startTime":2202.256,"endTime":2204.525,"body":"that we have at our disposal."},{"startTime":2205.155,"endTime":2207.645,"body":"It's a far finer gambit, he said than any chess gambit."},{"startTime":2208.045,"endTime":2212.005,"body":"A chess player may offer the sacrifice of a porn or even a piece, but"},{"startTime":2212.005,"endTime":2213.325,"body":"a mathematician offers the game."},{"startTime":2213.505,"endTime":2217.365,"body":"We can risk everything to see if our, if, if our hypothesis is true."},{"startTime":2218.225,"endTime":2220.645,"body":"So I'm gonna show you just one other proof by contradiction."},{"startTime":2220.645,"endTime":2225.102,"body":"Now we know what the technique is and again, it's a, it's an old proof,"},{"startTime":2225.102,"endTime":2228.965,"body":"but it's just so amazingly ingenious that it's, it's gonna make you smile."},{"startTime":2230.015,"endTime":2230.805,"body":"Makes me smile."},{"startTime":2231.755,"endTime":2232.045,"body":"Okay?"},{"startTime":2232.045,"endTime":2234.845,"body":"So we are gonna prove that the square root of two is irrational."},{"startTime":2235.825,"endTime":2239.556,"body":"So a rational number is one that you can write as a fraction of whole"},{"startTime":2239.556,"endTime":2239.805,"body":"numbers."},{"startTime":2240.115,"endTime":2242.525,"body":"Something like A over B where A and B are whole numbers."},{"startTime":2243.225,"endTime":2246.005,"body":"If you can't do that, then we say the number is irrational."},{"startTime":2246.785,"endTime":2251.788,"body":"Now there's no one you've, you know, when people first started thinking about mathematics and"},{"startTime":2251.788,"endTime":2256.125,"body":"numbers, there was no suspicion that such numbers could exist, but they do."},{"startTime":2256.225,"endTime":2259.775,"body":"And even in the most sort of commonly occurring places, I mean the square root"},{"startTime":2259.775,"endTime":2263.325,"body":"of two is just the length of the diagonal of a square of side one."},{"startTime":2263.465,"endTime":2266.033,"body":"You know, we haven't had to delve down the back of the sofa for this"},{"startTime":2266.033,"endTime":2266.205,"body":"thing."},{"startTime":2266.405,"endTime":2270.405,"body":"In some weird esoteric situation, it's really naturally occurring."},{"startTime":2270.425,"endTime":2274.042,"body":"The square root of two, um, you can't avoid it even in the most basic"},{"startTime":2274.042,"endTime":2274.525,"body":"of geometry."},{"startTime":2274.665,"endTime":2278.769,"body":"So the idea that this thing is can't be written as a fraction, um, is,"},{"startTime":2278.769,"endTime":2280.685,"body":"is very strange and exciting and exhilarating."},{"startTime":2281.505,"endTime":2282.565,"body":"And, but how do we prove it?"},{"startTime":2282.565,"endTime":2286.565,"body":"Again, it's this, how do we know we can't check all possible fractions?"},{"startTime":2286.825,"endTime":2288.125,"body":"So we've gotta use a different argument."},{"startTime":2288.125,"endTime":2290.485,"body":"And the argument is this, proof by contradiction."},{"startTime":2290.495,"endTime":2294.645,"body":"Let's imagine a world in which actually the square root of two is rational."},{"startTime":2295.515,"endTime":2296.365,"body":"What would that mean?"},{"startTime":2296.505,"endTime":2300.261,"body":"It means we can write it as a fraction, but once you've got a fraction,"},{"startTime":2300.261,"endTime":2302.765,"body":"you can always cancel it down to its lowest form."},{"startTime":2302.765,"endTime":2305.405,"body":"If there's any common factors on the top and the bottom, you can cancel them."},{"startTime":2305.745,"endTime":2309.855,"body":"So if there, if there is a fraction that is route two, we can assume"},{"startTime":2309.855,"endTime":2313.965,"body":"that we've just canceled it down, cancel any common factors and so we can um,"},{"startTime":2313.965,"endTime":2318.075,"body":"be happy that we would be able to write route two as some A over"},{"startTime":2318.075,"endTime":2319.445,"body":"B in its lowest form."},{"startTime":2319.705,"endTime":2321.525,"body":"No common factors in A and B, they're co prime."},{"startTime":2322.355,"endTime":2324.605,"body":"Alright, so we dunno anything about route two."},{"startTime":2324.605,"endTime":2328.205,"body":"So let's square both sides so we don't have to think about that scary thing."},{"startTime":2328.205,"endTime":2331.487,"body":"If we square both sides, we get that two is equal to A squared over"},{"startTime":2331.487,"endTime":2331.925,"body":"B squared."},{"startTime":2332.785,"endTime":2337.165,"body":"Um, so we can rearrange that A squared equals two B squared."},{"startTime":2337.745,"endTime":2339.525,"body":"Now at this point, let's think about numbers."},{"startTime":2339.885,"endTime":2341.765,"body":"A is a whole number so is B."},{"startTime":2342.385,"endTime":2344.285,"body":"So the square of A is an even number."},{"startTime":2345.465,"endTime":2349.324,"body":"Now if you take an odd number and multiply it by itself, the answer is"},{"startTime":2349.324,"endTime":2351.125,"body":"still odd, odd times odd equals odd."},{"startTime":2351.585,"endTime":2353.925,"body":"So a can't be an odd number, it must be an even number."},{"startTime":2354.545,"endTime":2357.365,"body":"So A is twice some other whole number C let's say."},{"startTime":2358.235,"endTime":2360.765,"body":"Okay, we put that back into what we've got."},{"startTime":2360.865,"endTime":2364.965,"body":"So two B squared is, is equal to a squared A is two C."},{"startTime":2365.225,"endTime":2369.485,"body":"So a squared is two C all squared, which is four C squared."},{"startTime":2370.705,"endTime":2373.525,"body":"And now we've got two B squared is equal to four C squared."},{"startTime":2373.525,"endTime":2377.462,"body":"We can cancel a a two there and find that B squared is twice C"},{"startTime":2377.462,"endTime":2377.725,"body":"squared."},{"startTime":2378.385,"endTime":2381.437,"body":"Now with this looks a bit familiar 'cause we just sort of had something like"},{"startTime":2381.437,"endTime":2381.845,"body":"that already."},{"startTime":2382.685,"endTime":2386.663,"body":"B must be an even number because it's square is even the square of an"},{"startTime":2386.663,"endTime":2387.725,"body":"old number is odd."},{"startTime":2388.145,"endTime":2391.045,"body":"So B must be even, but oh, hang on a minute."},{"startTime":2391.825,"endTime":2395.965,"body":"Didn't we start by saying A and B had no common factors and now we've"},{"startTime":2395.965,"endTime":2398.725,"body":"proved they're both even that can't be right, that's impossible."},{"startTime":2398.745,"endTime":2399.845,"body":"That's a contradiction."},{"startTime":2401.075,"endTime":2405.319,"body":"Okay, so our initial assumption that route two was rational led us to something that"},{"startTime":2405.319,"endTime":2405.885,"body":"obviously false."},{"startTime":2406.785,"endTime":2410.005,"body":"And so, so it must be all along that route two is irrational."},{"startTime":2410.065,"endTime":2412.325,"body":"So that's the power of a proof by contradiction."},{"startTime":2412.945,"endTime":2416.955,"body":"It allows you to deal with cases where there's infinitely many possible things that you"},{"startTime":2416.955,"endTime":2420.965,"body":"would have to check and deal with it in a, in a really efficient way."},{"startTime":2421.595,"endTime":2422.885,"body":"Wonderful proof technique."},{"startTime":2424.315,"endTime":2427.405,"body":"Alright, so we talked about some lovely proofs."},{"startTime":2427.695,"endTime":2432.645,"body":"We've talked about making conjecture and then we have to prove it, but you can't"},{"startTime":2432.645,"endTime":2433.965,"body":"get something for nothing."},{"startTime":2434.855,"endTime":2438.135,"body":"Every proof relies on either facts you already know are true, like an odd number"},{"startTime":2438.135,"endTime":2439.885,"body":"times an odd number is an odd number."},{"startTime":2440.585,"endTime":2446.027,"body":"Um, or, or even further back than that basic axioms facts that we are just,"},{"startTime":2446.027,"endTime":2448.205,"body":"we don't even try to prove."},{"startTime":2448.205,"endTime":2451.485,"body":"We say these are sort of truths we hold to be self-evident."},{"startTime":2451.865,"endTime":2456.302,"body":"An example of this is in geometry, of course we all learn very early on"},{"startTime":2456.302,"endTime":2458.965,"body":"that the SSON triangle add up to 180 degrees."},{"startTime":2459.745,"endTime":2462.605,"body":"The proof of that, or one proof goes like this, very simple."},{"startTime":2462.905,"endTime":2467.731,"body":"You take your triangle, you draw a line through a parallel to BC that gives"},{"startTime":2467.731,"endTime":2472.557,"body":"you, so the, the angles I've marked as green, those are equal 'cause their alternating"},{"startTime":2472.557,"endTime":2473.845,"body":"angles are equal, right?"},{"startTime":2474.075,"endTime":2476.765,"body":"Same with the blue angles, equal 'cause alternating angles are equal."},{"startTime":2477.225,"endTime":2480.141,"body":"And then we say, well, we know the angles on a straight line add up"},{"startTime":2480.141,"endTime":2480.725,"body":"to 180 degrees."},{"startTime":2481.265,"endTime":2484.925,"body":"So that means the green plus red plus blue is 180."},{"startTime":2485.015,"endTime":2487.085,"body":"Those are the angles in my triangle done."},{"startTime":2489.105,"endTime":2495.815,"body":"But the very first thing we said was draw a line through a parallel to"},{"startTime":2495.815,"endTime":2502.525,"body":"BC that is one formulation of the famous fifth postulate, the parallel postulate of Euclid."},{"startTime":2502.875,"endTime":2504.445,"body":"It's an axiom that he uses."},{"startTime":2504.705,"endTime":2508.727,"body":"He says, we're gonna begin with these axioms that are just obviously true as some"},{"startTime":2508.727,"endTime":2510.605,"body":"discussion as to how obvious that is."},{"startTime":2511.105,"endTime":2515.201,"body":"But if you don't, if you remove any of those basic ground rules, I mean"},{"startTime":2515.201,"endTime":2518.205,"body":"if you remove the parallel postulate, this proof does not work."},{"startTime":2518.505,"endTime":2520.245,"body":"So you, you can't even start this proof."},{"startTime":2521.105,"endTime":2525.862,"body":"So it opens up the possibility for maybe other things can happen if we, if"},{"startTime":2525.862,"endTime":2527.765,"body":"we put ourselves into different geometries."},{"startTime":2528.465,"endTime":2532.995,"body":"So it's not that this theorem is false or that the proof is false, but"},{"startTime":2532.995,"endTime":2537.525,"body":"really there's an invisible asterisk that says assuming the axioms of Euclid and geometry, right?"},{"startTime":2537.735,"endTime":2540.685,"body":"We've got, if those are our ground rules, then we can prove this."},{"startTime":2541.345,"endTime":2544.805,"body":"If those are not our ground rules, we get other things that can happen."},{"startTime":2545.385,"endTime":2550.347,"body":"So if you start looking at the geometry on the surface of a sphere, on"},{"startTime":2550.347,"endTime":2555.31,"body":"a curved surface, you get triangles whose angles can add up to things other than"},{"startTime":2555.31,"endTime":2556.965,"body":"180 degrees, which is wonderful."},{"startTime":2557.385,"endTime":2559.885,"body":"Um, you might think, well how there aren't even any straight lines."},{"startTime":2559.885,"endTime":2560.885,"body":"How can I make a triangle?"},{"startTime":2561.465,"endTime":2564.615,"body":"But what, what we have to think about what, what a straight line actually is"},{"startTime":2564.615,"endTime":2565.245,"body":"doing for us."},{"startTime":2565.315,"endTime":2569.365,"body":"What is the kind of philosophical definition of a straight line?"},{"startTime":2569.385,"endTime":2574.58,"body":"If that's not too esoteric on a plane between two points, the straight line between"},{"startTime":2574.58,"endTime":2577.005,"body":"them is the shortest path between them."},{"startTime":2577.425,"endTime":2582.817,"body":"So if we take that as really the motivating reason to think about lines on"},{"startTime":2582.817,"endTime":2588.209,"body":"the surface of its sphere, the shortest path between two points, the shortest paths there,"},{"startTime":2588.209,"endTime":2591.085,"body":"um, are arcs of great circles like equators."},{"startTime":2591.465,"endTime":2595.827,"body":"And that's why planes flying long haul will follow those curved paths rather than going"},{"startTime":2595.827,"endTime":2598.445,"body":"in what looks like straight lines on the map."},{"startTime":2598.825,"endTime":2602.662,"body":"So if you take those as our kind of spherical lines, then you can think"},{"startTime":2602.662,"endTime":2604.965,"body":"about triangles where, which are bounded by three lines."},{"startTime":2605.625,"endTime":2608.565,"body":"Now that all sounds like it's gonna get extremely complicated and horrible."},{"startTime":2608.945,"endTime":2614.513,"body":"Um, spherical geometry sounds scary, but there is one proof spherical geometry that is so"},{"startTime":2614.513,"endTime":2614.885,"body":"lovely."},{"startTime":2615.385,"endTime":2617.803,"body":"It tells us something about triangles that I want, I want to show it to"},{"startTime":2617.803,"endTime":2617.965,"body":"you."},{"startTime":2618.065,"endTime":2619.485,"body":"And this is what we can do."},{"startTime":2619.825,"endTime":2623.992,"body":"We, we, we are not any longer having to involve the, the parallel postulate that"},{"startTime":2623.992,"endTime":2627.605,"body":"is not valid in the serp on the surface of a sphere, okay?"},{"startTime":2627.665,"endTime":2630.525,"body":"So it involves us knowing this shape."},{"startTime":2630.985,"endTime":2632.885,"body":"So this is the shape you can't really have on a plane."},{"startTime":2633.305,"endTime":2636.965,"body":"Um, it's the shape bounded by two, its spherical lines, it's called a loon."},{"startTime":2637.155,"endTime":2639.125,"body":"It's a bit like, you know, a crescent moon or something."},{"startTime":2639.745,"endTime":2644.915,"body":"Um, and there's one that I've highlighted, but actually every pair of lines will create"},{"startTime":2644.915,"endTime":2645.605,"body":"two loons."},{"startTime":2645.605,"endTime":2649.177,"body":"There's, there's one kind of a shadow loon on the round going round the back"},{"startTime":2649.177,"endTime":2650.845,"body":"that's also created by these two lines."},{"startTime":2651.625,"endTime":2653.445,"body":"So take this loon here, we can work out."},{"startTime":2653.545,"endTime":2656.685,"body":"Its, its area based on the surface area of the sphere."},{"startTime":2656.715,"endTime":2661.229,"body":"It's very easy if you, if the angle there is X between the lines, then"},{"startTime":2661.229,"endTime":2665.744,"body":"that same proportion of the total area, um, will be, will be used by the"},{"startTime":2665.744,"endTime":2666.045,"body":"loop."},{"startTime":2666.905,"endTime":2669.685,"body":"So we don't need to know the formula for the surface area of a sphere."},{"startTime":2669.685,"endTime":2674.926,"body":"If you do great, let's just call it s uh, then the area of that"},{"startTime":2674.926,"endTime":2680.167,"body":"loon with that angle is just x three sixtieths of the total surface area, whatever"},{"startTime":2680.167,"endTime":2681.565,"body":"that happens to be."},{"startTime":2682.875,"endTime":2685.685,"body":"Okay, so now with that in mind, let's imagine a spherical triangle."},{"startTime":2685.785,"endTime":2689.125,"body":"So that is the region bounded by three of these spherical lines."},{"startTime":2689.785,"endTime":2693.885,"body":"Any pair of those lines defines a loom, well actually one loon at the front"},{"startTime":2693.885,"endTime":2695.525,"body":"and one loon at the back."},{"startTime":2696.185,"endTime":2699.75,"body":"So there's my spherical triangle, the blue one, and then I've got three loons that"},{"startTime":2699.75,"endTime":2702.365,"body":"are created by the three possible pairs, green, red, and yellow."},{"startTime":2703.225,"endTime":2707.408,"body":"And actually at the back, that whole thing is repeated sort of anti typically, if"},{"startTime":2707.408,"endTime":2708.245,"body":"that's a word."},{"startTime":2708.625,"endTime":2712.282,"body":"Um, you've got three loons kind of going round to the back and they all,"},{"startTime":2712.282,"endTime":2714.965,"body":"uh, overlap in another copy of that triangle at the back."},{"startTime":2715.715,"endTime":2716.205,"body":"Alright?"},{"startTime":2716.225,"endTime":2719.405,"body":"So we are gonna now think about that sphere."},{"startTime":2719.625,"endTime":2723.96,"body":"The whole surface of the sphere is completely covered by these six loons that we've"},{"startTime":2723.96,"endTime":2725.405,"body":"defined, but there's some overlap."},{"startTime":2725.405,"endTime":2729.116,"body":"So we can't say that the surface area of the sphere is just the sum"},{"startTime":2729.116,"endTime":2732.085,"body":"of the areas of these six loons 'cause of the overlapping bits."},{"startTime":2732.085,"endTime":2733.205,"body":"We've gotta take those into account."},{"startTime":2733.745,"endTime":2737.885,"body":"The triangle at the front there has, is covered by those three loons."},{"startTime":2738.585,"endTime":2742.081,"body":"So we've counted that three times and the triangle at the back is also covered"},{"startTime":2742.081,"endTime":2744.645,"body":"by the, the, the sort of shadow loons at the back."},{"startTime":2745.105,"endTime":2746.845,"body":"So we've counted that three times as well."},{"startTime":2747.465,"endTime":2751.005,"body":"So we've counted six triangle areas instead of actually just two."},{"startTime":2751.665,"endTime":2755.565,"body":"So if we take away four of those, then we'll everything ought to match up."},{"startTime":2756.145,"endTime":2760.395,"body":"So what I'm saying is the surface area of the sphere is two lots of"},{"startTime":2760.395,"endTime":2761.245,"body":"the red loon."},{"startTime":2761.245,"endTime":2765.303,"body":"So the red one and its shadow two green loons green one in its shadow"},{"startTime":2765.303,"endTime":2765.845,"body":"two yellows."},{"startTime":2766.385,"endTime":2771.058,"body":"And then we take away four whatever the area of the triangle is, which we"},{"startTime":2771.058,"endTime":2774.485,"body":"don't need to worry about, and that, that those are equal."},{"startTime":2774.915,"endTime":2775.205,"body":"Okay?"},{"startTime":2775.205,"endTime":2776.805,"body":"We know how to find the area of a loon."},{"startTime":2777.545,"endTime":2781.907,"body":"If the angles in the triangle X, y, and Z, then we're gonna get X"},{"startTime":2781.907,"endTime":2784.525,"body":"three sixtieths, Y three sixtieths and Z three sixtieths."},{"startTime":2784.615,"endTime":2787.605,"body":"Those are gonna be the areas of our loo, our loos."},{"startTime":2788.065,"endTime":2792.161,"body":"So we get this expression and what we're gonna find out is something about the,"},{"startTime":2792.161,"endTime":2795.165,"body":"the angle sum X plus Y plus Z on the plane."},{"startTime":2795.165,"endTime":2797.285,"body":"That would be 180 degrees on the sphere."},{"startTime":2797.545,"endTime":2802.669,"body":"You can rearrange this and you find that the angle cell x plus Y plus"},{"startTime":2802.669,"endTime":2806.085,"body":"Z is bigger than 180, it's 180 plus a bit."},{"startTime":2807.305,"endTime":2812.798,"body":"And the amount by which it exceeds 180 is dependent on the area of the"},{"startTime":2812.798,"endTime":2813.165,"body":"triangle."},{"startTime":2813.385,"endTime":2816.605,"body":"How much of the surface of the sphere is taken up by this triangle."},{"startTime":2817.345,"endTime":2819.885,"body":"And that intuitively makes a lot of sense."},{"startTime":2820.425,"endTime":2825.209,"body":"If you think about a very big triangle like the one we saw, angles are"},{"startTime":2825.209,"endTime":2826.485,"body":"much bigger than 180."},{"startTime":2827.065,"endTime":2831.395,"body":"But if you imagine a really small triangle, make it smaller and smaller and smaller"},{"startTime":2831.395,"endTime":2835.725,"body":"and smaller, we are getting closer and closer to the 180 until in the limit."},{"startTime":2835.815,"endTime":2840.676,"body":"Maybe you talk about drawing a triangle on a, a sphere of infinite radius, IEA"},{"startTime":2840.676,"endTime":2844.565,"body":"plane \u003claugh\u003e, you recover the, the Euclidean 180 that we all know."},{"startTime":2844.905,"endTime":2846.285,"body":"So I just love this argument."},{"startTime":2846.365,"endTime":2851.045,"body":"I think it's so beautifully simple and gives a fascinating fact about spheres."},{"startTime":2852.425,"endTime":2856.395,"body":"So we've seen lots of different proofs and some of these proofs are thousands of"},{"startTime":2856.395,"endTime":2856.925,"body":"years old."},{"startTime":2857.385,"endTime":2858.805,"body":"Is there anything new under the sun?"},{"startTime":2858.805,"endTime":2860.165,"body":"Well, there's one huge question."},{"startTime":2860.275,"endTime":2861.765,"body":"There's one huge development."},{"startTime":2862.705,"endTime":2865.565,"body":"Um, and that is computers."},{"startTime":2866.345,"endTime":2872.085,"body":"So we can ask ourselves to finish with can computers ever do proofs?"},{"startTime":2872.625,"endTime":2874.205,"body":"Can computers prove theorems?"},{"startTime":2875.185,"endTime":2877.085,"body":"And there are sort of three parts to this."},{"startTime":2877.695,"endTime":2880.125,"body":"There are three ways in which we might employ computers."},{"startTime":2880.505,"endTime":2887.165,"body":"The first has been happening for, for decades already, and that is computer assisted proofs."},{"startTime":2887.615,"endTime":2891.803,"body":"Maybe you've got some, most famously the four color film if you, if you're familiar"},{"startTime":2891.803,"endTime":2895.992,"body":"with that, you've got some thing you're trying to prove and maybe you find a"},{"startTime":2895.992,"endTime":2900.18,"body":"way to say, if I could just check these, I don't know, 200,000 cases \u003claugh\u003e,"},{"startTime":2900.18,"endTime":2904.369,"body":"um, and, and make sure that it holds in all these cases, then they would"},{"startTime":2904.369,"endTime":2906.045,"body":"be representative of the whole thing."},{"startTime":2906.045,"endTime":2908.725,"body":"And I, I could prove really rigorously that it would work."},{"startTime":2909.025,"endTime":2912.805,"body":"You know, kind of con everything might be reducible to one of these configurations."},{"startTime":2913.545,"endTime":2916.323,"body":"And of course you aren't gonna be able to check those cases by hand, but"},{"startTime":2916.323,"endTime":2917.805,"body":"you can program a machine to check them."},{"startTime":2918.145,"endTime":2922.268,"body":"So that has, that is happening and have been happening for many decades and I'm,"},{"startTime":2922.268,"endTime":2926.391,"body":"you know, okay, ideally we'd all love a half page beautiful, elegant proof, but you"},{"startTime":2926.391,"endTime":2929.965,"body":"know, if we can prove our fear of at all, that's good \u003claugh\u003e."},{"startTime":2930.225,"endTime":2933.864,"body":"So these things, I think they're completely valid and if you worry like how do"},{"startTime":2933.864,"endTime":2935.805,"body":"you know the computer hasn't made a mistake?"},{"startTime":2936.275,"endTime":2938.285,"body":"Well, how do you know you haven't made a mistake?"},{"startTime":2938.285,"endTime":2943.46,"body":"You know, like do humans make more or fewer mistake than computers that are correctly"},{"startTime":2943.46,"endTime":2943.805,"body":"programmed?"},{"startTime":2944.065,"endTime":2948.881,"body":"You know, discuss, I think the chance of a, a strain neutrino changing the ting"},{"startTime":2948.881,"endTime":2953.697,"body":"to one and making, making, uh, the computer output be wrong is small and probably"},{"startTime":2953.697,"endTime":2958.513,"body":"smaller than an error in a proof of 200 pages or something, which some mathematical"},{"startTime":2958.513,"endTime":2962.045,"body":"theorems, they, they are that long or even longer to prove."},{"startTime":2962.585,"endTime":2968.016,"body":"So that's one computer assisted proofs two, which is kind of starting to happen and"},{"startTime":2968.016,"endTime":2973.085,"body":"it's a really interesting area of research is computer or machine verification of proofs."},{"startTime":2973.345,"endTime":2977.834,"body":"So a human does a proof and then you turn it into kind of language"},{"startTime":2977.834,"endTime":2981.725,"body":"that the machine can understand and it checks the proof for logical consistency."},{"startTime":2982.025,"endTime":2983.245,"body":"And I think that's very interesting."},{"startTime":2983.785,"endTime":2986.085,"body":"Um, you know, we have spellcheckers why not proof checkers."},{"startTime":2986.705,"endTime":2991.636,"body":"The final, the final frontier of this and we are really in the infancy is"},{"startTime":2991.636,"endTime":2991.965,"body":"computers."},{"startTime":2992.385,"endTime":2998.845,"body":"Can they find new theorems and prove them without human kind of intervention?"},{"startTime":2999.785,"endTime":3004.135,"body":"And that is, you know, not really happening yet, I would say, but maybe the"},{"startTime":3004.135,"endTime":3007.325,"body":"aggression professor in a hundred years \u003claugh\u003e can revisit this question."},{"startTime":3009.145,"endTime":3014.941,"body":"So the concept of proof then, as we've seen, I hope I've convinced you it's"},{"startTime":3014.941,"endTime":3020.738,"body":"really absolutely vital to mathematics and it makes mathematics different really from other disciplines in"},{"startTime":3020.738,"endTime":3021.125,"body":"science."},{"startTime":3021.715,"endTime":3025.165,"body":"What we can do is test our hypotheses really, really stringently."},{"startTime":3025.545,"endTime":3029.575,"body":"We can never absolutely a hundred percent prove something because something could always turn up"},{"startTime":3029.575,"endTime":3031.725,"body":"and, and turn over our, our preconceived ideas."},{"startTime":3032.105,"endTime":3035.93,"body":"You know, Newton's laws work brilliantly unless you happen to be going almost at the"},{"startTime":3035.93,"endTime":3039.245,"body":"speed of light and then you need to bring in other, other possibilities."},{"startTime":3039.595,"endTime":3043.936,"body":"Doesn't make make Newton's laws less good in, in everyday life, but we know we"},{"startTime":3043.936,"endTime":3048.277,"body":"need to refine our ideas and that can happen when we, um, when we are"},{"startTime":3048.277,"endTime":3052.619,"body":"doing science, we test things with experiments, you know, in a court of law we"},{"startTime":3052.619,"endTime":3054.645,"body":"have to prove things beyond reasonable doubt."},{"startTime":3055.225,"endTime":3058.805,"body":"And in mathematics we have to prove things beyond unreasonable doubt."},{"startTime":3059.105,"endTime":3061.365,"body":"So, you know, I think that's the, the highest standard of proof."},{"startTime":3061.945,"endTime":3065.578,"body":"Um, well I hope you've enjoyed this and all my lectures in my series, it's"},{"startTime":3065.578,"endTime":3068.485,"body":"been a real honor to serve as the question professor of Geometry."},{"startTime":3068.615,"endTime":3069.405,"body":"Thank you very much."}]}