{"version":"1.0.0","segments":[{"startTime":5.335,"endTime":12.985,"body":"Today I'm going to share with you some mathematical illusions, seeming proofs that one equals"},{"startTime":12.985,"endTime":16.045,"body":"zero, or that fractions don't exist."},{"startTime":16.915,"endTime":17.845,"body":"Many others like that."},{"startTime":18.945,"endTime":24.115,"body":"But behind the scenes what's going on, uh, with some of the seeming paradoxes that"},{"startTime":24.115,"endTime":29.285,"body":"we are going to see has actually caused real difficulty for mathematicians over the years."},{"startTime":29.465,"endTime":33.605,"body":"And sorting it all out has led to some fascinating new mathematical developments."},{"startTime":34.455,"endTime":39.306,"body":"We're going to begin with three truly startling proofs, but I'm not gonna reveal the"},{"startTime":39.306,"endTime":44.158,"body":"tricks of those straight away because it's fun to try and work out what's what's"},{"startTime":44.158,"endTime":44.805,"body":"gone wrong."},{"startTime":44.835,"endTime":48.879,"body":"However, I do promise you that by the end of the lecture, we will have"},{"startTime":48.879,"endTime":51.845,"body":"resolved all of the problems and everything's going to be fine."},{"startTime":51.985,"endTime":56.268,"body":"So try and watch out and see where you think my little mathematical slight of"},{"startTime":56.268,"endTime":57.125,"body":"hand is happening."},{"startTime":58.785,"endTime":62.115,"body":"Will we ever be able to see beyond the current observable universe?"},{"startTime":62.535,"endTime":63.675,"body":"Oh, it's much worse than that."},{"startTime":63.775,"endTime":65.595,"body":"The observable universe is shrinking."},{"startTime":66.135,"endTime":70.418,"body":"If you want to know what are my odds of winning the lottery, you come"},{"startTime":70.418,"endTime":71.275,"body":"straight to probability."},{"startTime":71.345,"endTime":75.435,"body":"Yeah, because probability is all about how likely or not events are to happen."},{"startTime":76.035,"endTime":80.6,"body":"I think the chance of there being an undiscovered second species, very like humans out"},{"startTime":80.6,"endTime":83.035,"body":"there in the world today is pretty slender."},{"startTime":83.065,"endTime":83.555,"body":"However,"},{"startTime":84.175,"endTime":85.995,"body":"And these are the pictures if you haven't seen them."},{"startTime":86.115,"endTime":87.315,"body":"I mean, New York was orange."},{"startTime":87.575,"endTime":88.595,"body":"The air was orange."},{"startTime":88.785,"endTime":92.475,"body":"They said one day out in that air was like smoking a pack of cigarettes."},{"startTime":92.575,"endTime":95.875,"body":"It had the same effect on the lungs as smoking a number of cigarettes."},{"startTime":95.875,"endTime":99.001,"body":"So people who'd never smoked in their life were suddenly going to suffer some of"},{"startTime":99.001,"endTime":99.835,"body":"the same health effects."},{"startTime":100.695,"endTime":105.673,"body":"Any further questions is a brand new podcast from Gresham College, A place where we"},{"startTime":105.673,"endTime":110.652,"body":"ask our speakers all of your questions that went unanswered following their lecture guests have"},{"startTime":110.652,"endTime":114.635,"body":"included Ronald Hutton, Robin May, crystalline, TOTT, Sarah Hart and Maggie snowing."},{"startTime":115.095,"endTime":116.035,"body":"Any further questions?"},{"startTime":116.255,"endTime":119.155,"body":"All episodes are available wherever you listen to your podcasts."},{"startTime":125.735,"endTime":130.595,"body":"And don't worry if it feels difficult to believe these impossible things."},{"startTime":131.545,"endTime":133.875,"body":"Just follow the advice of the Red Queen."},{"startTime":134.735,"endTime":136.955,"body":"She says to Alice, it's just all about practice."},{"startTime":138.465,"endTime":141.826,"body":"When I was your age, she says, I always did it for half an hour"},{"startTime":141.826,"endTime":142.275,"body":"every day."},{"startTime":142.465,"endTime":145.795,"body":"Sometimes I've believed as many as six impossible things before breakfast."},{"startTime":146.095,"endTime":149.677,"body":"So, you know, we've all had our breakfast, so this should be much more achievable"},{"startTime":149.677,"endTime":150.155,"body":"for us."},{"startTime":150.615,"endTime":152.995,"body":"So we're gonna begin with the first three impossible things."},{"startTime":153.405,"endTime":153.955,"body":"Let's see."},{"startTime":155.365,"endTime":158.635,"body":"First, impossible fact one equals zero."},{"startTime":158.855,"endTime":161.873,"body":"So this might come as a surprise, but you know, we are gonna see a"},{"startTime":161.873,"endTime":162.075,"body":"proof."},{"startTime":162.095,"endTime":165.155,"body":"And I'm a mathematician, so you know, I'm gonna be very convincing, I hope."},{"startTime":165.415,"endTime":166.715,"body":"So here's, here's the argument."},{"startTime":166.815,"endTime":167.875,"body":"Why does one equals zero?"},{"startTime":168.025,"endTime":174.835,"body":"Well, we're just gonna consider and think about this innocuous little, little series of numbers."},{"startTime":175.095,"endTime":179.16,"body":"One minus one plus one minus one plus one minus one, and so on."},{"startTime":179.505,"endTime":183.921,"body":"Now, it's pretty obvious what this'll all finish with because we, it, it all comes"},{"startTime":183.921,"endTime":184.805,"body":"in pairs, right?"},{"startTime":184.865,"endTime":188.205,"body":"One minus one, another one minus another one."},{"startTime":188.535,"endTime":190.445,"body":"We're just adding, but then taking away repeatedly."},{"startTime":191.105,"endTime":194.973,"body":"So this thing is just a bunch of one takeaway once, so we're adding a"},{"startTime":194.973,"endTime":196.005,"body":"load of zeros together."},{"startTime":196.345,"endTime":197.885,"body":"So bullet, it was zero."},{"startTime":198.985,"endTime":202.642,"body":"Now there's another way of looking at it, the sort of bird in a hand"},{"startTime":202.642,"endTime":205.325,"body":"argument, which is to say we start off with one, right?"},{"startTime":205.325,"endTime":206.085,"body":"That we've got one."},{"startTime":206.465,"endTime":207.325,"body":"And then what do we do?"},{"startTime":207.385,"endTime":208.885,"body":"We take away one, but we add it again."},{"startTime":209.145,"endTime":211.986,"body":"We take away one, but we add it again, minus one, plus one, minus one"},{"startTime":211.986,"endTime":212.365,"body":"plus one."},{"startTime":212.545,"endTime":216.005,"body":"So really what we have here is one plus a bunch of zeros."},{"startTime":216.745,"endTime":218.485,"body":"So one, there we go."},{"startTime":218.485,"endTime":222.179,"body":"So we've, we've worked out the same thing two different ways and we've got zero"},{"startTime":222.179,"endTime":223.165,"body":"and we've got one."},{"startTime":223.345,"endTime":224.965,"body":"So those must be equal."},{"startTime":225.485,"endTime":226.685,"body":"QED one equals zero."},{"startTime":227.305,"endTime":229.245,"body":"So I'm sure you're all happily believing."},{"startTime":229.245,"endTime":233.485,"body":"Now that one equals zero, just in case you are not entirely convinced."},{"startTime":234.285,"endTime":238.045,"body":"I will give you actually a couple more proofs of this as we go along."},{"startTime":238.905,"endTime":243.213,"body":"As I say, I'm not gonna tell you exactly why this doesn't work right now,"},{"startTime":243.213,"endTime":247.521,"body":"but we might suspect something murky is going on with these.dot dots, \u003claugh\u003e that they"},{"startTime":247.521,"endTime":249.245,"body":"had a lot of, uh, problems."},{"startTime":249.545,"endTime":253.95,"body":"So we'll come back and think more about infinite series later on, but let's have"},{"startTime":253.95,"endTime":255.125,"body":"our second impossible fact."},{"startTime":256.825,"endTime":258.845,"body":"The smallest positive number is one."},{"startTime":259.625,"endTime":263.285,"body":"Now, this is actually very good pedagogically, right?"},{"startTime":263.525,"endTime":268.046,"body":"'cause teachers tell us, research has shown that one of the major stumbling blocks for"},{"startTime":268.046,"endTime":271.965,"body":"youngsters learning mathematics, and some of them never get beyond it, it's fractions."},{"startTime":272.545,"endTime":276.922,"body":"So once we've proved the smallest positive number is one, there'll be no need for"},{"startTime":276.922,"endTime":278.965,"body":"fractions anymore and everyone can be happy."},{"startTime":279.505,"endTime":282.845,"body":"So let's prove that the smallest positive number is one."},{"startTime":283.185,"endTime":285.045,"body":"So currently we don't know what it is, right?"},{"startTime":285.045,"endTime":286.525,"body":"So let's investigate it a bit."},{"startTime":287.105,"endTime":288.125,"body":"So here comes the proof."},{"startTime":288.385,"endTime":290.285,"body":"So we dunno what this smallest positive number is."},{"startTime":290.325,"endTime":291.525,"body":"Let's call it X for the moment."},{"startTime":292.115,"endTime":297.756,"body":"Obviously X is at most one 'cause one is a positive number and X is"},{"startTime":297.756,"endTime":298.885,"body":"the smallest one."},{"startTime":298.885,"endTime":302.645,"body":"So X is certainly less than or equal to one, okay?"},{"startTime":302.645,"endTime":303.845,"body":"But X is a positive number."},{"startTime":303.865,"endTime":307.933,"body":"If you take the square of a positive number, you still have a positive number,"},{"startTime":307.933,"endTime":308.205,"body":"right?"},{"startTime":308.625,"endTime":311.085,"body":"So x squared is, is also a positive."},{"startTime":311.085,"endTime":314.205,"body":"Number X is the smallest positive number."},{"startTime":314.825,"endTime":317.125,"body":"So X squared can't be any smaller than X."},{"startTime":317.785,"endTime":320.845,"body":"So that tells us that X is less than or equal to x squared."},{"startTime":321.155,"endTime":321.445,"body":"Okay?"},{"startTime":321.705,"endTime":326.285,"body":"Now x less than equal to x squared, X is positive."},{"startTime":326.285,"endTime":326.965,"body":"It's not zero."},{"startTime":327.185,"endTime":330.953,"body":"So we can divide through by X without, I'm not dividing by zero or anything"},{"startTime":330.953,"endTime":331.205,"body":"weird."},{"startTime":331.675,"endTime":336.071,"body":"This is okay, I can divide that, that little equation through by X on both"},{"startTime":336.071,"endTime":336.365,"body":"sides."},{"startTime":336.385,"endTime":338.125,"body":"And I get one is less than wrinkle to X."},{"startTime":338.745,"endTime":343.245,"body":"And now we've got a number sandwich, best kind of sandwich, uh, because we see,"},{"startTime":343.245,"endTime":347.745,"body":"we've showed that one is less than equal to x, but X is also less"},{"startTime":347.745,"endTime":349.245,"body":"than or equal to one."},{"startTime":349.745,"endTime":352.005,"body":"So X is trapped, it has to equal one."},{"startTime":352.905,"endTime":356.605,"body":"So we've shown that the smallest positive number is in fact one, which is great."},{"startTime":356.605,"endTime":359.925,"body":"No more fractions, mass has become suddenly a lot easier."},{"startTime":360.665,"endTime":362.765,"body":"So again, have a thi what do you think has gone wrong there?"},{"startTime":362.765,"endTime":363.925,"body":"Well, we'll come back to that later."},{"startTime":364.815,"endTime":368.165,"body":"Final of these three, uh, impossible factors coming up."},{"startTime":368.305,"endTime":371.885,"body":"And I thought as I'm the G regression, progressive geometry, we should do some geometry."},{"startTime":372.105,"endTime":376.125,"body":"So we're gonna do some geometry around congruent triangles."},{"startTime":376.705,"endTime":381.3,"body":"Um, I'll show you before then a couple of warnings because I don't want to"},{"startTime":381.3,"endTime":383.445,"body":"like, you know, lead you astray here."},{"startTime":383.505,"endTime":386.885,"body":"We just have to say ahead of this, watch out."},{"startTime":388.065,"endTime":392.481,"body":"Um, Saint Augustine does tell us to be aware of mathematicians and all those who"},{"startTime":392.481,"endTime":393.365,"body":"make empty prophecies."},{"startTime":393.525,"endTime":396.565,"body":"But okay, well, like, you know, well, I dunno."},{"startTime":396.585,"endTime":399.405,"body":"See, see what you like Saint Augustine, you know, be honest with us."},{"startTime":400.245,"endTime":404.09,"body":"Er, on the other hand, I dunno if this is more insulting to mathematicians or"},{"startTime":404.09,"endTime":407.935,"body":"to Frenchmen, um, but he says, whatever you say to either of them, uh, they"},{"startTime":407.935,"endTime":411.525,"body":"translate it into their own language and forth with it means something entirely different."},{"startTime":411.905,"endTime":413.485,"body":"So those are the warnings."},{"startTime":413.665,"endTime":416.445,"body":"Uh, but now let's proceed and see what we can see about triangles."},{"startTime":416.905,"endTime":419.045,"body":"Uh, okay, so there's the basic setup."},{"startTime":419.115,"endTime":421.205,"body":"I've got any old triangle, A, B, C, right?"},{"startTime":421.205,"endTime":422.645,"body":"Take a triangle, your favorite triangle."},{"startTime":423.145,"endTime":424.765,"body":"Now what I've, I've drawn in two lines."},{"startTime":424.765,"endTime":425.605,"body":"We're gonna have some more lines."},{"startTime":425.685,"endTime":426.805,"body":"I thought I'd do it step by step."},{"startTime":427.065,"endTime":429.765,"body":"So we've bisected that angle."},{"startTime":430.045,"endTime":430.165,"body":"A."},{"startTime":430.185,"endTime":433.358,"body":"So bisecting an angle is something we've probably all learned to do at school with"},{"startTime":433.358,"endTime":434.205,"body":"a ruler and compass."},{"startTime":434.585,"endTime":436.245,"body":"You can, you can do that construction."},{"startTime":436.665,"endTime":439.405,"body":"So bisect the angle A and that line is coming down."},{"startTime":439.985,"endTime":443.885,"body":"And then we're gonna cut the bottom, the opposite side in half."},{"startTime":443.905,"endTime":446.125,"body":"So the perpendicular bisector of that site."},{"startTime":446.425,"endTime":447.885,"body":"So we've cut it into two halves."},{"startTime":448.265,"endTime":451.933,"body":"Um, then we point is it D and then we've got that perpendicular coming up"},{"startTime":451.933,"endTime":455.602,"body":"and they're meeting at O and we're gonna add some lines from O we're gonna"},{"startTime":455.602,"endTime":459.271,"body":"add in, we're just gonna join O to the other two sides, O to B,"},{"startTime":459.271,"endTime":460.005,"body":"O to C."},{"startTime":460.305,"endTime":462.645,"body":"I'm not claiming those angles are being bisector or anything."},{"startTime":462.645,"endTime":463.925,"body":"We're just joining those lines."},{"startTime":464.585,"endTime":467.245,"body":"And we're gonna drop perpendiculars to the remaining sides."},{"startTime":467.415,"endTime":472.006,"body":"Again, I'm not claiming that that chops those sides in half, but we can still"},{"startTime":472.006,"endTime":472.925,"body":"drop those perpendiculars."},{"startTime":473.315,"endTime":475.605,"body":"Okay, that's a bit of complicated setup."},{"startTime":475.705,"endTime":479.534,"body":"So now watch me like a hawk, 'cause I'm going to do some things with"},{"startTime":479.534,"endTime":480.045,"body":"congruent triangles."},{"startTime":480.045,"endTime":483.165,"body":"And you've gotta see where I may or may not be lying to."},{"startTime":483.635,"endTime":483.925,"body":"Okay?"},{"startTime":484.065,"endTime":486.165,"body":"So to, we're gonna take them in pairs."},{"startTime":486.165,"endTime":489.365,"body":"We've divided this triangle into six little smaller triangles."},{"startTime":489.615,"endTime":491.325,"body":"Let's look at the two blue ones at the bottom."},{"startTime":491.355,"endTime":491.885,"body":"That pair."},{"startTime":492.985,"endTime":494.485,"body":"So I claim they're congruent."},{"startTime":494.625,"endTime":495.285,"body":"How do I know?"},{"startTime":495.595,"endTime":501.831,"body":"Well, we've got a common side OD and BD and CD are equal because we,"},{"startTime":501.831,"endTime":506.405,"body":"we, what we've got od there is the, the bisector, right?"},{"startTime":506.425,"endTime":508.125,"body":"So BD equals cd."},{"startTime":508.125,"endTime":509.365,"body":"So those two sides are equal."},{"startTime":510.265,"endTime":513.205,"body":"And between them, we've got the same angle."},{"startTime":513.205,"endTime":514.205,"body":"Happens to be a right angle."},{"startTime":514.784,"endTime":517.965,"body":"So one of the rules for congruent triangles is side, angle side, okay?"},{"startTime":518.025,"endTime":521.325,"body":"So by that rule, these two blue triangles are congruent to each other."},{"startTime":521.914,"endTime":526.085,"body":"Alright, next pair of triangles, the red ones, what have we got here?"},{"startTime":526.425,"endTime":529.765,"body":"The angles at a are, are equal by construction."},{"startTime":529.765,"endTime":533.815,"body":"We made the angle by sector, uh, at p and at Q we've got right"},{"startTime":533.815,"endTime":534.085,"body":"angles."},{"startTime":534.665,"endTime":536.405,"body":"So we've got two matching angles."},{"startTime":536.405,"endTime":538.645,"body":"And then of course the third angles must be the same."},{"startTime":538.865,"endTime":540.965,"body":"So all the angles are the same in these triangles."},{"startTime":541.145,"endTime":546.525,"body":"So they're definitely similar triangles, but also they have a common side, uh, ao."},{"startTime":546.985,"endTime":548.245,"body":"So that makes them congruent."},{"startTime":548.245,"endTime":548.485,"body":"Okay?"},{"startTime":548.485,"endTime":550.085,"body":"So it's another pair of congruent triangles."},{"startTime":551.505,"endTime":555.605,"body":"And now finally, think about the two yellow triangles there."},{"startTime":555.635,"endTime":556.805,"body":"They're the ones that are left."},{"startTime":557.305,"endTime":560.045,"body":"Uh, and so these ones, what can we say about them?"},{"startTime":560.065,"endTime":561.565,"body":"So they're right angled triangles."},{"startTime":561.765,"endTime":562.845,"body":"P and A Q got RightAngle."},{"startTime":563.905,"endTime":566.205,"body":"The hypo uses of those are the same."},{"startTime":566.585,"endTime":571.508,"body":"OB equals OC because the blue triangles are congruent and we've got another pair of"},{"startTime":571.508,"endTime":572.165,"body":"matching sides."},{"startTime":572.765,"endTime":576.925,"body":"P, o and QO are equal because the red triangles are congruent."},{"startTime":577.825,"endTime":583.881,"body":"So RightAngle, side, side, another of the criteria for being congruent, the yellow triangles are"},{"startTime":583.881,"endTime":584.285,"body":"congruent."},{"startTime":584.345,"endTime":586.565,"body":"So we've now got these three pairs of congruent triangles."},{"startTime":586.985,"endTime":588.125,"body":"So what can we say from this?"},{"startTime":589.035,"endTime":596.414,"body":"Well think about the side AB now from, because the red triangles are congruent, AP"},{"startTime":596.414,"endTime":603.793,"body":"is equal to aq, and because the yellow triangles are congruent, PB is equal to"},{"startTime":603.793,"endTime":604.285,"body":"qc."},{"startTime":605.145,"endTime":608.405,"body":"So AP equals aq, PB equals qc."},{"startTime":609.185,"endTime":610.605,"body":"Now you might see where we're going."},{"startTime":611.705,"endTime":619.045,"body":"AB is AP plus pb, but AP equals aq and PB uh, equals qc."},{"startTime":620.185,"endTime":623.935,"body":"So we've got the AB is equal, equal to in length AQ plus qc, which"},{"startTime":623.935,"endTime":624.685,"body":"is just ac."},{"startTime":625.505,"endTime":631.005,"body":"So what we've shown is that AB equals ac, so it's an is OES triangle,"},{"startTime":631.005,"endTime":633.205,"body":"but this was any old triangle."},{"startTime":633.865,"endTime":637.605,"body":"So we have proved the amazing theorem that all triangles are isosceles."},{"startTime":638.355,"endTime":638.965,"body":"Fantastic."},{"startTime":639.185,"endTime":643.473,"body":"So if you are puzzled by that, good, we'll come back and think more about"},{"startTime":643.473,"endTime":644.045,"body":"it later."},{"startTime":645.235,"endTime":649.393,"body":"Okay, well by this point you might be starting to ask yourself many things about"},{"startTime":649.393,"endTime":652.165,"body":"your life choices that led you to this room today."},{"startTime":652.385,"endTime":655.445,"body":"But you might say, well, okay, what actually then is a proof?"},{"startTime":655.765,"endTime":659.31,"body":"'cause these proofs perhaps are, seem to prove something that isn't true and that's not"},{"startTime":659.31,"endTime":660.965,"body":"really what we want in a proof."},{"startTime":661.665,"endTime":664.405,"body":"Uh, so what is a proof?"},{"startTime":664.425,"endTime":667.005,"body":"And the idea of proof goes back thousands of years."},{"startTime":667.235,"endTime":670.445,"body":"It's one of the things that makes mathematics, mathematics."},{"startTime":672.025,"endTime":675.148,"body":"So we can sort of loosely say that it, what are we doing when we"},{"startTime":675.148,"endTime":675.565,"body":"prove something?"},{"startTime":675.795,"endTime":681.076,"body":"Well, we, it's, it's a sequence of logical deductions starting from some things that we"},{"startTime":681.076,"endTime":682.485,"body":"all believe are true."},{"startTime":682.485,"endTime":685.285,"body":"You know, we might call them axioms, some, some initial assumptions."},{"startTime":685.595,"endTime":690.62,"body":"Then there's a sequence of logical deductions, everything following from what's gone before that establishes"},{"startTime":690.62,"endTime":692.965,"body":"the truth of, of a given statement."},{"startTime":693.985,"endTime":700.108,"body":"Um, so we, we, we started with three impossible facts, three proofs, and we've had"},{"startTime":700.108,"endTime":700.925,"body":"two quotes."},{"startTime":701.305,"endTime":704.211,"body":"So now we need to have one terrible joke because that has to be the"},{"startTime":704.211,"endTime":704.405,"body":"ratio."},{"startTime":704.785,"endTime":708.325,"body":"Um, there's another meaning of the word proof, half a percent of alcohol."},{"startTime":708.945,"endTime":712.565,"body":"And occasionally combining those things can make things go terribly wrong."},{"startTime":712.825,"endTime":714.405,"body":"So here's the advice for today."},{"startTime":714.935,"endTime":716.965,"body":"Don't drink and derive there."},{"startTime":716.965,"endTime":717.525,"body":"We carry that."},{"startTime":717.615,"endTime":718.085,"body":"Thank you."},{"startTime":718.315,"endTime":721.507,"body":"Yeah, I know I, you know, I thought I, I'm gonna, I I've only got"},{"startTime":721.507,"endTime":724.699,"body":"one more lecture after this as ion professor and I don't think I've told that"},{"startTime":724.699,"endTime":725.125,"body":"joke yet."},{"startTime":725.225,"endTime":726.645,"body":"So, you know, it had to happen."},{"startTime":727.185,"endTime":730.845,"body":"Anyway, let's carry on and think about the first definition of proof."},{"startTime":731.425,"endTime":734.685,"body":"Not involving any percent of our goal at all, if you're doing it properly."},{"startTime":735.025,"endTime":739.489,"body":"Um, where we make deductions following on from our initial assumptions and we go through"},{"startTime":739.489,"endTime":740.085,"body":"the proof."},{"startTime":740.385,"endTime":746.049,"body":"So there's a direction of travel we start here, everything follows from what's gone over,"},{"startTime":746.049,"endTime":746.805,"body":"gone before."},{"startTime":747.345,"endTime":752.765,"body":"Now getting that the wrong way round is something that can cause big issues."},{"startTime":753.145,"endTime":754.125,"body":"And I'll give you an example."},{"startTime":754.705,"endTime":758.525,"body":"And this is something that, a mistake that is a really common pitfall in fact."},{"startTime":759.185,"endTime":763.049,"body":"So you might get asked to show that some two algebraic expressions are equal to"},{"startTime":763.049,"endTime":763.565,"body":"each other."},{"startTime":764.195,"endTime":768.746,"body":"It's very tempting to make that the first line of your, of your proof and"},{"startTime":768.746,"endTime":773.297,"body":"say, right, let's start here and then mess around with the algebra and multiply things"},{"startTime":773.297,"endTime":775.725,"body":"out and cancel things and do some stuff."},{"startTime":775.865,"endTime":777.445,"body":"And then we end up with zero equals zero."},{"startTime":777.715,"endTime":778.405,"body":"Well, that's true."},{"startTime":779.015,"endTime":779.365,"body":"Great."},{"startTime":779.385,"endTime":780.525,"body":"So the original thing was true."},{"startTime":781.185,"endTime":782.965,"body":"Now this is very dangerous."},{"startTime":783.635,"endTime":785.485,"body":"It's dangerous thing in this case."},{"startTime":785.685,"endTime":790.98,"body":"I mean the, the original statement does happen to be true, but deducing something true,"},{"startTime":790.98,"endTime":793.805,"body":"um, is possible even starting from something false."},{"startTime":794.025,"endTime":798.771,"body":"So you've gotta make sure your direction of travel is, is, is in the right"},{"startTime":798.771,"endTime":802.885,"body":"direction, because if you don't do that, you risk proving things like this."},{"startTime":803.625,"endTime":806.285,"body":"So here I'm proving that one equals minus one."},{"startTime":806.785,"endTime":810.685,"body":"Um, so I start with one equals minus one, and then I square both sides."},{"startTime":810.945,"endTime":813.005,"body":"One squared will equal minus one squared."},{"startTime":813.355,"endTime":816.73,"body":"Well then on the left, you get one on the right, you get one, so"},{"startTime":816.73,"endTime":817.405,"body":"one equals one."},{"startTime":817.405,"endTime":818.285,"body":"Well, I know that's true."},{"startTime":818.465,"endTime":819.885,"body":"So the original thing must have been true."},{"startTime":820.145,"endTime":821.925,"body":"Now \u003claugh\u003e, of course, that's nonsense."},{"startTime":822.225,"endTime":826.565,"body":"We can see that it's nonsense because I've proved, proved, uh, something false here."},{"startTime":827.065,"endTime":831.155,"body":"But it doesn't look like nonsense necessarily in the thing above because it just so"},{"startTime":831.155,"endTime":835.245,"body":"happened that what we're aiming to do is it happens to be a true statement."},{"startTime":835.545,"endTime":839.064,"body":"But in mathematics we don't only want to be able to trust proofs of things"},{"startTime":839.064,"endTime":841.645,"body":"we already know are true, rather destroys the purpose of this."},{"startTime":841.665,"endTime":846.355,"body":"So it's so, so important to make sure that yeah, we, our, our, our proofs"},{"startTime":846.355,"endTime":851.045,"body":"are going from beginning assumptions to the end and, uh, not the other way around."},{"startTime":852.675,"endTime":852.965,"body":"Okay?"},{"startTime":853.105,"endTime":858.285,"body":"Uh, let's have our fourth impossible fact, and it's a good one."},{"startTime":858.555,"endTime":862.445,"body":"It's gonna help all of our, all of our financial, uh, calculations there."},{"startTime":862.465,"endTime":863.765,"body":"One penny equals one pound."},{"startTime":864.835,"endTime":865.125,"body":"Okay?"},{"startTime":865.145,"endTime":865.765,"body":"Here's the proof."},{"startTime":865.765,"endTime":867.245,"body":"It's a very nice short proof."},{"startTime":867.505,"endTime":870.365,"body":"So one penny, well what is that in pounds?"},{"startTime":870.365,"endTime":872.485,"body":"That's N point N one, okay?"},{"startTime":872.665,"endTime":876.325,"body":"Now N point N one is n 0.1 squared."},{"startTime":876.505,"endTime":877.565,"body":"So we just make that change."},{"startTime":878.995,"endTime":879.285,"body":"Okay?"},{"startTime":879.285,"endTime":885.205,"body":"So now I've got N 0.1 squared, but n 0.1 pounds, put that back again."},{"startTime":885.205,"endTime":887.165,"body":"That's 10 P yes, 10 pence."},{"startTime":887.785,"endTime":889.845,"body":"Um, but now of course we've gotta square it now."},{"startTime":889.865,"endTime":893.605,"body":"So 10 squared is a hundred and a hundred pence is one pound."},{"startTime":894.085,"endTime":894.245,"body":"Hooray."},{"startTime":894.705,"endTime":899.148,"body":"So I, I'm pretty sure this is how the Bank of England does quantitative easing"},{"startTime":899.148,"endTime":899.445,"body":"\u003claugh\u003e."},{"startTime":899.835,"endTime":902.445,"body":"It's at least as good as any other explanation I've heard."},{"startTime":902.905,"endTime":908.485,"body":"Um, but okay, what's going on here is there's something up with the units."},{"startTime":908.955,"endTime":914.285,"body":"Something's going wrong when we trying to change between, uh, pennies and pounds."},{"startTime":915.105,"endTime":921.685,"body":"And actually problems with units have caused real world issues on many, many occasions."},{"startTime":921.765,"endTime":924.965,"body":"I wanted to just mention my top three of these 'cause it, it does matter."},{"startTime":925.125,"endTime":929.061,"body":"I mean, we can see that this is wrong, but sometimes there are, there are"},{"startTime":929.061,"endTime":931.685,"body":"issues that arise when people don't notice problems with units."},{"startTime":931.825,"endTime":933.445,"body":"So the first of these, what are the most famous?"},{"startTime":934.355,"endTime":940.625,"body":"Yeah, the Mars, the Mars Climate Orbiter, which, uh, on very unfortunately that mission failed,"},{"startTime":940.625,"endTime":944.805,"body":"uh, at a cost of gosh, something like $120 million."},{"startTime":945.435,"endTime":951.137,"body":"Because, because, um, some bits of the collection of, uh, you know, companies and of"},{"startTime":951.137,"endTime":956.839,"body":"people who were doing this, some people were expecting to receive some numbers that are"},{"startTime":956.839,"endTime":962.541,"body":"needed to calculate various, you know, thrust in the, in the, uh, craft wanted to"},{"startTime":962.541,"endTime":968.243,"body":"receive the numbers and were thinking they were receiving the numbers in Newton's of thrust"},{"startTime":968.243,"endTime":972.805,"body":"and they were actually being sent the numbers in pounds of thrust."},{"startTime":973.265,"endTime":975.365,"body":"And those two do not give you the same answer."},{"startTime":975.385,"endTime":980.706,"body":"So unfortunately, because of that me mess up between metric and imperial, uh, the, the,"},{"startTime":980.706,"endTime":982.125,"body":"the mission sadly failed."},{"startTime":982.905,"endTime":984.045,"body":"Um, we can go back."},{"startTime":984.045,"endTime":985.285,"body":"This is not a new phenomenon."},{"startTime":985.345,"endTime":987.005,"body":"So it's not all the fault of metric."},{"startTime":987.745,"endTime":990.005,"body":"Uh, this is a much older example."},{"startTime":990.105,"endTime":996.788,"body":"So this is the Swedish warship vasa, uh, which sank, you know, mere moments after"},{"startTime":996.788,"endTime":998.125,"body":"launch in 1628."},{"startTime":998.625,"endTime":1002.065,"body":"And then they raised it in the 1980s and they had a look at what"},{"startTime":1002.065,"endTime":1005.505,"body":"might have gone wrong and what they found, or one of the, one of the"},{"startTime":1005.505,"endTime":1008.946,"body":"problems was they found on the, on the ship, you know, it just literally just"},{"startTime":1008.946,"endTime":1009.405,"body":"been launched."},{"startTime":1009.425,"endTime":1014.543,"body":"So there was still a few bits and bobs of their construction, uh, materials and"},{"startTime":1014.543,"endTime":1019.662,"body":"they found some rulers, they found four rulers and they were able to sort of"},{"startTime":1019.662,"endTime":1024.781,"body":"deduce that there seemed to have been two teams of carpenters or ship builders working"},{"startTime":1024.781,"endTime":1025.805,"body":"on this ship."},{"startTime":1026.223,"endTime":1028.243,"body":"One on the port side, one on the starboard side."},{"startTime":1029.065,"endTime":1033.525,"body":"And on one side the rulers were measuring Swedish feet."},{"startTime":1033.944,"endTime":1036.444,"body":"And on the other side they were measuring Amsterdam feet."},{"startTime":1037.385,"endTime":1041.866,"body":"And one of those is about 28 centimeters and the other one is about 29"},{"startTime":1041.866,"endTime":1042.165,"body":"centimeters."},{"startTime":1042.185,"endTime":1046.733,"body":"And that discrepancy was enough to make the, the weight of the ship unbalanced like"},{"startTime":1046.733,"endTime":1049.765,"body":"one half that the, the things were thicker than others."},{"startTime":1049.825,"endTime":1052.365,"body":"And so it was unbalanced and catastrophe."},{"startTime":1052.665,"endTime":1056.186,"body":"So even if you are both using feet, you need to make sure you're both"},{"startTime":1056.186,"endTime":1057.125,"body":"using the same feet."},{"startTime":1057.945,"endTime":1061.565,"body":"Now the final example, uh, is a slightly different one."},{"startTime":1061.625,"endTime":1063.725,"body":"So this is the High Rhine Bridge."},{"startTime":1064.665,"endTime":1070.586,"body":"Uh, it was built to connect the city, the Swiss city of Laufenberg to the"},{"startTime":1070.586,"endTime":1072.165,"body":"German city of Laufenberg."},{"startTime":1072.225,"endTime":1074.965,"body":"So both are the same name, but they're on different sides of the Rhine."},{"startTime":1074.965,"endTime":1076.325,"body":"One's in Germany, one's in Switzerland."},{"startTime":1077.115,"endTime":1077.405,"body":"Okay?"},{"startTime":1077.425,"endTime":1084.159,"body":"So this, uh, was a problem because in Switzerland the um, altitude is measured from"},{"startTime":1084.159,"endTime":1090.445,"body":"a base point, which is related to essentially a big rock in Lake Geneva."},{"startTime":1091.185,"endTime":1096.196,"body":"And on the German side altitude, um, sort of mean altitude is measured, um, based"},{"startTime":1096.196,"endTime":1100.205,"body":"on a measurement that's sort of mean water level somewhere in Amsterdam."},{"startTime":1100.395,"endTime":1100.685,"body":"Okay?"},{"startTime":1100.775,"endTime":1101.925,"body":"Those are different from each other."},{"startTime":1102.065,"endTime":1105.445,"body":"Now they knew this, they did know this, you know, they got complete idiots."},{"startTime":1105.545,"endTime":1108.085,"body":"Um, the difference is about 27 centimeters."},{"startTime":1108.265,"endTime":1111.133,"body":"So they said, okay, well we just have to take account of this in our"},{"startTime":1111.133,"endTime":1111.325,"body":"calculations."},{"startTime":1112.105,"endTime":1116.811,"body":"And at this point my heart goes out to them because I too, uh, often"},{"startTime":1116.811,"endTime":1121.205,"body":"miss out minus signs and then have to go back and find 'em again."},{"startTime":1121.525,"endTime":1121.765,"body":"\u003claugh\u003e, right?"},{"startTime":1122.335,"endTime":1126.205,"body":"Rogue minus signs of the bans of many people's lives including mine."},{"startTime":1126.345,"endTime":1127.965,"body":"Um, a minus sign went wrong."},{"startTime":1128.585,"endTime":1133.591,"body":"And so instead of getting rid of the 27 degree, uh, centimeter discrepancy, they doubled"},{"startTime":1133.591,"endTime":1133.925,"body":"it."},{"startTime":1133.925,"endTime":1137.765,"body":"And when the halves at the bridge were kind of coming together and closer to"},{"startTime":1137.765,"endTime":1141.605,"body":"meet, they found there were 54 centimeters apart from each other and sort of emergency"},{"startTime":1141.605,"endTime":1142.885,"body":"changes had to be made."},{"startTime":1143.345,"endTime":1147.845,"body":"So in that case it was, yeah, we, we using the same measurements, but we"},{"startTime":1147.845,"endTime":1152.345,"body":"are starting from different levels and the rogue minus sign messed everything up, which uh,"},{"startTime":1152.345,"endTime":1153.845,"body":"they seem often to do."},{"startTime":1154.505,"endTime":1159.205,"body":"So if we think back to, so units mattering, if we think back to our,"},{"startTime":1159.205,"endTime":1163.906,"body":"our impossible proof that one penny equals one pound, it's easier to spot what's gone"},{"startTime":1163.906,"endTime":1168.607,"body":"wrong if you hadn't already, if we were to try and do this same thing,"},{"startTime":1168.607,"endTime":1173.308,"body":"but with centimeters and meters because we straightaway can see that we, some of the"},{"startTime":1173.308,"endTime":1178.009,"body":"things that we are saying are equal to each other aren't even measuring the same"},{"startTime":1178.009,"endTime":1182.71,"body":"thing because we've got meters not 0.1 meters and we are claiming that that is"},{"startTime":1182.71,"endTime":1187.411,"body":"equal to some number of meters squared, but meters are a unit of length and"},{"startTime":1187.411,"endTime":1189.605,"body":"meters squared are a unit of area."},{"startTime":1189.705,"endTime":1191.485,"body":"So there's no way those can be the same thing."},{"startTime":1192.305,"endTime":1196.147,"body":"Um, it's harder to spot that with pounds and pennies 'cause we don't often think"},{"startTime":1196.147,"endTime":1198.965,"body":"about square roots of a pound or square of a penny."},{"startTime":1199.025,"endTime":1203.035,"body":"But so what went wrong here was that we just sort of glibly put these"},{"startTime":1203.035,"endTime":1207.045,"body":"units inside squares and square roots and, and didn't pay attention to what was happening."},{"startTime":1207.585,"endTime":1212.685,"body":"So yes, units important, uh, and that, that is a warning to us all."},{"startTime":1215.275,"endTime":1221.954,"body":"Okay, so let's think now again about one equals zero, just in case you didn't"},{"startTime":1221.954,"endTime":1222.845,"body":"believe me."},{"startTime":1223.185,"endTime":1224.805,"body":"Um, I'm gonna give you two more proofs."},{"startTime":1224.805,"endTime":1227.438,"body":"That one equals zero 'cause I really want you to believe it by the end"},{"startTime":1227.438,"endTime":1227.965,"body":"of the day."},{"startTime":1228.465,"endTime":1232.205,"body":"And they're, they're increasing in mathematical, uh, sophistication."},{"startTime":1232.705,"endTime":1235.285,"body":"So that, so we just use ones and minus ones before."},{"startTime":1235.505,"endTime":1237.925,"body":"Now we're gonna bring exponents into it."},{"startTime":1238.825,"endTime":1240.805,"body":"So just a couple of observations."},{"startTime":1240.805,"endTime":1242.285,"body":"We all know the laws of powers."},{"startTime":1242.355,"endTime":1244.525,"body":"Yeah, anything to the power zero is one, right?"},{"startTime":1244.525,"endTime":1246.805,"body":"So X to the nor is one for any value of X."},{"startTime":1247.785,"endTime":1251.757,"body":"But on the other hand, if we take zero and raise it to any power,"},{"startTime":1251.757,"endTime":1254.405,"body":"you just get zero, multiply some zeros together, still zero."},{"startTime":1254.465,"endTime":1258.125,"body":"So zero raised to the power Y equals zero for any Y."},{"startTime":1258.705,"endTime":1261.685,"body":"So I simply set X and Y both to be zero."},{"startTime":1262.385,"endTime":1266.504,"body":"And you'll see that from the first expression I've got that, uh, zero to zero"},{"startTime":1266.504,"endTime":1269.525,"body":"is one, and from the second one it clearly equals zero."},{"startTime":1270.025,"endTime":1273.254,"body":"So it clearly is equal to both one and zero, which are therefore equal to"},{"startTime":1273.254,"endTime":1273.685,"body":"each other."},{"startTime":1273.865,"endTime":1275.565,"body":"So one equals zero, okay?"},{"startTime":1275.635,"endTime":1277.405,"body":"Marvelous, easy."},{"startTime":1278.195,"endTime":1283.079,"body":"Just in case you're still not convinced, I'm gonna, I'm gonna bring calculus into it,"},{"startTime":1283.079,"endTime":1283.405,"body":"okay?"},{"startTime":1283.605,"endTime":1285.405,"body":"'cause calculus can do anything."},{"startTime":1286.065,"endTime":1289.215,"body":"So this is where, this is the our only glimpse of calculus in this, in"},{"startTime":1289.215,"endTime":1290.685,"body":"this talk, but it's a good one."},{"startTime":1290.905,"endTime":1295.036,"body":"So this isn't calculus yet, but look, I'm just showing you here what we mean"},{"startTime":1295.036,"endTime":1296.965,"body":"by something squared, which we all know."},{"startTime":1296.985,"endTime":1300.885,"body":"So two squared is two plus two, two lots of two three squared is three"},{"startTime":1300.885,"endTime":1301.925,"body":"plus three plus three."},{"startTime":1302.275,"endTime":1304.005,"body":"Four squared would be four, lots of four."},{"startTime":1304.345,"endTime":1308.35,"body":"So in general for any number XX squared is just x plus x plus x"},{"startTime":1308.35,"endTime":1309.685,"body":"plus x like x times."},{"startTime":1311.155,"endTime":1311.445,"body":"Okay?"},{"startTime":1311.545,"endTime":1313.045,"body":"So now here comes the calculus."},{"startTime":1313.435,"endTime":1315.125,"body":"Just differentiate on both sides."},{"startTime":1315.465,"endTime":1318.69,"body":"If you remember your calculus, don't worry if you don't, you can just be, use"},{"startTime":1318.69,"endTime":1321.485,"body":"one of the other proofs, that one equals zero, you won't lose anything."},{"startTime":1321.865,"endTime":1323.365,"body":"Uh, differentiate on both sides."},{"startTime":1323.365,"endTime":1326.87,"body":"And we all know when we differentiate X with respect to X, when we differentiate"},{"startTime":1326.87,"endTime":1327.805,"body":"x, we get one."},{"startTime":1328.465,"endTime":1330.325,"body":"So a bunch of ones on the left hand side."},{"startTime":1330.545,"endTime":1333.885,"body":"And when we differentiate x squared, you may remember the answer is two x."},{"startTime":1334.465,"endTime":1338.428,"body":"So now on the left we've got X lots of one, and on the right"},{"startTime":1338.428,"endTime":1339.485,"body":"we've got two x."},{"startTime":1340.425,"endTime":1343.485,"body":"So for any value of x, we just have x equals two x."},{"startTime":1343.865,"endTime":1348.005,"body":"So I'm gonna substitute X equals one and that gives you one equals two."},{"startTime":1348.385,"endTime":1351.445,"body":"And then you just subtract one from both sides, one equals zero."},{"startTime":1352.105,"endTime":1353.805,"body":"So listen, I've given you three proofs."},{"startTime":1354.405,"endTime":1358.305,"body":"I think it would frankly be childish if you didn't now believe that one equals"},{"startTime":1358.305,"endTime":1358.565,"body":"zero."},{"startTime":1359.145,"endTime":1360.965,"body":"So let's assume you do believe that one equals zero."},{"startTime":1362.435,"endTime":1365.565,"body":"That allows us to do something very interesting."},{"startTime":1367.025,"endTime":1369.205,"body":"We can now prove that everything is true."},{"startTime":1369.975,"endTime":1374.766,"body":"Everything it's a corollary, this is a a, a theorem that's a consequence of a"},{"startTime":1374.766,"endTime":1375.405,"body":"previous theorem."},{"startTime":1375.465,"endTime":1380.645,"body":"So here's the corollary, let s be any statement then s is true."},{"startTime":1380.985,"endTime":1384.165,"body":"So this solves all of our worries and we can prove everything."},{"startTime":1384.665,"endTime":1385.925,"body":"So here, let's do that."},{"startTime":1386.505,"endTime":1390.445,"body":"So take any false statement, whatever you like, any false statement at all."},{"startTime":1391.125,"endTime":1396.333,"body":"F uh, now if I happen to know, if I've managed to prove that this"},{"startTime":1396.333,"endTime":1401.542,"body":"thing that at least one of f and s is true, if I know that"},{"startTime":1401.542,"endTime":1406.75,"body":"that statement is true, then the thing that's true must be S because F is"},{"startTime":1406.75,"endTime":1407.445,"body":"false, right?"},{"startTime":1408.185,"endTime":1412.396,"body":"So if I can prove that at least one of s and F is true,"},{"startTime":1412.396,"endTime":1415.485,"body":"then we will be able to deduce that S is true."},{"startTime":1415.835,"endTime":1416.125,"body":"Okay?"},{"startTime":1416.395,"endTime":1416.885,"body":"Alright."},{"startTime":1417.065,"endTime":1419.925,"body":"Now let's consider a particular false statement."},{"startTime":1421.025,"endTime":1421.725,"body":"One equals zero."},{"startTime":1422.865,"endTime":1424.045,"body":"So that's a false statement."},{"startTime":1424.165,"endTime":1428.142,"body":"I think we probably probably agree, but we also now have to agree 'cause I've"},{"startTime":1428.142,"endTime":1431.325,"body":"proved it three different ways that one equals nor is also true."},{"startTime":1432.105,"endTime":1433.325,"body":"We just proved it three times."},{"startTime":1434.305,"endTime":1436.485,"body":"So this happens to be also true."},{"startTime":1437.065,"endTime":1441.165,"body":"So f is true in that case it's definitely true to say that at least"},{"startTime":1441.165,"endTime":1444.445,"body":"one of f and F is true 'cause F is true marvelous."},{"startTime":1444.445,"endTime":1448.186,"body":"But we just said earlier and nobody told me I couldn't, that that if we"},{"startTime":1448.186,"endTime":1451.927,"body":"can prove that statement that at least one of s and F is true, then"},{"startTime":1451.927,"endTime":1452.925,"body":"s must be true."},{"startTime":1453.665,"endTime":1456.525,"body":"So you have to now agree that s is true, okay?"},{"startTime":1456.765,"endTime":1458.845,"body":"QED every statement is true."},{"startTime":1459.425,"endTime":1462.405,"body":"So this is how high the stakes are, right?"},{"startTime":1462.745,"endTime":1469.485,"body":"If we let even one false statement into, into mathematics, then it just renders everything."},{"startTime":1470.105,"endTime":1475.045,"body":"Uh, you know, all bets are off, everything is true."},{"startTime":1475.425,"endTime":1479.605,"body":"So we gotta be really, really careful and try not to prove anything that's false."},{"startTime":1480.485,"endTime":1482.765,"body":"I, I try to do that in my mathematical life."},{"startTime":1483.635,"endTime":1483.925,"body":"Okay?"},{"startTime":1484.145,"endTime":1486.125,"body":"So with that in mind, we've already proved everything is true."},{"startTime":1486.185,"endTime":1489.045,"body":"So, you know, maybe we could stop there, but let's carry on, let's carry on."},{"startTime":1489.265,"endTime":1493.3,"body":"Um, 'cause I wanna, I wanna show you six impossible facts to, 'cause I admire"},{"startTime":1493.3,"endTime":1494.645,"body":"the red queen so much."},{"startTime":1494.705,"endTime":1499.355,"body":"You know, we try and practice these things every day, okay, so impossible fact number"},{"startTime":1499.355,"endTime":1502.765,"body":"five, um, it turns out that zero is a positive number."},{"startTime":1502.865,"endTime":1503.845,"body":"Didn't know that before."},{"startTime":1503.985,"endTime":1504.485,"body":"Now we do."},{"startTime":1504.485,"endTime":1506.805,"body":"Zero is greater than zero, okay?"},{"startTime":1506.805,"endTime":1508.245,"body":"So let's, let's prove that."},{"startTime":1510.075,"endTime":1513.165,"body":"Okay, we're gonna think about these two theories."},{"startTime":1513.385,"endTime":1516.685,"body":"So X is gonna be one plus a third, plus a fifth plus a seven"},{"startTime":1516.685,"endTime":1517.125,"body":"plus a."},{"startTime":1517.345,"endTime":1520.365,"body":"So kind of the odd one over the odd numbers added together."},{"startTime":1520.825,"endTime":1524.213,"body":"And y it's gonna be one over the even numbers added together a half plus"},{"startTime":1524.213,"endTime":1526.925,"body":"a quarter plus a a six plus an eighth and so on."},{"startTime":1527.825,"endTime":1529.045,"body":"So there are those two things."},{"startTime":1529.065,"endTime":1531.645,"body":"We don't know what those ah, but let's just call the X and Y."},{"startTime":1532.555,"endTime":1533.045,"body":"Alright?"},{"startTime":1534.915,"endTime":1536.405,"body":"What happens if we double Y?"},{"startTime":1537.345,"endTime":1539.525,"body":"So now we've got just double each term."},{"startTime":1539.585,"endTime":1542.725,"body":"So one plus a half plus a third plus a quarter and so on."},{"startTime":1543.385,"endTime":1546.995,"body":"So this is actually the combination of X and Y because we've got one over"},{"startTime":1546.995,"endTime":1550.125,"body":"all of the odd numbers and interleaved with one over all even numbers."},{"startTime":1550.545,"endTime":1553.245,"body":"So two y is actually equal to x plus Y."},{"startTime":1555.545,"endTime":1558.605,"body":"So that's, if we solve that, well not solve it."},{"startTime":1558.605,"endTime":1562.931,"body":"But if we rearrange, just take away two y from both sides, we get X"},{"startTime":1562.931,"endTime":1564.085,"body":"minus y equals zero."},{"startTime":1564.715,"endTime":1565.005,"body":"Okay?"},{"startTime":1565.385,"endTime":1569.045,"body":"So that's all good, but there's a slight different way we can think about it."},{"startTime":1569.865,"endTime":1570.765,"body":"Here's the different way."},{"startTime":1571.065,"endTime":1575.048,"body":"So now think about x minus Y and just, you know, x and YX is"},{"startTime":1575.048,"endTime":1575.845,"body":"above Y here."},{"startTime":1575.845,"endTime":1579.885,"body":"So we can just do like compare the terms or take away the terms pairwise."},{"startTime":1580.265,"endTime":1584.06,"body":"So we can have one minus a half and a third minus a quarter and"},{"startTime":1584.06,"endTime":1586.085,"body":"a fifth minus a sixth, dot, dot dot."},{"startTime":1586.265,"endTime":1587.605,"body":"That's also x minus y."},{"startTime":1588.505,"endTime":1593.45,"body":"Now the reason I put that bracketing there is because if you look at each"},{"startTime":1593.45,"endTime":1598.396,"body":"bracket, then each bracket you've got something that's strictly greater than zero, one minus a"},{"startTime":1598.396,"endTime":1600.045,"body":"half is bigger than zero."},{"startTime":1600.685,"endTime":1602.565,"body":"A third minus a quarter is bigger than zero, right?"},{"startTime":1602.765,"endTime":1605.345,"body":"A fifth is bigger than a sixth, a seventh is bigger than an eighth and"},{"startTime":1605.345,"endTime":1606.205,"body":"so on and so on."},{"startTime":1606.205,"endTime":1610.445,"body":"So we are adding together infinitely many things that are each bigger than zero."},{"startTime":1611.305,"endTime":1615.16,"body":"So if we add together infinitely any positive numbers, we're gonna end up with something"},{"startTime":1615.16,"endTime":1616.445,"body":"bigger than zero for sure."},{"startTime":1617.025,"endTime":1620.584,"body":"So X minus Y then is bigger than zero, but hang on a minute, we"},{"startTime":1620.584,"endTime":1622.245,"body":"just saw above that it equals zero."},{"startTime":1622.585,"endTime":1627.285,"body":"So again, we've got, we've got this fantastic result that zero is bigger than zero."},{"startTime":1627.985,"endTime":1630.365,"body":"So that's, that's that proof all very nice."},{"startTime":1630.785,"endTime":1636.515,"body":"So I think it's now time, uh, we already saw an infinite sum doing something"},{"startTime":1636.515,"endTime":1642.246,"body":"a little bit suspicious, let's say right at the beginning here we've got another bit"},{"startTime":1642.246,"endTime":1645.685,"body":"of something odd going on with infinite, infinite sums."},{"startTime":1645.985,"endTime":1650.525,"body":"So it's now time I think to talk about the challenge of infinite series."},{"startTime":1650.635,"endTime":1652.805,"body":"What, how can we understand what's going on?"},{"startTime":1653.145,"endTime":1656.285,"body":"We have, we really proved that zero is greater than zero, I hope not."},{"startTime":1656.725,"endTime":1658.885,"body":"\u003claugh\u003e, I hope we haven't proved that one equals an all."},{"startTime":1658.945,"endTime":1661.725,"body":"So what, what do we do with infinite series?"},{"startTime":1662.305,"endTime":1666.014,"body":"And the basic, the problem, one of the problems is if you think about this,"},{"startTime":1666.014,"endTime":1669.724,"body":"this one, so one plus two plus four plus eight plus what powers of two,"},{"startTime":1669.724,"endTime":1672.445,"body":"we can see that that's getting bigger and bigger and bigger."},{"startTime":1673.265,"endTime":1676.805,"body":"It feels like intuitively, like the sum of that ought to be infinity."},{"startTime":1677.315,"endTime":1677.605,"body":"Okay?"},{"startTime":1677.605,"endTime":1680.165,"body":"We are just getting, we're adding more and more and bigger and bigger numbers."},{"startTime":1680.625,"endTime":1683.805,"body":"Um, we could, we could plausibly say that equals infinity."},{"startTime":1684.505,"endTime":1690.355,"body":"Um, well okay, that's our x if we work out one plus two x, um,"},{"startTime":1690.355,"endTime":1693.085,"body":"that's one plus twice everything in there."},{"startTime":1693.105,"endTime":1694.925,"body":"So one plus two plus four, plus eight plus so on."},{"startTime":1695.345,"endTime":1699.809,"body":"And if you multiply all those things by two, it becomes 2, 4, 8, 16,"},{"startTime":1699.809,"endTime":1700.405,"body":"so on."},{"startTime":1700.405,"endTime":1703.125,"body":"So actually one plus two x is just X again."},{"startTime":1703.945,"endTime":1706.285,"body":"So now we've got one plus two x equals x."},{"startTime":1706.425,"endTime":1709.245,"body":"And if you solve that, you get x equals minus one."},{"startTime":1710.265,"endTime":1714.125,"body":"So am I saying that infinity equals minus one?"},{"startTime":1714.325,"endTime":1716.885,"body":"I mean computer scientists say that, don't they?"},{"startTime":1716.935,"endTime":1718.565,"body":"Maybe, maybe, maybe that's true."},{"startTime":1718.905,"endTime":1721.725,"body":"Uh, but it, it feels like that isn't quite right."},{"startTime":1721.905,"endTime":1723.445,"body":"So what could, what could we do?"},{"startTime":1723.715,"endTime":1727.485,"body":"What which bit of this reasoning is, is something that we should disallow."},{"startTime":1727.945,"endTime":1732.53,"body":"So we could say, well let's not do, let's say we are not allowed to"},{"startTime":1732.53,"endTime":1734.365,"body":"do calculations with infinity for example."},{"startTime":1734.395,"endTime":1735.725,"body":"That would, that would get us outta it."},{"startTime":1735.725,"endTime":1739.965,"body":"Just say we, this is banned because it leads us to weird paradoxical ideas."},{"startTime":1740.065,"endTime":1740.685,"body":"You could do that."},{"startTime":1741.825,"endTime":1746.285,"body":"Um, but that wouldn't help you with the first example we saw."},{"startTime":1746.825,"endTime":1751.802,"body":"It wouldn't help you with this ones and one minus one, one minus one issue"},{"startTime":1751.802,"endTime":1754.125,"body":"because there nothing zooming off to infinity."},{"startTime":1754.745,"endTime":1758.748,"body":"We, we did it one way and got zero, we did it another way and"},{"startTime":1758.748,"endTime":1761.685,"body":"got one, but there's no infinities in there in the calculation."},{"startTime":1761.705,"endTime":1767.165,"body":"So that feels like we could banish infinity, but that doesn't fully fix the problem."},{"startTime":1768.105,"endTime":1771.245,"body":"Uh, so maybe it's something else that we need to do."},{"startTime":1772.025,"endTime":1776.628,"body":"Um, you could say, well I'm just not going to allow adding up infinitely many"},{"startTime":1776.628,"endTime":1780.005,"body":"numbers that's clearly problematic and I'm just going to ban it."},{"startTime":1780.505,"endTime":1782.845,"body":"Um, it doesn't sound very intellectually curious."},{"startTime":1782.985,"endTime":1783.725,"body":"So we don't wanna do that."},{"startTime":1783.725,"endTime":1788.1,"body":"But also we can't really afford to do that because we do need these things,"},{"startTime":1788.1,"endTime":1790.725,"body":"not these ones precisely, but we need infinite series."},{"startTime":1791.185,"endTime":1792.165,"body":"We use them all the time."},{"startTime":1792.785,"endTime":1797.364,"body":"Um, if we want to work out, it's kind of useful to be able to"},{"startTime":1797.364,"endTime":1801.943,"body":"know and think that three tenths plus three hundredths plus three thousandths plus three 10000th"},{"startTime":1801.943,"endTime":1803.165,"body":"plus.dot, what is that?"},{"startTime":1803.185,"endTime":1806.285,"body":"The decimal N 0.33, three n 0.3 recurring."},{"startTime":1806.465,"endTime":1808.325,"body":"That's the decimal expansion of one third."},{"startTime":1808.795,"endTime":1810.485,"body":"It's kind of useful to have that."},{"startTime":1810.905,"endTime":1814.516,"body":"Or you might say, okay, well we won't do that, we'll just have fractions, we'll"},{"startTime":1814.516,"endTime":1817.165,"body":"just write a third, you know, we've got something right there."},{"startTime":1817.545,"endTime":1822.485,"body":"But of course that will not work for numbers like pie, which are irrational."},{"startTime":1822.485,"endTime":1825.965,"body":"There is no fraction, uh, that exactly is pie."},{"startTime":1826.465,"endTime":1829.645,"body":"So we need these, these.dot dots."},{"startTime":1829.825,"endTime":1834.103,"body":"We need to somehow incorporate them into mathematics in a way that leaves us, puts"},{"startTime":1834.103,"endTime":1838.382,"body":"us on a firm footing where we don't accidentally prove that n equals one, which"},{"startTime":1838.382,"endTime":1841.805,"body":"is takes the shine off your afternoon if you accidentally prove that."},{"startTime":1842.065,"endTime":1843.445,"body":"Um, or maybe it makes it more fun."},{"startTime":1844.025,"endTime":1849.029,"body":"So we need to be able to work with infinite series and therefore we have"},{"startTime":1849.029,"endTime":1852.365,"body":"to work out how to avoid these, these apparent paradoxes."},{"startTime":1854.025,"endTime":1855.965,"body":"So how do we do this?"},{"startTime":1856.795,"endTime":1858.525,"body":"Well, let's think about an example."},{"startTime":1858.665,"endTime":1864.145,"body":"So if I took a half plus a quarter plus an eight plus a 16th,"},{"startTime":1864.145,"endTime":1869.625,"body":"so one over the powers of two add those together plus.dot, um, what we do"},{"startTime":1869.625,"endTime":1875.106,"body":"with this or other series is we say, well we obviously we can't actually add"},{"startTime":1875.106,"endTime":1879.125,"body":"infinitely many numbers together so we make better and better approximations."},{"startTime":1880.025,"endTime":1885.5,"body":"And what we do is to work out the sum to a given point call,"},{"startTime":1885.5,"endTime":1887.325,"body":"we'll call that partial sum."},{"startTime":1887.465,"endTime":1891.336,"body":"We work out these partial sums, so the sum to one term, sum to two"},{"startTime":1891.336,"endTime":1892.885,"body":"terms, three terms and so on."},{"startTime":1893.105,"endTime":1896.609,"body":"So if we do this, I've actually plotted the first few of these on a"},{"startTime":1896.609,"endTime":1898.245,"body":"graph just to show you what happens."},{"startTime":1898.785,"endTime":1900.285,"body":"So the first term is just a half."},{"startTime":1900.305,"endTime":1904.067,"body":"If we had a quarter, the sum is three quarters, a half plus a quarter,"},{"startTime":1904.067,"endTime":1907.83,"body":"plus an eighth is seven eighths, then we get 15 sixteens and so on and"},{"startTime":1907.83,"endTime":1909.085,"body":"so on and so on."},{"startTime":1909.665,"endTime":1912.536,"body":"And you can see I've, I've done, so this is the graph of the partial"},{"startTime":1912.536,"endTime":1913.685,"body":"sums that we are working out."},{"startTime":1913.985,"endTime":1917.79,"body":"And you can see it looks really rather like we are getting closer and closer"},{"startTime":1917.79,"endTime":1918.805,"body":"and closer to one."},{"startTime":1919.945,"endTime":1923.651,"body":"And in fact if you can work out an expression for what is the nth"},{"startTime":1923.651,"endTime":1927.357,"body":"partial sum sum to end terms and it's one minus one over two to the"},{"startTime":1927.357,"endTime":1927.605,"body":"N."},{"startTime":1927.985,"endTime":1931.979,"body":"So as n gets bigger, that one over two to the end is getting smaller"},{"startTime":1931.979,"endTime":1933.045,"body":"and smaller and smaller."},{"startTime":1933.345,"endTime":1939.005,"body":"We really are getting closer and closer and closer and staying close to one."},{"startTime":1940.385,"endTime":1944.244,"body":"So I mean I, you know, we could give a a formal very formal definition"},{"startTime":1944.244,"endTime":1946.045,"body":"of, of what we want to say."},{"startTime":1946.465,"endTime":1952.405,"body":"But the basic idea is we would say that this series does have a limit."},{"startTime":1952.865,"endTime":1955.525,"body":"The limit is one and it converges to one."},{"startTime":1956.025,"endTime":1961.407,"body":"And the way we prove that is that we show that however tiny tinier number"},{"startTime":1961.407,"endTime":1966.79,"body":"you pick as as a distance away from one at some point, eventually those postal"},{"startTime":1966.79,"endTime":1972.173,"body":"sums will stay, will become and stay as close as you like to that the,"},{"startTime":1972.173,"endTime":1974.685,"body":"the limit that we think it is."},{"startTime":1974.825,"endTime":1978.595,"body":"So here we think the limit is one and you can prove that however tiny"},{"startTime":1978.595,"endTime":1982.365,"body":"a number you give me, I can guarantee that there will be a point beyond"},{"startTime":1982.365,"endTime":1985.885,"body":"which all of those sums stay closer to one than your tiny teeny number."},{"startTime":1987.385,"endTime":1992.278,"body":"And so that's our, I mean informally I've defined the concept of convergence there that"},{"startTime":1992.278,"endTime":1996.845,"body":"this, there is convergence, uh, to this limit, which in this case is one."},{"startTime":1997.665,"endTime":2001.793,"body":"So that's the basic idea and we can see that this does now help us"},{"startTime":2001.793,"endTime":2003.445,"body":"with our one minus one problem."},{"startTime":2004.395,"endTime":2005.085,"body":"What about this?"},{"startTime":2005.355,"endTime":2009.216,"body":"Well, if we work out those partial sums, first one is one and then it's"},{"startTime":2009.216,"endTime":2013.077,"body":"zero and then it's one again and then it's zero again, it just keeps flip"},{"startTime":2013.077,"endTime":2014.365,"body":"flopping between zero and one."},{"startTime":2014.875,"endTime":2019.916,"body":"It's never gonna converge because it never settles down to being very, very close to"},{"startTime":2019.916,"endTime":2020.925,"body":"a single number."},{"startTime":2021.265,"endTime":2025.365,"body":"It just keeps flip flopping between, um, after an even number of terms it's zero."},{"startTime":2025.385,"endTime":2026.885,"body":"And after an odd number of terms it's one."},{"startTime":2027.305,"endTime":2031.565,"body":"So this thing does not converge with this, with this definition we're working with."},{"startTime":2031.985,"endTime":2034.405,"body":"Um, so if it doesn't converge, what's the opposite of that?"},{"startTime":2034.665,"endTime":2035.525,"body":"It diverges."},{"startTime":2035.525,"endTime":2039.245,"body":"This is a divergent series even though it doesn't zoom off to infinity, but it"},{"startTime":2039.245,"endTime":2040.485,"body":"never settles on one value."},{"startTime":2040.985,"endTime":2043.805,"body":"So it's meaningless to say it equals zero or equals one."},{"startTime":2044.185,"endTime":2047.805,"body":"Um, it never act doesn't equal anything 'cause it never, it never sits still."},{"startTime":2049.755,"endTime":2050.245,"body":"Alright?"},{"startTime":2050.545,"endTime":2054.021,"body":"So we can think, well when, when would things, how do we know if something"},{"startTime":2054.021,"endTime":2054.485,"body":"does converge?"},{"startTime":2054.605,"endTime":2056.885,"body":"I mean that's clearly quite important."},{"startTime":2057.864,"endTime":2061.901,"body":"So certainly if we are adding up all these things and we've got to stay"},{"startTime":2061.901,"endTime":2065.938,"body":"within a smaller and smaller distance from our, what we think is the limit, um,"},{"startTime":2065.938,"endTime":2069.975,"body":"the things that we are adding on are gonna have to get smaller and smaller"},{"startTime":2069.975,"endTime":2070.245,"body":"themselves."},{"startTime":2071.985,"endTime":2076.364,"body":"Uh, so if we think about something like this, this is called the harmonic series."},{"startTime":2076.665,"endTime":2080.215,"body":"So we're adding, um, one plus a half plus a third plus a quarter plus"},{"startTime":2080.215,"endTime":2080.925,"body":"a fifth plus."},{"startTime":2082.425,"endTime":2085.9,"body":"If you look at the partial sums here, and I've plotted the first about a"},{"startTime":2085.9,"endTime":2089.376,"body":"hundred of them, you can see that it kind of won and then one and"},{"startTime":2089.376,"endTime":2091.925,"body":"a half, three over two, and then you add a third."},{"startTime":2092.025,"endTime":2093.925,"body":"So the bits you're adding on get less less each time."},{"startTime":2094.195,"endTime":2096.165,"body":"Kind of the grainy of this thing is settling down."},{"startTime":2096.745,"endTime":2099.93,"body":"It hasn't settled down to anything yet, but it sort of looks as if maybe"},{"startTime":2099.93,"endTime":2101.205,"body":"it might give it enough time."},{"startTime":2101.225,"endTime":2103.365,"body":"You know, it's getting flatter and flatter the line."},{"startTime":2103.745,"endTime":2105.485,"body":"So it might, it might settle down to something."},{"startTime":2106.025,"endTime":2107.845,"body":"Um, but it doesn't."},{"startTime":2108.465,"endTime":2110.965,"body":"And I, let me explain, lemme explain why it doesn't."},{"startTime":2111.545,"endTime":2115.045,"body":"So if we think about this, this series, we can group the terms."},{"startTime":2115.985,"endTime":2119.753,"body":"Um, so we've got one and then a half and then, and then we're gonna"},{"startTime":2119.753,"endTime":2123.522,"body":"group the next two terms, a third and a quarter, then the next four terms"},{"startTime":2123.522,"endTime":2127.291,"body":"up to one eighth, then the next eight terms, the next 16 and so on"},{"startTime":2127.291,"endTime":2128.045,"body":"and so on."},{"startTime":2128.265,"endTime":2133.65,"body":"The reason we're doing this is because inside each bracket everything before the last term"},{"startTime":2133.65,"endTime":2135.445,"body":"is strictly greater than that."},{"startTime":2135.465,"endTime":2138.896,"body":"So one third is bigger than a quarter quarter, uh, one fifth, one sixth, one"},{"startTime":2138.896,"endTime":2139.125,"body":"seventh."},{"startTime":2139.125,"endTime":2140.365,"body":"Those are all bigger than one eighth."},{"startTime":2141.305,"endTime":2146.211,"body":"So whatever this sum is, it's bigger than what I've just written here where we've"},{"startTime":2146.211,"endTime":2151.117,"body":"replaced a third with a quarter, we've replaced everything in that bracket by one eighth"},{"startTime":2151.117,"endTime":2156.023,"body":"and so on, and four lots of one eighth is half, two, lots of quarter"},{"startTime":2156.023,"endTime":2157.005,"body":"is a half."},{"startTime":2157.545,"endTime":2160.085,"body":"All of the brackets add up to a half now."},{"startTime":2160.345,"endTime":2164.13,"body":"So we know whatever this sum is, it's bigger than one plus a half, plus"},{"startTime":2164.13,"endTime":2165.645,"body":"a half plus a half forever."},{"startTime":2166.625,"endTime":2170.045,"body":"And that's clearly going to be bigger than any number you like to mention."},{"startTime":2170.425,"endTime":2174.205,"body":"If you want this to be bigger than a million, you just do 2 million"},{"startTime":2174.205,"endTime":2176.725,"body":"of these brackets and you are, you're bigger than that."},{"startTime":2176.785,"endTime":2179.525,"body":"So even though it slows down, it grows slower and slower."},{"startTime":2180.035,"endTime":2183.645,"body":"This series gets to be if you wait long enough, bigger than any number."},{"startTime":2184.465,"endTime":2185.805,"body":"So this cannot converge."},{"startTime":2185.835,"endTime":2191.432,"body":"This is also a divergent series, so it's not enough to just have the terms"},{"startTime":2191.432,"endTime":2192.925,"body":"getting smaller and smaller."},{"startTime":2193.105,"endTime":2194.005,"body":"We need more than that."},{"startTime":2194.585,"endTime":2195.965,"body":"So you do have to be quite careful."},{"startTime":2196.815,"endTime":2198.645,"body":"There is a tweak we can make to this series."},{"startTime":2199.545,"endTime":2204.518,"body":"Um, if we alternate pluses and minuses, you get what we call the alternating for"},{"startTime":2204.518,"endTime":2205.845,"body":"obvious reasons harmonic series."},{"startTime":2205.845,"endTime":2209.587,"body":"So this is one minus a half plus a third minus a quarter plus a"},{"startTime":2209.587,"endTime":2211.085,"body":"fifth minus a six and saw."},{"startTime":2211.905,"endTime":2216.255,"body":"Now again, I've plotted some partial sums here and I've gone up to, uh, the"},{"startTime":2216.255,"endTime":2217.125,"body":"200th partial sum."},{"startTime":2217.585,"endTime":2221.719,"body":"And you can see this isn't proving anything, but it is looking quite good for"},{"startTime":2221.719,"endTime":2225.853,"body":"converging 'cause it's sort of oscillating around and then settling down to, to a number"},{"startTime":2225.853,"endTime":2228.885,"body":"that seems to be somewhere just slightly less than not 0.7."},{"startTime":2229.585,"endTime":2235.45,"body":"And you can actually prove, uh, properly that if you work this out, this does"},{"startTime":2235.45,"endTime":2241.316,"body":"converge, we'll call it s it converges to uh, the logarithm of two, log two,"},{"startTime":2241.316,"endTime":2244.445,"body":"uh, which is about 0.69, something like that."},{"startTime":2245.345,"endTime":2250.773,"body":"So this does converge, it's a convergence series, but there's an issue and mathematicians discovered"},{"startTime":2250.773,"endTime":2256.201,"body":"this are, they're looking at this and they said, yes, it converges to to log"},{"startTime":2256.201,"endTime":2256.925,"body":"two fine."},{"startTime":2257.625,"endTime":2261.325,"body":"But let's think about calculating that in a slightly different way."},{"startTime":2261.325,"endTime":2263.365,"body":"Is there is a different way of calculating this thing."},{"startTime":2263.985,"endTime":2267.379,"body":"So, um, we just, we're gonna add all the same things up but just in"},{"startTime":2267.379,"endTime":2268.285,"body":"a slightly different order."},{"startTime":2269.025,"endTime":2271.298,"body":"Um, like we, you know, we know we can do this if we're adding two"},{"startTime":2271.298,"endTime":2272.965,"body":"or three numbers, it doesn't matter which one we add first."},{"startTime":2273.065,"endTime":2276.045,"body":"So, you know, seems, okay, so what have I done?"},{"startTime":2276.165,"endTime":2278.797,"body":"I want to, I want to explain or I want you to be convinced that"},{"startTime":2278.797,"endTime":2281.429,"body":"we have got all of the numbers in there, all, all of the terms in"},{"startTime":2281.429,"endTime":2281.605,"body":"there."},{"startTime":2281.625,"endTime":2284.125,"body":"So each of these brackets has three terms."},{"startTime":2285.025,"endTime":2290.165,"body":"The first term that I'm highlighting, uh, in green is one over the odd numbers."},{"startTime":2290.465,"endTime":2294.627,"body":"So we've got, um, in the, in the original ordering it's one minus a half"},{"startTime":2294.627,"endTime":2297.125,"body":"plus a third minus a quarter plus a fifth."},{"startTime":2297.225,"endTime":2300.167,"body":"So we've got, we are adding all the one over odds and you can see"},{"startTime":2300.167,"endTime":2302.325,"body":"that they appear at the beginning of each of these brackets."},{"startTime":2302.325,"endTime":2305.625,"body":"So you've got one plus a third, plus a fifth, plus a seventh and so"},{"startTime":2305.625,"endTime":2305.845,"body":"on."},{"startTime":2306.265,"endTime":2310.165,"body":"And then the other two terms are the one over even numbers just in the"},{"startTime":2310.165,"endTime":2312.245,"body":"original order, but we're putting them in pairs."},{"startTime":2312.245,"endTime":2316.32,"body":"So minus the half minus a quarter in the first bracket, minus six, minus an"},{"startTime":2316.32,"endTime":2318.765,"body":"eighth, minus a 10th, minus 12th and so on."},{"startTime":2318.905,"endTime":2320.045,"body":"So we have got everybody."},{"startTime":2320.945,"endTime":2322.405,"body":"Now let's think about these brackets then."},{"startTime":2322.625,"endTime":2327.068,"body":"Uh, the first two terms in each bracket, one minus a half is just a"},{"startTime":2327.068,"endTime":2327.365,"body":"half."},{"startTime":2328.025,"endTime":2328.885,"body":"Um, what have we got?"},{"startTime":2328.885,"endTime":2332.875,"body":"We've got a third minus sixth, that's a sixth, a fifth minus a 10th is"},{"startTime":2332.875,"endTime":2334.205,"body":"a 10th and so on."},{"startTime":2334.545,"endTime":2335.805,"body":"So we can simplify a bit."},{"startTime":2336.115,"endTime":2341.765,"body":"This is still the same sum and now we've got the, this calculation to do."},{"startTime":2342.545,"endTime":2346.67,"body":"You'll notice that all the denominators now are even, so we can bring out a"},{"startTime":2346.67,"endTime":2350.795,"body":"factor of a half and then we'll see we've got a half of one uh,"},{"startTime":2350.795,"endTime":2354.645,"body":"minus a half plus a third minus a quarter plus a fifth minus six."},{"startTime":2355.245,"endTime":2356.245,"body":"Hmm, that sounds familiar."},{"startTime":2356.625,"endTime":2359.405,"body":"That's our original sequence or aboriginal uh, series."},{"startTime":2359.865,"endTime":2364.128,"body":"So we just seem to have shown that S is equal to half of S"},{"startTime":2364.128,"endTime":2368.391,"body":"Um, but if s is a number that sort of implies that S is zero,"},{"startTime":2368.391,"endTime":2372.655,"body":"which have we just proved that the logarithm of two is zero, we hope we"},{"startTime":2372.655,"endTime":2374.645,"body":"haven't \u003claugh\u003e that that would be bad."},{"startTime":2374.945,"endTime":2379.99,"body":"And this was a genuine issue that mathematicians realized, oh my goodness, we worked out"},{"startTime":2379.99,"endTime":2385.035,"body":"this thing and then somehow now it also seems to be equal to zero what"},{"startTime":2385.035,"endTime":2386.045,"body":"is going on?"},{"startTime":2386.595,"endTime":2389.832,"body":"Because as I've said, you can, if you've got, like if you're doing one plus"},{"startTime":2389.832,"endTime":2393.07,"body":"two plus three, you can definitely, you can rearrange that, you can do it as"},{"startTime":2393.07,"endTime":2395.445,"body":"three plus two plus one or three plus one plus two."},{"startTime":2395.585,"endTime":2396.565,"body":"You get the same answer."},{"startTime":2397.105,"endTime":2399.245,"body":"So why can't we do this now?"},{"startTime":2400.155,"endTime":2403.245,"body":"Well sometimes you seem to be able to get away with it, sometimes you don't."},{"startTime":2403.245,"endTime":2407.565,"body":"What's going on and what is going on was finally worked out by uh, the"},{"startTime":2407.565,"endTime":2411.885,"body":"great mathematician Bernard Reman, uh, in what is now known as the Reman series theorem."},{"startTime":2411.885,"endTime":2413.565,"body":"So this is gonna solve all our problems."},{"startTime":2413.875,"endTime":2418.005,"body":"Well perhaps just the ones about infinite series, not like all our problems."},{"startTime":2418.345,"endTime":2419.965,"body":"Uh, but it's a great theorem."},{"startTime":2420.245,"endTime":2424.771,"body":"I need to give you one bit of terminology and it's this, if you've got"},{"startTime":2424.771,"endTime":2429.297,"body":"a series like the ones we've seen, um, there's this concept of being absolutely convergent"},{"startTime":2429.297,"endTime":2433.823,"body":"and that's not like really convergent \u003claugh\u003e, you know, you can either be convergent or"},{"startTime":2433.823,"endTime":2434.125,"body":"not."},{"startTime":2434.465,"endTime":2439.742,"body":"Um, but absolutely convergent is indicating taking the absolute values of the terms of the"},{"startTime":2439.742,"endTime":2442.205,"body":"series, the absolute value of the number."},{"startTime":2442.595,"endTime":2445.405,"body":"It's just like it's magnitude ignoring the signs."},{"startTime":2445.425,"endTime":2448.925,"body":"So the absolute value of minus seven is seven, for instance."},{"startTime":2449.505,"endTime":2452.965,"body":"So we say that a series is absolutely convergent."},{"startTime":2453.585,"endTime":2457.917,"body":"If the series where you add the absolute values, then the ModuLite of the numbers"},{"startTime":2457.917,"endTime":2460.805,"body":"is convergent in the way we've talked about with limits."},{"startTime":2461.465,"endTime":2467.671,"body":"So as an example, that alternating harmonic series we saw that's convergent to log two"},{"startTime":2467.671,"endTime":2468.085,"body":"possibly."},{"startTime":2468.345,"endTime":2472.895,"body":"Um, but it's not absolutely convergent because if you turn those minor signs into plus"},{"startTime":2472.895,"endTime":2477.445,"body":"signs you retrieve the harmonic series that we saw was divergent went off to infinity."},{"startTime":2478.185,"endTime":2479.605,"body":"So that's the definition we need."},{"startTime":2479.665,"endTime":2481.205,"body":"And now here comes the theorem."},{"startTime":2481.905,"endTime":2487.71,"body":"It says if a series is convergent but not absolutely convergent, like our alternating harmonic"},{"startTime":2487.71,"endTime":2489.645,"body":"series, this is really amazing."},{"startTime":2489.825,"endTime":2495.875,"body":"It can be rearranged to converge to any limit at all or even to not"},{"startTime":2495.875,"endTime":2497.085,"body":"converge to diverge."},{"startTime":2497.785,"endTime":2501.405,"body":"So we saw a rearrangement of the alternating harmonic series."},{"startTime":2501.405,"endTime":2507.335,"body":"We started off, they converged to log two, then we rearranged it and it seemed"},{"startTime":2507.335,"endTime":2513.265,"body":"to end up being zero actually you couldn't rearrange it to sum to anything at"},{"startTime":2513.265,"endTime":2518.405,"body":"all to converge to any limit at all, which is pretty amazing \u003claugh\u003e."},{"startTime":2519.065,"endTime":2522.805,"body":"Now what, what's does this mean that this, this log two thing is wrong?"},{"startTime":2523.385,"endTime":2526.565,"body":"No, what it means is that when you rearrange it, you get a different answer."},{"startTime":2526.785,"endTime":2531.942,"body":"So if, if you've got this kind of series that converges but not isn't absolutely"},{"startTime":2531.942,"endTime":2534.005,"body":"convergent, you cannot rearrange the terms."},{"startTime":2534.525,"endTime":2538.389,"body":"'cause if you do it, you know, any arguments you make become null and void,"},{"startTime":2538.389,"endTime":2540.965,"body":"you have changed the sequence of partial sums by rearranging."},{"startTime":2541.385,"endTime":2542.765,"body":"So you're just not allowed to do that."},{"startTime":2542.765,"endTime":2545.465,"body":"If you don't do that, then you're okay and you can trust the answer that"},{"startTime":2545.465,"endTime":2546.365,"body":"you get at the end."},{"startTime":2546.945,"endTime":2551.711,"body":"By contrast, these absolutely convergent series are brilliant because for them you can rearrange it"},{"startTime":2551.711,"endTime":2555.525,"body":"as much as you like and it will not affect the answer."},{"startTime":2555.625,"endTime":2558.845,"body":"So they're like the gold standard of series, that's what you want."},{"startTime":2558.845,"endTime":2561.507,"body":"You want something to be absolutely convergent, then you can do whatever you like to"},{"startTime":2561.507,"endTime":2561.685,"body":"it."},{"startTime":2561.905,"endTime":2566.77,"body":"So this really resolved everything and it means we can avoid these scary apparent paradoxes"},{"startTime":2566.77,"endTime":2569.365,"body":"and accidentally proving that that zero equals one."},{"startTime":2570.145,"endTime":2573.005,"body":"So this deals with the kind of infinite series sort of problem."},{"startTime":2574.805,"endTime":2577.778,"body":"I wanna give you one more impossible fact and then I know that I owe"},{"startTime":2577.778,"endTime":2579.365,"body":"you a couple more explanations from the beginning."},{"startTime":2580.145,"endTime":2582.885,"body":"So this one, uh, we're gonna prove that half equals a third."},{"startTime":2583.505,"endTime":2585.845,"body":"Um, now seems unlikely."},{"startTime":2586.595,"endTime":2593.129,"body":"This proof is based on one that was devised by a French mathematician, uh, Joseph"},{"startTime":2593.129,"endTime":2597.485,"body":"Bert Hong, who wanted to illustrate some difficulties with probability."},{"startTime":2598.305,"endTime":2599.845,"body":"So it think about this question."},{"startTime":2600.065,"endTime":2604.67,"body":"So you take a circle and you have to find the probability p that's a"},{"startTime":2604.67,"endTime":2606.205,"body":"random chord of that circle."},{"startTime":2606.345,"endTime":2611.373,"body":"So just a line drawn between two points on the circumference that any random chord"},{"startTime":2611.373,"endTime":2614.725,"body":"is longer than the side of the inscribed equilateral triangle."},{"startTime":2615.235,"endTime":2615.525,"body":"Okay?"},{"startTime":2615.745,"endTime":2619.101,"body":"What's the probability that a random chord will be longer than the side of that"},{"startTime":2619.101,"endTime":2619.325,"body":"trunk?"},{"startTime":2620.025,"endTime":2623.645,"body":"So there's, I'm gonna show you how to, how we can work this out."},{"startTime":2624.025,"endTime":2626.205,"body":"So there's take your random chord, how do you do that?"},{"startTime":2626.205,"endTime":2629.205,"body":"Well, you pick two random points on the circumference, right?"},{"startTime":2629.425,"endTime":2633.775,"body":"So pick your first point and then you pick another point and that gives you"},{"startTime":2633.775,"endTime":2634.645,"body":"a chord, okay?"},{"startTime":2634.745,"endTime":2639.323,"body":"So to, to think about whether this is gonna be longer than the side of"},{"startTime":2639.323,"endTime":2641.765,"body":"the triangle, I'm gonna just rotate the triangle."},{"startTime":2641.875,"endTime":2644.825,"body":"Doesn't matter if I rotate that picture on the right to make it fit in"},{"startTime":2644.825,"endTime":2646.005,"body":"with whatever I need to do."},{"startTime":2646.225,"endTime":2650.296,"body":"So I'm gonna do that so that one vertex of the triangle coincides with the,"},{"startTime":2650.296,"endTime":2651.925,"body":"the first point of my chord."},{"startTime":2652.705,"endTime":2656.538,"body":"And then we can see it's quite easy because the chords gonna be longer than"},{"startTime":2656.538,"endTime":2659.605,"body":"the side of that triangle precisely for the other end of it."},{"startTime":2659.605,"endTime":2664.827,"body":"That the second point that we chose lies in this arc, which is between the"},{"startTime":2664.827,"endTime":2669.005,"body":"two other triangle vertices, and that's a third of the the circumference."},{"startTime":2669.385,"endTime":2671.22,"body":"So there's a one in three chance of that happening."},{"startTime":2671.505,"endTime":2674.605,"body":"So clearly the probability is one third of this thing happening."},{"startTime":2674.755,"endTime":2675.445,"body":"Okay, great."},{"startTime":2676.025,"endTime":2677.445,"body":"Now I'm gonna give you a different argument."},{"startTime":2678.385,"endTime":2679.805,"body":"So the different argument goes like this."},{"startTime":2680.105,"endTime":2683.09,"body":"We can, we can define a chord in a slightly different way or choose our"},{"startTime":2683.09,"endTime":2684.085,"body":"chord in a different way."},{"startTime":2684.665,"endTime":2689.745,"body":"Uh, because if you take a chord of a circle, if you bisect it taking"},{"startTime":2689.745,"endTime":2694.826,"body":"perpendicular bisect, you might remember that uh, line will go through the center of the"},{"startTime":2694.826,"endTime":2695.165,"body":"circle."},{"startTime":2695.665,"endTime":2698.765,"body":"So you can define a random chord in a slightly different way."},{"startTime":2699.025,"endTime":2705.125,"body":"You can pick a random radius and pick a random point along that radius and"},{"startTime":2705.125,"endTime":2710.005,"body":"draw a perpendicular at that point that will give you a chord."},{"startTime":2710.425,"endTime":2715.275,"body":"And so every chord is uniquely determined, um, by picking a radius and a point"},{"startTime":2715.275,"endTime":2716.245,"body":"along the radius."},{"startTime":2716.755,"endTime":2718.445,"body":"Okay, so how, what's the criteria then?"},{"startTime":2718.545,"endTime":2721.751,"body":"How do we know if that chord is longer than the side of the inscribed"},{"startTime":2721.751,"endTime":2721.965,"body":"triangle?"},{"startTime":2722.275,"endTime":2725.365,"body":"Well, again, I can, I can put my inscribed triangle wherever I like."},{"startTime":2725.365,"endTime":2729.44,"body":"It doesn't change the size, so I'm gonna put it so that, uh, its base"},{"startTime":2729.44,"endTime":2731.885,"body":"is at right angles to this radius I've got."},{"startTime":2732.385,"endTime":2736.965,"body":"And then we can see that the cord will be longer than that side precisely"},{"startTime":2736.965,"endTime":2740.325,"body":"when the point that I choose, uh, is inside that triangle."},{"startTime":2740.505,"endTime":2743.245,"body":"So above wherever the triangle meets the radius."},{"startTime":2743.745,"endTime":2745.725,"body":"So where does the triangle meet the radius?"},{"startTime":2746.315,"endTime":2748.685,"body":"Well, uh, let's quickly do a bit of trigonometry."},{"startTime":2748.785,"endTime":2752.365,"body":"Um, if we join a little radius there, we can see what we've created."},{"startTime":2752.365,"endTime":2756.665,"body":"There is a RightAngle triangle which has a 30 degree angle and sign of 30"},{"startTime":2756.665,"endTime":2757.525,"body":"is a half."},{"startTime":2757.945,"endTime":2760.605,"body":"And so the height of that little triangle is half the radius."},{"startTime":2761.265,"endTime":2764.565,"body":"So the triangle meets this radius halfway along."},{"startTime":2765.305,"endTime":2769.275,"body":"Any point when we are making the chord, if we draw the point, um, in"},{"startTime":2769.275,"endTime":2773.245,"body":"the first half of the radius, it'll be longer than the side of the triangle."},{"startTime":2773.245,"endTime":2775.365,"body":"Otherwise not so clearly."},{"startTime":2775.385,"endTime":2776.325,"body":"The probability is a half."},{"startTime":2777.065,"endTime":2781.494,"body":"And I'm convinced by both those arguments and there are arguments that we can see"},{"startTime":2781.494,"endTime":2782.085,"body":"as well."},{"startTime":2782.515,"endTime":2783.725,"body":"Neither of them is wrong."},{"startTime":2783.835,"endTime":2784.965,"body":"Neither of them is right."},{"startTime":2785.385,"endTime":2788.245,"body":"The problem here is that we haven't defined the problem correctly."},{"startTime":2788.425,"endTime":2789.965,"body":"And this is a real issue in probability."},{"startTime":2790.465,"endTime":2795.013,"body":"If you've got an infinite amount of things to choose from, when you're trying to"},{"startTime":2795.013,"endTime":2799.562,"body":"select randomly, you have to be very, very careful because the way you are making"},{"startTime":2799.562,"endTime":2801.685,"body":"your choices often will affect the answer."},{"startTime":2801.865,"endTime":2804.605,"body":"So we have two perfectly valid bits of reasoning."},{"startTime":2805.105,"endTime":2809.322,"body":"If we choose our chord by picking two random points of circumference, the prob is"},{"startTime":2809.322,"endTime":2809.885,"body":"one third."},{"startTime":2810.345,"endTime":2814.122,"body":"If we choose our chord by this other way of, um, radii and point on"},{"startTime":2814.122,"endTime":2815.885,"body":"a radius, the probability is a half."},{"startTime":2817.095,"endTime":2821.885,"body":"We've got to define how we're choosing those random things from this infinite collection."},{"startTime":2822.065,"endTime":2826.161,"body":"And this is one of the, you know, a major step in understanding probability theory,"},{"startTime":2826.161,"endTime":2830.258,"body":"that actually sometimes we don't have enough information and it doesn't make sense to define"},{"startTime":2830.258,"endTime":2830.805,"body":"a thing."},{"startTime":2830.865,"endTime":2835.765,"body":"And this deals as well with that one of the proofs that one equals zero."},{"startTime":2835.845,"endTime":2839.302,"body":"I gave where I said that, you know, zero to the zero is one and"},{"startTime":2839.302,"endTime":2840.685,"body":"zero to the zero is zero."},{"startTime":2841.505,"endTime":2846.565,"body":"Now both of those kind of make sense that they're, they're reasonable things to say."},{"startTime":2846.905,"endTime":2851.184,"body":"It doesn't mean that one equals zero because actually if you don't say how we"},{"startTime":2851.184,"endTime":2855.463,"body":"are going to define zero to the past zero, then you can't, you can't trust"},{"startTime":2855.463,"endTime":2856.605,"body":"the answers you get."},{"startTime":2856.745,"endTime":2859.925,"body":"So normally, in fact, what we say is zero to the zero is undefined."},{"startTime":2859.945,"endTime":2864.001,"body":"We just don't define it because we'd like, you know, we want it to fit"},{"startTime":2864.001,"endTime":2866.165,"body":"in with something to the X is one."},{"startTime":2866.585,"endTime":2870.056,"body":"Um, sorry, X to zero is one, but we'd also like to fit in with"},{"startTime":2870.056,"endTime":2871.445,"body":"zero to the anything it zero."},{"startTime":2871.625,"endTime":2873.285,"body":"And you can't make both of those happen."},{"startTime":2873.545,"endTime":2876.525,"body":"So we just say we are not gonna define that there be dragons."},{"startTime":2876.665,"endTime":2877.885,"body":"And sometimes you have to do that."},{"startTime":2877.885,"endTime":2879.445,"body":"That's the only thing that makes sense."},{"startTime":2880.475,"endTime":2883.445,"body":"Okay, so we've got a few errors to clear up in the last few minutes."},{"startTime":2883.985,"endTime":2887.365,"body":"We, um, we talked about whether the smallest number really is one."},{"startTime":2888.345,"endTime":2891.765,"body":"Uh, I, I'm sorry to have to say it isn't \u003claugh\u003e, it isn't."},{"startTime":2891.765,"endTime":2892.685,"body":"But what went wrong?"},{"startTime":2893.155,"endTime":2896.432,"body":"Well, what went wrong is really what we've got here is a proof that there"},{"startTime":2896.432,"endTime":2897.525,"body":"is no smallest positive number."},{"startTime":2897.835,"endTime":2902.061,"body":"It's a proof by contradiction, a reductive ad absurd because we started with an assumption"},{"startTime":2902.061,"endTime":2905.725,"body":"that there was a smallest number and it led us to something ridiculous."},{"startTime":2906.225,"endTime":2908.885,"body":"It didn't actually prove the smallest positive number is one."},{"startTime":2909.265,"endTime":2911.725,"body":"Um, so we could rephrase this as an actual theorem."},{"startTime":2911.725,"endTime":2913.245,"body":"There is no smallest positive number."},{"startTime":2914.105,"endTime":2917.461,"body":"And the proof is we say, well let's imagine there were and do a thought"},{"startTime":2917.461,"endTime":2917.685,"body":"experiment."},{"startTime":2918.305,"endTime":2922.545,"body":"Uh, so let's assume there isn't, sorry, let's assume that there is a smallest positive"},{"startTime":2922.545,"endTime":2926.786,"body":"number and we'll call it X and then we did our little deduction and show"},{"startTime":2926.786,"endTime":2928.765,"body":"that X would have to be one."},{"startTime":2929.145,"endTime":2933.061,"body":"But that's clearly ridiculous because you know, a half is a positive number that is"},{"startTime":2933.061,"endTime":2933.845,"body":"smaller than one."},{"startTime":2934.225,"endTime":2937.925,"body":"So we, we ended up in the realm of something false, a contradiction."},{"startTime":2938.105,"endTime":2942.178,"body":"And so our assumption that there is a smallest positive number must have been wrong"},{"startTime":2942.178,"endTime":2945.165,"body":"because it led us to this, this false statement, this contradiction."},{"startTime":2945.625,"endTime":2949.525,"body":"So proof by contradiction are really wonderful things in mathematics."},{"startTime":2949.525,"endTime":2952.285,"body":"Some of the best proofs there are involve this technique."},{"startTime":2952.305,"endTime":2956.738,"body":"You just take an idea for a walk and if it fails, you know, once"},{"startTime":2956.738,"endTime":2960.285,"body":"you've eliminated the impossible, whatever remains, \u003claugh\u003e has to be the truth."},{"startTime":2960.705,"endTime":2965.045,"body":"So I didn't want this next thing I wanna show you another proof by contradiction."},{"startTime":2965.125,"endTime":2968.805,"body":"I didn't want to say it's an impossible fact 'cause I fully believe it's conclusion."},{"startTime":2969.625,"endTime":2971.165,"body":"All positive intes are interesting."},{"startTime":2971.305,"endTime":2972.925,"body":"So let's call this the optimist theorem."},{"startTime":2972.925,"endTime":2973.845,"body":"So let's prove this."},{"startTime":2974.455,"endTime":2976.445,"body":"Let's imagine a ridiculous thing to imagine."},{"startTime":2976.615,"endTime":2981.525,"body":"Let's imagine that's not true then there's at least one boring positive integer."},{"startTime":2982.425,"endTime":2984.712,"body":"But so if there's at least one of these, we can look at the whole"},{"startTime":2984.712,"endTime":2986.085,"body":"set of them and look at the smallest one."},{"startTime":2986.455,"endTime":2991.325,"body":"We'll call that n So n is the smallest of all the positive integers."},{"startTime":2991.565,"endTime":2995.605,"body":"N is the very smallest first boring positive integer."},{"startTime":2995.745,"endTime":3002.295,"body":"That's pretty interesting, pretty interesting fact about n So, so n is interesting which contradicts"},{"startTime":3002.295,"endTime":3003.605,"body":"what we said."},{"startTime":3003.665,"endTime":3008.481,"body":"So there, there, so they clearly, as we all, no no, there's no, uh, boring"},{"startTime":3008.481,"endTime":3009.445,"body":"numbers at all."},{"startTime":3009.445,"endTime":3010.125,"body":"They're all interesting."},{"startTime":3010.345,"endTime":3011.685,"body":"That's cast iron proof."},{"startTime":3012.275,"endTime":3017.736,"body":"Okay, so there's one proof left that we haven't dealt with and that was the"},{"startTime":3017.736,"endTime":3020.285,"body":"amazing fact that all triangles are isosceles."},{"startTime":3020.945,"endTime":3023.725,"body":"And here, well let's begin at the beginning."},{"startTime":3024.745,"endTime":3028.045,"body":"Let that, that diagram I drew here."},{"startTime":3028.045,"endTime":3031.832,"body":"Here's an iso leaves triangle and we're gonna see what happens when we move the"},{"startTime":3031.832,"endTime":3032.085,"body":"top."},{"startTime":3032.265,"endTime":3036.187,"body":"So, so A, B, C, we're gonna see what happens when we move that top"},{"startTime":3036.187,"endTime":3039.325,"body":"vertex a so that it stops being ISOs leaves the red line."},{"startTime":3039.325,"endTime":3042.045,"body":"You see that is bisecting the, the side bbc."},{"startTime":3042.045,"endTime":3043.085,"body":"So that's gonna stay fixed."},{"startTime":3043.385,"endTime":3047.189,"body":"But the angle bisector, which is currently hind behind the red line because when it's"},{"startTime":3047.189,"endTime":3049.725,"body":"an is OES triangle, those two lines are the same."},{"startTime":3050.025,"endTime":3054.337,"body":"The angle bi sector's gonna move and we're gonna look at where they intersect, where"},{"startTime":3054.337,"endTime":3056.925,"body":"the angle BI sector intersects this, uh, perpendicular bisect."},{"startTime":3057.065,"endTime":3063.536,"body":"So, um, we'll just watch this happen and you'll see that that place where they"},{"startTime":3063.536,"endTime":3066.125,"body":"meet isn't actually inside the triangle."},{"startTime":3066.985,"endTime":3070.485,"body":"Now you can see we, it gets close, closer, click on."},{"startTime":3070.485,"endTime":3073.245,"body":"No, it's not gonna make it, it is not inside the triangle."},{"startTime":3073.985,"endTime":3076.405,"body":"Now that on its own isn't necessarily an issue."},{"startTime":3076.685,"endTime":3080.83,"body":"'cause maybe all the, you know, the calculations or whatever we did didn't rely on"},{"startTime":3080.83,"endTime":3082.765,"body":"the diagram necessarily, but well we'll see."},{"startTime":3082.945,"endTime":3085.787,"body":"So the diagram ought to look a bit more like the thing on the right,"},{"startTime":3085.787,"endTime":3086.925,"body":"not the thing on the left."},{"startTime":3087.235,"endTime":3092.443,"body":"However, all of those calculations or or deductions about the congruent triangles, they are all"},{"startTime":3092.443,"endTime":3093.485,"body":"still completely valid."},{"startTime":3093.735,"endTime":3094.565,"body":"Those were correct."},{"startTime":3094.945,"endTime":3097.365,"body":"All of those pairs of triangles really are congruent to each other."},{"startTime":3097.825,"endTime":3104.83,"body":"So we still genuinely, it's true that AP equals AQ and PB equals qc, but"},{"startTime":3104.83,"endTime":3107.165,"body":"the problem is guess what?"},{"startTime":3107.165,"endTime":3108.245,"body":"It's a rogue minus sign."},{"startTime":3108.525,"endTime":3109.925,"body":"I knew that there'd be one somewhere."},{"startTime":3110.185,"endTime":3115.399,"body":"So yes, ab if you look at the diagram on the right, AB does equal"},{"startTime":3115.399,"endTime":3120.614,"body":"AP plus pb and AP does equal aq and PB does equal cq, but that"},{"startTime":3120.614,"endTime":3122.005,"body":"does not equal ac."},{"startTime":3122.885,"endTime":3125.485,"body":"'cause Q is outside of the, of the triangle."},{"startTime":3125.545,"endTime":3127.965,"body":"Now we had to extend that line to drop a perpendicular."},{"startTime":3128.185,"endTime":3133.765,"body":"And so AC does not equal AQ plus qc, it equals AQ minus qc."},{"startTime":3133.785,"endTime":3136.045,"body":"And that of course \u003claugh\u003e changes everything."},{"startTime":3136.345,"endTime":3140.461,"body":"So here the diagram, it wasn't that it was wrong, but that it led us"},{"startTime":3140.461,"endTime":3143.205,"body":"to think something that was not guaranteed to be correct."},{"startTime":3143.385,"endTime":3147.789,"body":"So if you see some geometry, just make sure to draw a diagram that uh,"},{"startTime":3147.789,"endTime":3150.725,"body":"that's perhaps quite accurate and see, see what you get."},{"startTime":3151.225,"endTime":3156.325,"body":"Um, well let's, let's finish up with another theorem."},{"startTime":3156.735,"endTime":3159.005,"body":"There exists a triangle with three right angles."},{"startTime":3159.005,"endTime":3160.405,"body":"Now we've proved all triangles, right?"},{"startTime":3160.465,"endTime":3160.885,"body":"Softly."},{"startTime":3160.885,"endTime":3161.805,"body":"This should be a dole."},{"startTime":3162.315,"endTime":3170.574,"body":"Well, now this is kind of the final thing that I want us to think"},{"startTime":3170.574,"endTime":3171.125,"body":"about."},{"startTime":3171.155,"endTime":3175.525,"body":"That when we prove something, we start with initial assumptions."},{"startTime":3176.315,"endTime":3180.551,"body":"When we prove that the angles in a triangle add up to 180 degrees, you"},{"startTime":3180.551,"endTime":3184.787,"body":"may remember this proof, it relies on the parallel postulate 'cause you have your triangle"},{"startTime":3184.787,"endTime":3186.765,"body":"and you the top point of that."},{"startTime":3187.145,"endTime":3191.535,"body":"Um, you draw a line through that top vertex parallel to the opposite side, and"},{"startTime":3191.535,"endTime":3195.926,"body":"that then you do some stuff and you get very quickly the angles add up"},{"startTime":3195.926,"endTime":3196.805,"body":"to hundred 80."},{"startTime":3197.265,"endTime":3200.455,"body":"If you do not have the parallel postulate, then you can't prove, at least not"},{"startTime":3200.455,"endTime":3203.645,"body":"in that way, that the angles in a triangle add up to hundred 80 degrees."},{"startTime":3204.115,"endTime":3205.565,"body":"That particular proof doesn't work."},{"startTime":3205.665,"endTime":3209.556,"body":"And it turns out that there are geometries, for example, the one we have on"},{"startTime":3209.556,"endTime":3213.447,"body":"the surface of a sphere in which the angles in a triangle do not add"},{"startTime":3213.447,"endTime":3214.485,"body":"up to 180 degrees."},{"startTime":3214.945,"endTime":3217.005,"body":"And here is one that has three right angles."},{"startTime":3217.665,"endTime":3223.245,"body":"And that exploring the idea, the question, when is this true?"},{"startTime":3223.635,"endTime":3224.765,"body":"When could this be true?"},{"startTime":3225.115,"endTime":3227.685,"body":"It's actually one of those most powerful questions in mathematics."},{"startTime":3228.025,"endTime":3231.725,"body":"So it's really fun to explore when might this thing be true."},{"startTime":3232.345,"endTime":3236.371,"body":"Now, this is obviously a very non-nuclear juncture is a very deep and exciting and"},{"startTime":3236.371,"endTime":3237.445,"body":"interesting area of mathematics."},{"startTime":3237.805,"endTime":3241.885,"body":"I thought I'd give you a slightly less highfalutin puzzle to finish with, um, along"},{"startTime":3241.885,"endTime":3244.605,"body":"the line, what I call fake proofs of true facts."},{"startTime":3245.185,"endTime":3247.285,"body":"So fractions again, and sadly they exist."},{"startTime":3247.825,"endTime":3249.765,"body":"Um, we all know how to cancel fractions, right?"},{"startTime":3249.765,"endTime":3251.925,"body":"If it's the same thing on the top and the bottom, we can cancel it."},{"startTime":3252.105,"endTime":3256.645,"body":"So 16 over 64 does equal one quarter because we just cancel those sixes."},{"startTime":3256.645,"endTime":3260.285,"body":"Yeah, that's how cancellation isn't that?"},{"startTime":3260.945,"endTime":3261.165,"body":"No."},{"startTime":3261.165,"endTime":3261.965,"body":"Okay, we'll do another one."},{"startTime":3261.965,"endTime":3262.485,"body":"We'll do another one."},{"startTime":3262.805,"endTime":3265.085,"body":"I I, okay, 19 over 95, right?"},{"startTime":3265.085,"endTime":3266.045,"body":"Cancel the nines."},{"startTime":3266.745,"endTime":3266.965,"body":"Yes."},{"startTime":3267.265,"endTime":3271.765,"body":"So it works, uh, does not always work, but when does it work, right?"},{"startTime":3271.995,"endTime":3272.645,"body":"It's quite fun."},{"startTime":3272.665,"endTime":3276.208,"body":"In fact, you can extend this if you have a one and a bunch of"},{"startTime":3276.208,"endTime":3279.752,"body":"nines and then the same number of nines on the bottom and then a five,"},{"startTime":3279.752,"endTime":3283.296,"body":"you can cancel all of those pairs of nines and you do magically get the"},{"startTime":3283.296,"endTime":3284.005,"body":"answer one fifth."},{"startTime":3284.355,"endTime":3287.085,"body":"There's a little proof of that underneath very quick thing."},{"startTime":3287.475,"endTime":3289.005,"body":"It's actually a true proof for once."},{"startTime":3289.305,"endTime":3293.355,"body":"So you take n nines and then a five, that's 10 to the power NN"},{"startTime":3293.355,"endTime":3296.325,"body":"plus one, take away five, bring out the factor of five."},{"startTime":3296.995,"endTime":3300.045,"body":"Then you get two, uh, times 10 to the N."},{"startTime":3300.105,"endTime":3306.369,"body":"So two with n zeros minus A one, and then that equals exactly five, lots"},{"startTime":3306.369,"endTime":3307.205,"body":"of one."},{"startTime":3307.225,"endTime":3308.645,"body":"And then N nines."},{"startTime":3308.645,"endTime":3308.805,"body":"Okay?"},{"startTime":3308.805,"endTime":3309.925,"body":"So you can actually prove this."},{"startTime":3310.225,"endTime":3315.205,"body":"So the challenge is find other, uh, true facts with fake proofs like this."},{"startTime":3315.585,"endTime":3317.605,"body":"And in particular, I don't know what the answer to this is."},{"startTime":3317.785,"endTime":3321.419,"body":"Is it ever true that you can add fractions in the way that some eight"},{"startTime":3321.419,"endTime":3325.054,"body":"year olds and nine year olds would like to add fractions, which is A over"},{"startTime":3325.054,"endTime":3328.205,"body":"B plus C over D equals A plus C over B plus D."},{"startTime":3328.205,"endTime":3332.49,"body":"So that's a little challenge to keep you going until my next lecture, which is"},{"startTime":3332.49,"endTime":3336.776,"body":"my very fine aggression lecture Next time, uh, on June the fourth, we're gonna talk"},{"startTime":3336.776,"endTime":3341.062,"body":"about some real proofs and, and you know, the beautiful ideas that we can exploit"},{"startTime":3341.062,"endTime":3342.205,"body":"and enjoy in mathematics."},{"startTime":3342.385,"endTime":3345.325,"body":"And I'm gonna show you some of my favorite proofs, but we'll stop there."},{"startTime":3345.325,"endTime":3345.885,"body":"Thank you very much."},{"startTime":3356.095,"endTime":3356.765,"body":"Thank you Sarah."},{"startTime":3356.905,"endTime":3358.045,"body":"Um, I'm Martin Elliott."},{"startTime":3358.045,"endTime":3361.866,"body":"I'm the provost of the college and it's always an immense treat for me to"},{"startTime":3361.866,"endTime":3362.885,"body":"come here, Sarah, speak."},{"startTime":3363.225,"endTime":3367.256,"body":"I'm also very reassured that I now know how the chancellor works out the next"},{"startTime":3367.256,"endTime":3367.525,"body":"budget."},{"startTime":3367.845,"endTime":3372.845,"body":"\u003claugh\u003e."},{"startTime":3368.605,"endTime":3369.965,"body":"Tremendously powerful."},{"startTime":3370.345,"endTime":3375.832,"body":"I'm gonna take some questions online first, is there anything other than an infinite series"},{"startTime":3375.832,"endTime":3379.125,"body":"that can cause false statements to be proven true?"},{"startTime":3379.835,"endTime":3380.565,"body":"Well, uh, yes."},{"startTime":3380.585,"endTime":3384.845,"body":"Proven there's got to be a hole in your proof somewhere, but I mean, yes."},{"startTime":3384.865,"endTime":3387.525,"body":"And as we've, we've seen some of the ways that that can happen."},{"startTime":3388.005,"endTime":3391.955,"body":"I did not divide by zero at any point today, but that's another way that"},{"startTime":3391.955,"endTime":3394.325,"body":"you can often make false things happen to true."},{"startTime":3394.425,"endTime":3396.805,"body":"You disguise it, you do some algebraic calculation."},{"startTime":3396.805,"endTime":3400.623,"body":"You say, you know, let x equal Y at the beginning and then you wait"},{"startTime":3400.623,"endTime":3404.441,"body":"a few lines until everyone's forgotten and then you divide by X minus Y and"},{"startTime":3404.441,"endTime":3405.205,"body":"hope nobody notices."},{"startTime":3405.265,"endTime":3409.328,"body":"So there's a lot of ones that involve division by zero, but I thought this"},{"startTime":3409.328,"endTime":3413.392,"body":"is a very superior audience and they're not, they are gonna spot division by zero"},{"startTime":3413.392,"endTime":3414.205,"body":"just straight away."},{"startTime":3414.225,"endTime":3415.485,"body":"So I I didn't do that."},{"startTime":3416.015,"endTime":3418.565,"body":"Let's take some questions from the floor before I go back to those online."},{"startTime":3418.905,"endTime":3419.885,"body":"So anybody who have one"},{"startTime":3420.405,"endTime":3425.389,"body":"Y the, um, the, the one I, um, uh, hadn't seen before, and I'm not"},{"startTime":3425.389,"endTime":3429.045,"body":"quite sure of the, the problem, the, the differentiation one mm-Hmm."},{"startTime":3429.065,"endTime":3431.565,"body":"Um, the x plus X plus X equals x squared."},{"startTime":3432.065,"endTime":3432.925,"body":"Can you you say?"},{"startTime":3433.705,"endTime":3433.925,"body":"So?"},{"startTime":3434.005,"endTime":3438.518,"body":"I think so this is where you have x, lots of X and then on"},{"startTime":3438.518,"endTime":3440.925,"body":"the right hand side you have X squared."},{"startTime":3441.425,"endTime":3445.4,"body":"Now, one problem is that I had X, lots of X, and when I differentiated"},{"startTime":3445.4,"endTime":3449.375,"body":"all those little Xs, I got ones, but I didn't do anything about the fact"},{"startTime":3449.375,"endTime":3450.965,"body":"that I had X of them."},{"startTime":3451.585,"endTime":3453.805,"body":"And I think that's, that's the basic issue."},{"startTime":3453.825,"endTime":3457.909,"body":"So I ended up with X, lots of one, but it's not really clear how"},{"startTime":3457.909,"endTime":3461.993,"body":"you make a function that says here are X, lots of X and actually you"},{"startTime":3461.993,"endTime":3465.805,"body":"sort of force yourself to write X squared and that's \u003claugh\u003e, that's the issue."},{"startTime":3465.825,"endTime":3468.937,"body":"So I mean, if you were to try and do this from first principles and"},{"startTime":3468.937,"endTime":3472.05,"body":"properly represent the left hand side, I ignored the fact that I had x lots"},{"startTime":3472.05,"endTime":3474.125,"body":"of this thing and I didn't do anything about that."},{"startTime":3474.145,"endTime":3475.885,"body":"That's really what's gone wrong there."},{"startTime":3477.665,"endTime":3478.765,"body":"Any more questions than the before?"},{"startTime":3479.115,"endTime":3480.605,"body":"Otherwise I'll read you one out."},{"startTime":3481.145,"endTime":3485.261,"body":"Uh, why are there irrational numbers that can't be expressed as a ratio of two"},{"startTime":3485.261,"endTime":3486.085,"body":"other whole numbers?"},{"startTime":3486.545,"endTime":3488.525,"body":"And does it show us anything or prove anything?"},{"startTime":3489.345,"endTime":3493.504,"body":"Uh, well why are there, you know, it might have to ask a higher power"},{"startTime":3493.504,"endTime":3495.445,"body":"than me, but there definitely are \u003claugh\u003e."},{"startTime":3495.655,"endTime":3500.13,"body":"There are, and, and so you can see this, the, um, if you take it"},{"startTime":3500.13,"endTime":3504.606,"body":"just a square whose sides of one and the diagonal of that square cannot be"},{"startTime":3504.606,"endTime":3508.485,"body":"expressed as, as a fraction, however hard you try it, square of two."},{"startTime":3508.745,"endTime":3513.725,"body":"Um, and these, so these are called irrational numbers and they are everywhere."},{"startTime":3514.345,"endTime":3516.525,"body":"Uh, but you know, nothing to be afraid of."},{"startTime":3516.555,"endTime":3520.605,"body":"It's just that we, our, our human minds who thought of using fractions to write"},{"startTime":3520.605,"endTime":3523.845,"body":"things just didn't realize that there are these other numbers out there."},{"startTime":3523.945,"endTime":3527.485,"body":"So yeah, they, they, they are, they vastly outnumber."},{"startTime":3527.585,"endTime":3531.145,"body":"In fact, the numbers that come over written as fractions, almost every number that you"},{"startTime":3531.145,"endTime":3533.045,"body":"pick up random off the street is irrational."},{"startTime":3533.825,"endTime":3539.209,"body":"How, how frequently do these things happen in the conversion of pure mass as it"},{"startTime":3539.209,"endTime":3540.645,"body":"were into applied mass?"},{"startTime":3540.935,"endTime":3545.003,"body":"Apart from the first example you showed of us landing on Mars, how frequently disasters"},{"startTime":3545.003,"endTime":3547.445,"body":"happen when people get these sequence of assumptions wrong?"},{"startTime":3547.945,"endTime":3552.605,"body":"So I think we are quite lucky in that most of the time I would"},{"startTime":3552.605,"endTime":3556.645,"body":"say we, we've got away with things over the years, let's say \u003claugh\u003e."},{"startTime":3556.865,"endTime":3561.005,"body":"Um, so in the kind of 200, 300 years ago, people would kind of play"},{"startTime":3561.005,"endTime":3563.765,"body":"around with infinite series and assume everything would be fine."},{"startTime":3564.425,"endTime":3568.381,"body":"And it, because the series they were playing around with happened to be quite well"},{"startTime":3568.381,"endTime":3568.645,"body":"behaved."},{"startTime":3568.645,"endTime":3571.503,"body":"Mostly you can get away with it, but that's, I mean, that's no way to"},{"startTime":3571.503,"endTime":3571.885,"body":"do mathematics."},{"startTime":3571.985,"endTime":3575.925,"body":"But it took a while for to really come up against these issues for real."},{"startTime":3576.225,"endTime":3580.54,"body":"And I think one risk that can happen if you're transferring kind of from pure"},{"startTime":3580.54,"endTime":3584.856,"body":"mathematics to applied mathematics is maybe the pure mathematicians have this cast iron thing that"},{"startTime":3584.856,"endTime":3588.885,"body":"says, under these conditions and these conditions alone, you are allowed to do this."},{"startTime":3589.505,"endTime":3592.085,"body":"And then perhaps that may slightly get lost sometimes."},{"startTime":3592.185,"endTime":3593.845,"body":"And then you just say, oh, I know I can do that."},{"startTime":3593.905,"endTime":3594.885,"body":"And I'm sure it's fine."},{"startTime":3595.315,"endTime":3597.125,"body":"Just every so often it isn't fine."},{"startTime":3597.665,"endTime":3602.232,"body":"Um, but you know, in the, in the real world, of course, uh, you, you"},{"startTime":3602.232,"endTime":3606.8,"body":"have mathematical models, you have approximations and you can test and everything and make sure"},{"startTime":3606.8,"endTime":3609.845,"body":"that, that it will, it will work in real life."},{"startTime":3609.945,"endTime":3613.857,"body":"But we, it is important to, to make sure, yeah, that we are not gonna"},{"startTime":3613.857,"endTime":3616.205,"body":"end up with \u003claugh\u003e with, with proving impossible facts."},{"startTime":3616.265,"endTime":3617.965,"body":"So yeah, occasionally it crops up."},{"startTime":3618.505,"endTime":3619.245,"body":"Uh, thanks for that."},{"startTime":3619.305,"endTime":3622.525,"body":"Is there a governing body or anything to accept a proof?"},{"startTime":3622.905,"endTime":3623.725,"body":"Oh, okay."},{"startTime":3623.725,"endTime":3624.685,"body":"Yeah, there should be."},{"startTime":3624.785,"endTime":3628.433,"body":"Um, well what, what in practice, what happens is when you're a mathematician and you,"},{"startTime":3628.433,"endTime":3632.081,"body":"you think you've got a theorem, you will write that up as a, an academic"},{"startTime":3632.081,"endTime":3632.325,"body":"paper."},{"startTime":3632.325,"endTime":3636.595,"body":"You will send it to a journal and the journals then will send that out"},{"startTime":3636.595,"endTime":3637.165,"body":"to referees."},{"startTime":3637.165,"endTime":3641.415,"body":"So, so you know, people, other mathematicians who will look independently at what you've written"},{"startTime":3641.415,"endTime":3645.665,"body":"and try and follow your argument and if, if there's something that they don't understand"},{"startTime":3645.665,"endTime":3647.365,"body":"or why, why is that allowed?"},{"startTime":3647.365,"endTime":3650.612,"body":"Then they send it back and say, can you explain how you got from here"},{"startTime":3650.612,"endTime":3651.045,"body":"to here?"},{"startTime":3651.225,"endTime":3653.125,"body":"And if you can't, maybe you realize there's a mistake."},{"startTime":3653.145,"endTime":3657.728,"body":"So the idea is, you know, peer review will capture, um, any errors that you"},{"startTime":3657.728,"endTime":3658.645,"body":"might have made."},{"startTime":3658.945,"endTime":3663.38,"body":"And you know, as you go on in your mathematical career, hopefully you make fewer"},{"startTime":3663.38,"endTime":3667.816,"body":"gross, logical, uh, \u003claugh\u003e mistakes, you know, it still can happen and, and you know,"},{"startTime":3667.816,"endTime":3671.365,"body":"not, not perhaps not these kind of things, but maybe other things."},{"startTime":3671.385,"endTime":3675.445,"body":"And of course you can always, everyone can make mistakes, um, accidentally."},{"startTime":3675.445,"endTime":3679.495,"body":"So yeah, it's peer review that happens and then things are published and your other"},{"startTime":3679.495,"endTime":3680.845,"body":"fellow mathematicians will read them."},{"startTime":3680.905,"endTime":3682.285,"body":"So that's the closest we get."},{"startTime":3682.365,"endTime":3682.605,"body":"I guess"},{"startTime":3682.995,"endTime":3686.365,"body":"Last thing you said there, a great example of that was Andrew Wildes."},{"startTime":3686.585,"endTime":3686.805,"body":"Yes,"},{"startTime":3687.625,"endTime":3687.845,"body":"Yes."},{"startTime":3687.865,"endTime":3692.391,"body":"So Andrew Welles, so famously the proof of firm last theory in which, which came"},{"startTime":3692.391,"endTime":3696.917,"body":"some, some years after firmer claimed he'd, uh, got a nice little thing that didn't"},{"startTime":3696.917,"endTime":3698.125,"body":"fit in the margin."},{"startTime":3698.465,"endTime":3703.853,"body":"Um, but Andrew Wells approved firm's lost theorem, but his first proof as submitted had"},{"startTime":3703.853,"endTime":3707.805,"body":"a fatal flaw and happily that was able to be fixed."},{"startTime":3708.065,"endTime":3712.48,"body":"But it, the first original proof was that like 93, um, had this, had this"},{"startTime":3712.48,"endTime":3716.896,"body":"mistake in it that was, it wasn't like, oh, you've forgotten to do something, some"},{"startTime":3716.896,"endTime":3717.485,"body":"trivial thing."},{"startTime":3717.485,"endTime":3721.19,"body":"It was genuinely a fatal flaw in the proof, but happily he did manage to"},{"startTime":3721.19,"endTime":3721.685,"body":"correct that."},{"startTime":3721.905,"endTime":3726.21,"body":"It took a lot of work and then the correct version was, was submitted and"},{"startTime":3726.21,"endTime":3727.645,"body":"everyone does now believe it."},{"startTime":3728.905,"endTime":3733.789,"body":"So I've got one last question, which borders on the philosophical, oh, \u003claugh\u003e, the basis"},{"startTime":3733.789,"endTime":3737.045,"body":"of mathematics is the concept of zero unity and infinity."},{"startTime":3738.225,"endTime":3740.885,"body":"Do those concepts apply universally?"},{"startTime":3742.355,"endTime":3747.764,"body":"Alright, well, uh, if you accept, certainly those are very important concepts, zero, one and"},{"startTime":3747.764,"endTime":3748.125,"body":"infinity."},{"startTime":3748.385,"endTime":3752.565,"body":"And you can almost, so I remember, uh, James Joyce, famous mathematician \u003claugh\u003e."},{"startTime":3753.145,"endTime":3756.461,"body":"He, he had a little notebook and he, he wrote these sort of links between"},{"startTime":3756.461,"endTime":3757.125,"body":"these three concepts."},{"startTime":3757.245,"endTime":3761.645,"body":"So zero divided by one is zero, one divided by zero is infinity."},{"startTime":3762.105,"endTime":3765.747,"body":"And you know, infinity divided by one is, so all of these things you can"},{"startTime":3765.747,"endTime":3767.205,"body":"sort of relate to each other."},{"startTime":3767.265,"endTime":3770.508,"body":"So they are, they're wonderful concepts and in a way they're all linked to each"},{"startTime":3770.508,"endTime":3770.725,"body":"other."},{"startTime":3771.345,"endTime":3777.707,"body":"But, um, I say I think that they are universal because, you know, and not"},{"startTime":3777.707,"endTime":3783.645,"body":"zero IE you don't have anything that's clearly a thing that can happen anywhere."},{"startTime":3783.905,"endTime":3788.898,"body":"Um, one, the basic unit from which everything else is built, I feel, you know,"},{"startTime":3788.898,"endTime":3793.891,"body":"that's got to, if you have any alien civilization and they count, they're gonna start"},{"startTime":3793.891,"endTime":3798.885,"body":"at one like we do, you know, these things, these things are not really choices."},{"startTime":3799.185,"endTime":3804.222,"body":"And there's a gone who, what's the name of that mathematician who said that, uh,"},{"startTime":3804.222,"endTime":3808.925,"body":"God invented the, the natural numbers and everything else \u003claugh\u003e comes from man, right?"},{"startTime":3809.265,"endTime":3812.325,"body":"But yeah, I think these things are universal concepts."},{"startTime":3812.895,"endTime":3813.365,"body":"Thank you."},{"startTime":3813.445,"endTime":3817.327,"body":"Well, I'm sure you'll agree with me that Sarah is more than one in 1,000,001"},{"startTime":3817.327,"endTime":3817.845,"body":"in infinity."},{"startTime":3818.175,"endTime":3818.525,"body":"Thank."}]}