Well, I'm a mathematician and I like YouTube and I'm often inspired by stuff that I see on Youtube and recently I saw two clips that Gave me an idea. I just want to tell you about this Impressive, I'm sure they don't do anything else, but march that's the main occupation, right This is impressive, but what I found even more impressive was some Japanese people competing in a marching competition So here we go Now that's really impressive Anyway, so I had this idea It's basically squares marching around obviously as a mathematician. I like squares. I like circles, I like all sorts of these things, so I thought well, maybe What I want to do is... I had this idea If you had two (identical) squares kind of marching around and what I wanted to do is kind of devise some sort of choreography for these two squares merging into one big huge square, and so you can imagine the two squares kind of marching around there. And then eventually they they kind of come together and whoops you see like one big square marching, okay? It's a good idea, so let's let's see whether we can do this. Okay, first thing we have to figure out is how big do we want to make our squares, okay? So we could start with really really small ones like So 2 by 2, is that going to work? Let's have a look right. So we've got four marches and here another four marches in there. It's a total of eight marches But we need a square number of marchers, right? So that's not going to work all right so we can't really go really small. What about the next size up three? Times three and three times three so that's nine here, and that's nine there. That is together 18. Again that's not a square number, so it's not going to work. Let's keep on going 16 + 16 that's 32 not a square number, okay, then 25 Plus 25 That's 50 It's actually pretty close 49 [so] [it's] just one off so Well anyway, let's keep searching. So eventually we got to find something hopefully already pretty close here on let's use this It's actually not going to work, but eventually is going to work, and I just want something small It's a near miss. All right, so this is 12 squared. This is 12 squared - 12 squared is 144 Times 2 is a 288 and that 17 squared is 289. So it's also just one off. All right, and it's good enough to actually show us What's going to happen okay? So now let's not worry about actually finding the right example Let's just keep on going with our choreography. How we're going to actually merge those two squares together, okay? And also the other thing that I want to say since I'm having so much trouble finding these squares I'm going to use the smallest example eventually. It's going to be important for later on okay. So this let's just say this is the smallest example that exists okay, and so now we're going to do our choreography. We're going to merge those two squares together. So here we go. We're going to march these squares around okay, and these are actually the Guys who are marching and this is kind of just the template for us to see where they now actually have to go. Eventually these these blue dots have to fit into this grid here Okay what we're going to do is just march these two squares in from the left and the right okay like that And they're going to overlap here in the middle. It's a bit of a problem Maybe but not really you just have some marchers stand on their comrades' shoulders, right? So here we basically all double up, where the red points are all doubled up, right? Okay So what that means is in the square that we're seeing here [there] [is] nobody standing here There's just one person standing in the blue bits and on every one of the red bits There's two people doubled up right? So if I just take the guys who are standing on the shoulders They should now Fit into those Empty squares right because eventually you know whatever we had to start with is supposed to fill in all the spots, right? So what that means of course is that this red Square is? Exactly as big as those two little ones Combined together right. So that's a logical conclusion right, so this one plus that one equals that one. Okay, well that's great. But there's something really really really strange here, hmm... remember We started out with saying this is the smallest example where two [identical] smaller squares combine into a larger square But now we found even smaller ones How is that possible that's supposed to be the smallest example and now we found an even smaller one? Hmm Well, you can think about it, but actually the only way you can resolve this Paradox is to say well there is no smallest example because if there is a smallest example There can't be any smaller example. We basically start with an assumption that we can find the smallest example, but that shows us, this kind of paradox this contradiction shows us. there can't be a smallest example. So there is actually no way to merge two [identical] squares into a larger square So that's what we're just shown it's actually in mathematics is called a proof by contradiction You assume that something is the case okay? And then you make logical deductions until you come to a point where you can see there's some nonsense here But from something that's true Nonsense can't follow so that means that what you started with has to be [nonsense] in the first place and it means that whatever is the opposite of what you assumed has to be true. in this case two [identical] squares cannot possibly combine two a larger square [alright], so we can also express this mathematically or algebraically. Two integer squares can't add up to another integer square like that and then we can transform this a little bit, so Two times m squared is equal to n squared doesn't work and then if you divide here that one can't work and now if you do a square root you see that one can't work and what that says here is that the square root of 2 cannot be written as a fraction as a fraction of integers and what that says is that square root of 2 is irrational so actually our marching square scenario our marching square consideration also show that the square root of two is irrational≥ All right now. What else can we say here? Okay, this was actually the near-miss example that I used okay. So it was 12 squared plus 12 squared equals 17 squared minus 1 well, I Mean I just use this for illustration, but actually when you when you really do it And I'm just going to do it again in a nice symmetric way There's something nice coming out of this and I just want to show you that one, okay, so just kind of March these squares in Now this way, and then we get the same overlap as before all right Okay, and then actually Because we started out with a near-Miss I mean if this was exact then the green bit should be exactly as big as as these two combine together But since we started with a near-miss. Well, we should be closed again. We should [be] close again. We can actually check now We've got one two three four five and here we've got one two three four five six seven that's the other near-miss that we kind of encountered already at the very very beginning and that's 5 square plus 5 squared equals seven squared plus one okay, and why stop there all right? So we can just move those two guys into the middle now, and they're going to create an overlap Let's just see what we get here. Well. We've got a two squared there with our two squared That's eight and just [off] by one from nine. So that's another near miss here. All right, so we get another near miss there and What what else we don't have to stop there? It's a really simple one, but it's still a near-miss 1 squared plus 1 squared equals 1 squared plus 1 okay? right so so with one one of those near misses we kind of You know get all all sorts of other near misses when we kind of repeat this merging procedure. It's really really quite fun And actually get all of the smaller ones, but we can also kind of go backwards, and when we go backwards from here we actually get all the near misses that there are [okay], so let's just go backwards, so [kind] of go like this so unmerge right then we put the square around unmerge that gets us back to the beginning now [if] we want the next higher or the next bigger near-miss what we do is we just put a big square around this like so Measure how wide this is then there's another square inside. Measure, how far this is and we get 29 Squared plus 29 squared equals 41 squared plus 1 and Then you take those two squares again arrange them like this put a big square around it and we get another example and another example and another example, and it's actually going to be all examples of two squares Adding up to another square with the difference of just one okay, and here are a couple of our examples. There is more There's heaps more here on the left side and Well those near misses They are also really nice in another way, so let's for example look here, it was 1/1, 3/2 7/5 if you just actually look at them as fractions you get a 1 here you get a 1.5 there you get a 1.4 there and That one here and what do you think is happening here? well, they're getting closer and closer to square root of 2 And Especially this one, these are actually really really nice approximations really really good approximations to the square root of 2 So I mean people always obsess about Pi and like one of the things you know about pi Is that 22 over 7 is almost pi right? so if you're actually looking for a fraction that you know does the same for square of 2 you'd probably go for this one here nobody knows this one But 17 over 12 plays the same role for square root of 2 as 22 over 7 plays for pi
I managed to come across this which uses a similar method to prove the irrationality of the square root of 3, 6 and 10.
Is there a way to generalize the last bit into some sort of limit for the square-root of 2?
I'm really liking the videos on this channel.
So the underlying algebra is that if 2a2 = b2 then 2(b - a)2 = (2a - b)2 . But since b/2 < a < b (easy to show assuming all started positive), the latter is a smaller positive example than the former, so there is no smallest positive example of satisfying 'a's and 'b's.
Interesting. Haven't thought very hard about proofs that don't appeal to the fundamental theorem of arithmetic, but this seems to be a nice one that can be shown from first principles.
What do you mean by generalize?