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P7: Root for Squares (Irrationality of the Square Root of Two)
524 Plays4 years ago
Join Sofía and Gabriel as they discuss an old but great proof of the irrationality of the square root of two.
[Featuring: Sofía Baca, Gabriel Hesch]
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Transcript
Introduction to the Irrationality of √2
00:00:07
Speaker
I'm Sophia. And I'm Gabriel. And you're listening to a Breaking Math Problem episode about the irrationality of the square root of 2. There are many proofs about the irrationality of the square root of 2, some that are published even as recently as the 1950s. You can read more about that on the Wikipedia page. I found that very fascinating in the research for this episode.
Promotional Announcements
00:00:25
Speaker
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00:00:42
Speaker
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00:01:07
Speaker
We also have a sister podcast that is available on YouTube and also wherever podcasts are played. And that is found at youtube.com slash touring rabbit holes podcast. You can also search for the touring rabbit holes podcast on any podcast player as well. And that's done with myself as well as Dr. Alex Alaniz whose specialty is particle physics. Very good show.
00:01:27
Speaker
Oh yeah, absolutely.
The Legend of Hippasus and √2
00:01:29
Speaker
So yeah, I mean, so this goes back all the way to Archimedes and the legend of Hippaces of Metapontum who divulged that the square root of two was irrational and according to legend they murdered him for it. And so this was a pretty big deal to them because they thought that everything should be representable as the ratio between two whole numbers.
00:01:49
Speaker
Yeah, it's fascinating that some knowledge can be forbidden and that somebody could actually be murdered for it, but history is filled with that kind of thing. Oh, yeah, absolutely. And real quick, I know that you had mentioned that Archimedes kept this a secret. Rather, he kept it a secret that the square root of 2 is irrational. I'm wondering, is there evidence of that or is it assumed based on what we do know about him that he probably was aware of that fact?
00:02:13
Speaker
I can't remember actually where that part of the construction comes from. I think eventually it was accepted by the Archimedians. I think that's something about that. But honestly, I haven't seen the original historical records or anything about this. So I'm not totally sure. Interesting.
Why is √2 Irrational?
00:02:28
Speaker
Yeah, so the square root of 2 is irrational. That means that its decimal representation, for example, never repeats itself. Because 1 over 7, right, it's like 0.146 something, like has six digits that repeat over and over again. Yes.
00:02:43
Speaker
and 1 divided by 9 or like 15 divided by 99 is like 0.1111 or 0.151515, right? Yeah, so that one actually repeats, but a non-repeating decimal number is what we're talking about here.
00:02:56
Speaker
Yeah. And, uh, which means that you could write it down for, you could write it down until you have 30 hard drives full of digits. Eventually you'd run into a sequence that isn't anywhere on those hard drives if you just keep going. Yeah. Now there's irrationals that are pretty popular, including things like PI. One thing that was interesting in the research for this episode is that PI is a, what was the term? Transcendental number.
00:03:17
Speaker
It's the transcendental but also it's a universe creator something about oh, yeah It's uh, I mean some people have called it a universe factory. Um, there's an xkcd that I like where it's uh, it says like, um pi equals 3.1415926 and then it says help i'm trapped in the universe factory mine finite because the thing is about pi is because this is transcendental it has every uh
00:03:39
Speaker
decimal representation contained within it, meaning that in any way that you can write the universe down as a set of numbers is contained somewhere within pi. That is wild, that is absolutely wild. Including a universe where everybody is the same except for their British accents.
00:03:56
Speaker
Wow. Wow. That's the crazy thing about mathematics is the universality of it. Wow. Oh yeah. And just you run into these bizarre things, but square root of two is a GM is an algebraic number, meaning that it could be made with a compasses and a straight edge. Um, and therefore it is not transcendental. Yes. It is, uh, yeah, because, um, yeah, the, the algebraic numbers can actually be enumerated. Um, so you can have all the algebraic numbers in a row.
00:04:22
Speaker
Yes.
Proof by Contradiction Explained
00:04:23
Speaker
And as we said earlier, so this is a problems episode where we will be proving that the square root of two is irrational. And this is a, again, there's many different kinds of proofs here and we're doing an interesting one. I kind of like it. It involves evens and odds.
00:04:40
Speaker
Yeah. And the proof structure is this. It's an infinite descent proof, which is a type of proof by contradiction. So Gabriel, you've done proof by contradictions in the past, right? Do you want to say kind of what that is? Yes. A proof by contradiction is when you start off with an assumption with a truth claim about something, and then you go through the proof and you show that the proof couldn't be correct, or rather the conclusion of the proof is something we know to be incorrect. Therefore, the original assumption is incorrect.
00:05:07
Speaker
Yeah, and not only is it incorrect, it's perfectly incorrect, meaning in such a way that you have to assume it's opposite. So if me assuming, for example, that the listeners of the show are fish leads me to making a series of business decisions that are blunderous, like only promoting fish food on the show, things like that, then we have to make the assumption that my initial assumption was wrong. That was a little bit of a stretch.
00:05:34
Speaker
Again, even not necessarily, not to get knit picky here, but I see what you're trying to say about proof by a contradiction. We're going to assume to start with what we think is false and what we're going to show is false.
Infinite Descent Method
00:05:51
Speaker
That is that the square root of 2 is equal to p over q for some integers p and q. This might be a good time to talk about what infinite descent is. Infinite descent is a type of proof by contradiction where
00:06:03
Speaker
you assume that something is already in its most basic form. And you show that no matter what form it's in, there's a more basic form. And that means that you could just keep going on forever. And with integers, you can't go on forever, right? If you keep decreasing with integers, you're going to reach a point where you stop. Yes, yeah, exactly. And as you said earlier, we're starting off with the assumption that the square root of 2 can be shown or can be represented by two single integers, p and q. And they are relatively prime, right? Yes.
00:06:32
Speaker
Yes. And this is the definition of a rational number. So, you know, we know this is incorrect, but we're just going to say it's true for this proof. Yeah. Rational doesn't mean that these numbers like having talks about sound environmental policy or anything. It means that they're based on ratios, rational.
00:06:51
Speaker
Yes, yeah, yeah, exactly ratios. And again, as you said, infinite descent, that does sound almost fractal in a sense. Oh, yeah. And yeah, basically an infinite descent doesn't work if you have fractions and stuff that we're working with. But since we're working with integers, infinite descent works. It would also work with ordinals, for example, if you want to go back and listen to the ordinals episode.
00:07:10
Speaker
And we're just going to use a little bit of algebra in this proof, and we'll be done before you know it.
Detailed Proof of √2's Irrationality
00:07:15
Speaker
So if the square root of 2 equals p over q, what happens if we square both sides? In algebra, what you do to one side, you do to the other. So if you square the square root of 2, you get 2. And if you square the other side, which is p over q, you get p squared over q squared. So 2 equals p squared over q squared.
00:07:33
Speaker
And if you multiply both sides now by q squared to get to make sure we're we don't have q squared on the bottom, you get two q squared equals p squared. Brings me back to early middle school algebra. Getting rid of that denominator, two q squared equals p squared.
00:07:49
Speaker
Now, what does that say about p squared? It means that p squared is even because it can be divided by 2. Now, p squared is even probably means that p is even, right? That is correct. But we want to prove this because we're being, you know, as careful as we can because, you know, for all we know, q can be 1 and the square root of 2 can be some other integer based on what we have so far. I mean, we could actually probably get rid of that possibility by saying p equals, I mean, by showing that 1 is not the square root of 2. But let's say we don't. Yeah.
00:08:18
Speaker
Let's just go the route of, so if an integer is odd, we're gonna show that if an integer is odd, then its square is also odd. Yes, yes. You can think of any example in your head right now, but let's firmly prove it. The examples in your head, what, maybe say nine, we know that nine is odd. Yeah, and so 81 is also odd. Yes, and also not, and the square root of nine is three, which is also odd. You can't think of an odd number that has a square root in which the square root is even.
00:08:46
Speaker
Yeah. And what we're going to prove here is that specifically that if an integer is odd, then it's a square is odd. Because what we're going to do is we're going to look at kind of the opposite of that with evens. But first, let's go through this. Sure. So if an integer is odd, we could write it as 2 times some other integer plus 1, right? Correct. Yes. So for example, 7 is 2 times 3 plus 1. 123 is 2 times 61 plus 1. Yes. So we're going to say m equals 2n plus 1. OK. So what would m squared be?
00:09:15
Speaker
Then m squared would be, you'd have to square the other half of the equation. So then you'd have to square 2n plus 1. Quantity 2n plus 1 times quantity 2n plus 1. We know the FOIL method from algebra, and you'd get 4n squared plus 4n plus 1. And 4 is even, right? Yes. And an even number times an odd integer is an even number, right?
00:09:42
Speaker
Yeah, I'm sorry, an even number times an odd number is always even. And an even number times even number is even too, right? That is correct. So that means that 4n squared plus 4n is even. Yes. Meaning that 4n squared plus 4n plus 1, which is m squared, is odd. Yes, correct. So now we're going to talk about the contrapositive of something, because we're going to take the contrapositive of what we just proved, which is that if an integer is odd, its square is odd.
00:10:08
Speaker
So the contrapositive of I am a human, therefore I exist in the physical world. In the contrapositive of that would be I am an incorporeal being, therefore I'm not a human. Okay, there you go. So if we have our statement, an integer odd means that square is odd, what's the contrapositive of that statement? The contrapositive is that a square being even means that its square root is even.
00:10:33
Speaker
Yeah. And so we show that P squared is even, right? Which means that with the, via this lemma. So since P is even, we're going to say that P equals two K, right? Yes, exactly. So what would P squared be? Then P squared would just be four K squared for some value K. And we already said before that two Q squared is equal to P squared. And recall that P over Q was what we said the square root of two was. Yes. And we ended up proving that that means that two Q squared equals P squared was how we got to P squared mean even.
00:11:00
Speaker
And it's good to point out here that only that p squared and q squared cannot both be even, right? Correct. Because if they were, then p over q would not be at its most basic form. It'd mean that we could reduce it further. Yes, yeah, exactly. And since we can reduce it further, you have to keep reducing past one. Yes. And since there's no integers less than one, that's a contradiction and that's how infinite descent works. So now we have 2q squared equals 4k squared, right? Yes.
00:11:27
Speaker
and that we could divide both sides by 2 and then we get q squared equals 2k squared, which says what about q? It means that q must be even. But earlier we said it must be odd. Yeah, and we've shown by any number, I mean shown within a proof of eye conjugate within this and also just via infinite descent that there's no p integers p and q such that the square root of 2 is equal to p over q.
00:11:53
Speaker
Okay, excellent. Excellent. Yeah, yeah, there we go. There we go. So there's a proof by contradiction using infinite descent.
Alternative Proof Methods and Challenges
00:12:01
Speaker
These are some really cool buzzwords for you.
00:12:05
Speaker
I've noticed that all the proofs that I've seen for the Squirt of Two's irrationality, any prize by necessity because you're saying kind of what you can't do, that it's irrational are based on proof by contradiction and infinite descent. Another one that I like that I'll just talk about super quick is you take two squares, one that has twice the area of another.
00:12:27
Speaker
So then the ratio of the two sides of the squares would be the square root of two, right? Yes, correct. And we say that if they're integers, basically what you do is you prove that if they overlap, kind of like two napkins overlapping via the corner, that the overlap area would have to be twice, would have to equal the area of the two corner squares. And then you basically prove that this could be done infinitely, and then you get another proof by infinite descent.
00:12:57
Speaker
Yeah, I just thought there'd be a good proof to show y'all because it's something that we hear more or less kind of often in the math world that certain things are irrational. But I think it's good to show what it takes to prove that something is irrational. And a good thing to maybe prove at home for you too is that if the integer, I mean if the square root of something that is an integer is not an integer, then it must be irrational. And that might be something to prove at home.
00:13:23
Speaker
Are we now going to, you know, give our audience homework? Why not? Yes. And first one to solve this correctly, we'll get a poster if they're within the United States or the file of the poster if they're worldwide. And so just send your proof to breakingmouthpodcastatgmail.com and we'll read your proof on the show. Also I'd love for, I'd love it if anyone were to send us your favorite proofs, perhaps we'll talk about those on a future episode.
Conclusion and Listener Engagement
00:13:49
Speaker
So yeah, thank you for listening to Breaking Math and breakingmathpodcast.gmail.com or breakingmathpodcast.app is our website. And you can buy our poster at facebook.com slash breakingmathpodcast. Just click on shop until next time.