the beauty of root 2

本站添加: 2026-05-11

[00:00:01]Square root of two has a lot of interesting properties that I want to talk about. So, you all remember hopefully from high school the right angle triangle the side lengths of A B and C.

[00:00:21]And there's this rule called the Pythagorean theorem that says that A squared plus B squared has to equal C squared.

[00:00:30]Make sense? So, what if A is one and B is one?

[00:00:34]We're going to have A squared which is one plus B squared which is just going to be one equals C squared meaning that C squared is going to be equal to two.

[00:00:43]So, C will be equal to the square root of two.

[00:00:48]So, that's going to be the length.

[00:00:50]So, in real life if you wanted to intuit what root two length means, it'd be very very easy. Just take two things with an arbitrary length that are the same length and hold them at right angles to each other and then just measure from here to here, right? It's going to give you root two.

[00:01:06]Now, the square root of two has a lot of interesting properties that I want to talk about. And the main one is irrationality.

[00:01:15]Root two is an irrational number. Now, what do I mean by irrational? Well, a rational number is a number that can be expressed as a ratio. So, rational comes from the word ratio.

[00:01:31]Namely a ratio of integers.

[00:01:33]So, all of the classic numbers, you know, one, two, three, four, five, these are all rational. A half is rational, one over two, a third, a fourth, a fifth, two fifths, three fifths, whatever. If you can express it as a ratio of integers which are whole numbers, then the number has to be rational.

[00:01:50]So, what what is an irrational number?

[00:01:53]Well, a number is a irrational.

[00:01:58]Let's say that P is irrational if you can't express it in the form A over B.

[00:02:12]Formally, we would say that it is not true that there exists some pair of integers A and B where P can be expressed in the form A over B. It doesn't really matter. Um I'm not We're not too into the formal stuff here, so can't be expressed as a ratio of integers. Now, we're going to prove today that the square root of two is in fact irrational, that it can't be expressed in a form like this. There's no two numbers A and B that I can put on top of it one another and say that that equals the square root of two.

[00:02:54]Now, you might be thinking, "Well, that sounds quite hard to prove." Well, I'm going to try.

[00:02:59]We're going to use a really powerful method in proving which I'll probably going to talk about later on this channel called a proof by contradiction.

[00:03:08]I'm just going to write that down. Proof by contra diction.

[00:03:16]Now, a proof by contradiction proves that something is false by assuming it's true and then looking for a contradiction in that truth.

[00:03:25]So, typically, it's a lot easier to prove that something's true than that something's false. And you can think about this in real life, you know, if if someone had a pair of scissors in the room and someone asked me to prove that there are scissors in the room, it's quite straightforward. I just have to go and find the scissors. Okay, here you go, here are the scissors, they're in the room. I've proven they're in the room.

[00:03:45]Now, if you wanted to tell me to prove that they're not in the room, it's a lot harder, right? I can't just open the drawer and say that there's no scissors in the drawer cuz they could be in any drawer. They could be taped to the roof.

[00:03:57]In a sense, I have to show them every area in the room the scissors can be and show that there are no scissors in all the area. You can see how these are two different categories of proofs.

[00:04:07]A proof by contradiction would be something like to assume the thing that you that you know is false and look for how that breaks the universe.

[00:04:18]Um I'm probably not explaining this very well, but we're going to get into a math example.

[00:04:23]We're going to assume for the sake of argument that two is rational.

[00:04:30]Which means there exists Sorry, wrong symbol.

[00:04:34]There exists A and B which are both integers such that the square root of two can be expressed in the form A over B.

[00:04:49]That's the formula I wrote right here.

[00:04:50]All we really care about is is this part here. That's what we're going to be using. Two equals A over B.

[00:04:55]Now what do we know about A over B? Well they are both going to be integers.

[00:05:02]We have that written here.

[00:05:04]And are there any other properties of A over B or of A and B that we are interested in?

[00:05:08]Well, yes, there's one.

[00:05:11]A and B are what's known as co-prime.

[00:05:16]Now, hopefully everyone knows what a prime number is. It's a number that can't be divided by any number besides one and itself to yield another number. These are all in the family of integers, obviously.

[00:05:28]I should say positive integers.

[00:05:31]Um basically what it means is that A and B, their greatest common denominator GCD of A and B equals one.

[00:05:43]Basically, they don't share any factors besides one.

[00:05:47]Um or one of their own or one of themselves. So, let's say for example, that means that they don't both share a factor of two.

[00:05:56]Because if they both shared a factor of two, for example, 14/8, we could simplify the fraction further and we could say seven over four.

[00:06:07]So, what I'm getting at is that this fraction is going to be in its simplest form.

[00:06:14]And if a fraction isn't in its simplest form, you can keep dividing the two numbers by common factors to make it smaller and smaller. So, eventually, there's going to be a form for this fraction where the two numbers are coprime and the greatest common denominator is just [clears throat] one. So, we are at this We're going to say for the sake of argument that if root two is a is rational, can be expressed as a over b, where a and b are both coprime.

[00:06:39]Cool. And that's all we're going to really need for now. Now, I'm just going to do some algebraic stuff to this idea here. We'll start start over here.

[00:06:47]So, I'm going to square both sides to start off with. This is a pretty common intuition some of might have.

[00:06:56]Cool. And let's times both sides by b squared.

[00:07:01]Now, I end up with 2 * b squared equals a squared.

[00:07:07]Um now, b squared is going to be an integer, which means that a squared can be expressed as two times something.

[00:07:12]Now, if something can be expressed as two times something, that means that it has to be even. So, I'm going to say a squared is even.

[00:07:23]Now, what I'm trying to prove here is that a itself has to be even also.

[00:07:27]And we might have to sort of prove this separately, but the idea here is that if a is odd, a squared is going to be odd.

[00:07:33]And if a is even, then a squared is going to be even. So, we can say then that if a squared is even, then a has to be even also.

[00:07:40]Um we won't just take that for granted, so we're going to we're going to prove it here.

[00:07:44]Let's consider the two kinds of numbers, even numbers, 2k, and odd numbers, which can be written as 2k + 1.

[00:07:54]Hopefully, you can appreciate that every number falls under one of these two categories, an even or an odd.

[00:07:59]So, if I square an even number, basically, if I square 2k, I'm going to end up with 4k squared, which has a factor of two in it, because, you know, four has a factor of two in it, so this is even.

[00:08:14]If I square 2k + 1, I'm going to end up with 4k squared plus 4k plus one, which is the same thing as saying two times a bunch of stuff plus one, which is an odd number.

[00:08:31]So, I've shown that if I take an even number and square I get an even number.

[00:08:35]If I take an odd number and square, I get an odd number. Those are only two things that can happen.

[00:08:40]Therefore, if a squared is even, we can say that therefore a is even.

[00:08:47]Okay, so a has to be even. That's fine.

[00:08:50]Let's keep going.

[00:08:52]So, if a is even, I can express it in this form.

[00:08:56]I'm going to write it as two times q, where where q is some integer.

[00:09:08]Because it's even, I can write it this way. So, so I'm going to plug that in to my most recent equation to give me that 2b squared equals 2q squared.

[00:09:19]Now, expanding this out, we're going to square we're going to square this and then divide it by two, so it's going to end up with it's going to be 4q squared, half of that is 2q squared.

[00:09:31]Cool.

[00:09:32]Which means that b squared can now be expressed as two times something, cuz again, this is just going to be an integer. I'm sorry, but you've hit the time limit for this conversation. Please start a new chat to keep the conversation going. Oh my god, that scared me.

[00:09:49]ChatGPT is talking to me.

[00:09:52]That was from the other video.

[00:09:55]Oh, that scared me.

[00:09:57]Sorry, guys.

[00:09:59]Let's keep going.

[00:10:01]All right, so we've established that now b squared is even.

[00:10:07]And by the same logic I've just laid out before, if b squared is even, b is even.

[00:10:14]So, we've ended up establishing that both b is even and a is even.

[00:10:24]If two numbers are both even, they share a common factor of two.

[00:10:29]But, the largest common factor of a and b has to be one.

[00:10:35]And you can see here there's the idea of contradiction.

[00:10:38]We've now found a contradiction.

[00:10:41]So, to put it simply, the square root of two has to be irrational because if it was rational, you would have a complication where two greatest numbers have a common common greatest common divisor of both one and something higher than one.

[00:10:57]So, I'm just going to rub all this out.

[00:10:58]Hopefully, this is pretty easy to follow. But, now I can say with certainty something we know about the square root of two is that square root of two is irrational.

[00:11:15]Now, there are other numbers that are irrational, namely pi, e.

[00:11:23]These are the big ones.

[00:11:24]We can prove these two. They're harder to prove, so I'm not going to prove those.

[00:11:28]Now, a mathematician a very a very natural question a mathematician might have is well, what about square root of three?

[00:11:35]What about square root of four?

[00:11:37]Keep going.

[00:11:38]What about square root of What about square root of n?

[00:11:41]Right? For all let's say natural numbers n.

[00:11:50]Are these all going to be irrational?

[00:11:52]Well, obviously not, right? Because the square root of four is two.

[00:11:56]So, this is an irrational. It's two.

[00:11:58]Two's fine.

[00:11:59]Same will be true for all the square root of nine.

[00:12:02]Square root of 16, right? If if you're squaring a square number like all these ones, it's going to give you just the number itself. This gives you three.

[00:12:11]This gives you four. This gives you five.

[00:12:13]But, using a very similar argument we can show that if the number you're squaring isn't square it will give you an irrational number.

[00:12:24]And have a think about why that proof that I just gave works generally unless the number is squared. I'll leave that as an exercise to the viewers.

[00:12:32]But, our basic idea here is that every number the square root of any number is either going to be an integer like one, two, three, four, five, etc. or an irrational number.

[00:12:49]You're never going to get a rational integer like three quarters, something like this. A number that isn't whole, but it isn't irrational.

[00:13:00]Cool. And that's the first interesting thing I have to say about square root of two.

[00:13:04]And there's one more while we're talking about irrational numbers.

[00:13:08]I want to show you another proof that also requires the square root of two.

[00:13:12]And that is the proof that an irrational raised to the power of an irrational can be rational.

[00:13:36]Now, typically when people or math students see a question like this, their first instinct is going to be to do what's called a proof by construction.

[00:13:46]Which basically just means I want to show that a claim is true by giving an example of it being true.

[00:13:52]This claim is not very demanding. It's just saying that there exists one example of two irrational numbers where we raise them to each other's powers, you get a rational number.

[00:14:02]Um you know, it's similar to my analogy saying this is the scissors in the room.

[00:14:06]You find the scissors, you've proven they're in there.

[00:14:08]So, a contradiction wouldn't really work here cuz we're actually trying to prove something's true.

[00:14:15]Now, here's a really elegant argument that I have found. So, let's say an irrational to the power of irrational can be rational. Well, we know that the root two is irrational, right? We just proved that.

[00:14:28]So, for an example, I could think about root two to the power of root two.

[00:14:35]What's this?

[00:14:37]Well, if this is rational, we're done.

[00:14:42]We've proven it.

[00:14:45]If it's not rational, we've got more work to do. So, we have two cases here basically, right?

[00:14:49]And I'm going to demonstrate to you that in both cases, we can find a solution.

[00:14:56]So, case one is the easy case. In case one, root two to the power of root two is a member of our rational numbers.

[00:15:07]Q just means rational numbers.

[00:15:09]If that's If that's the case, then we have already proven our claim.

[00:15:16]Right? This is what you want to happen.

[00:15:17]This is This would This would be nice if this was true.

[00:15:23]But, and what I fear is going to be the case, let's say that the square root of two to the square root of two is it in the family of rational numbers.

[00:15:38]Well, that means that this is a rational itself.

[00:15:43]So, now I'm going to consider the square root of two to the power of the square root of two to the power square root of two.

[00:15:52]Now, in this second case, which I should be writing here as case two, this is another example of an irrational number to the power of an irrational number.

[00:16:03]Now, this we can solve.

[00:16:04]I can solve it quite easily.

[00:16:08]Two square root of two to the power square root of two to the power of two to the power square root of two.

[00:16:13]Let me just quickly make sure I'm doing this correctly.

[00:16:34]Yeah, we have our simple power rule where this is going to be the same thing as saying this.

[00:16:45]And if you have a power to another power, like A to the power B to the power C, then this way, we can just multiply these two powers together. This is This is going to end up being root two to the power of root two times root two, which is two.

[00:17:00]Which is two. Like this. Two is rational.

[00:17:06]So, in this case here, we have also proven it.

[00:17:12]This is what I really love about this proof, or or one thing I really love about this proof is the fact that it's not a proof by construction.

[00:17:22]We've shown that there are two irrational numbers such that when I raise one to the power of the other one I get a rational number.

[00:17:30]But, we haven't made any specific claim about what those irrational numbers have to be. I haven't found a specific example of numbers that fit my rule, but I have proven that no matter what happens, my rule is fit.

[00:17:44]Whether it's this example or this example, it doesn't actually matter.

[00:17:50]I've just shown that one of them has to be true.

[00:17:53]So, in either way, my rule has been proven.

[00:17:58]Yeah, and I think that's all I have to say about the square root of two. But, it's a cool number.

[00:18:02]Um and yeah, get excited for more math videos.

[00:18:06]Cheers, guys.