Transcendental Numbers - Numberphile

SIMON PAMPENA: It's mind blowing. I learned this when I was at uni, the existence of transcendental numbers. And the name was a selling point. Because I was like, transcendental. You know, it's a time when you're really interested in, like, out of body experiences and whatnot. But the idea that mathematicians gave this name to numbers, numbers, these are numbers that you're familiar with. Like pi, you can write down as a decimal expansion. You'll never get it right, but it's just a number that you're familiar with. Has this property that we just didn't know about. OK, we're going to play a game, and we're going to try and understand transcendental numbers with this game. The game is reducing down numbers to 0. That's what you want to do. So the rules are you can only use whole numbers to do it, and you can add, take, multiply, and put the whole thing to any power you like, but it has to be a whole number power. OK, so let's play the game. OK, so do you have a favorite number? BRADY HARAN: Well, I like the number 10. SIMON PAMPENA: 10? BRADY HARAN: Yeah, but that seems like quite an easy one. SIMON PAMPENA: Sure. That's fine. You mean 10 in base 10? BRADY HARAN: 10 in base 10. SIMON PAMPENA: Yeah, OK. So 10 in base 10. OK, here we go. Let's start the game. So we want to get this down to 0, so the first thing we could do is multiply it by 0. But that you can do with any number, because any number times 0 is-- BRADY HARAN: 0. SIMON PAMPENA: Bingo. You can do that, but that's not very interesting. But what is interesting is that if we try and use these rules, we can go, OK, what happens if I take away 10 from this? We're done. So there you go. So that sounds kind of trivial, but it's a really good start. So we used a whole number, and we used the take. What about something else? How about 3/4? First of all, let's multiply it by 4. OK, so these things will cancel. You get 3. Now we can take away 3, we'll get 0. Excellent. But what about something crazier? What about like a really crazy number? What about like, the square root of 2? I think you guys know about the square of 2. BRADY HARAN: Yes, we do. That's irrational, isn't it? SIMON PAMPENA: It's an irrational number, and irrational means it can't be expressed as a fraction. So the square root of 2 is kind of a very strange number, and so this little thing here, I often say this little thing here is like a little sentence. It says, what number multiplied by itself gives you this number? That's the way I think of the square root sign. So I don't know what number multiplied by itself gives me 2, but that doesn't matter. Now, what we'll do is to try and get this one down to 0, OK? First of all, we'll have to-- BRADY HARAN: OK, that, I reckon I can do that. SIMON PAMPENA: Well, tell me. BRADY HARAN: I reckon if we raise that to a power-- SIMON PAMPENA: Yep. What power? BRADY HARAN: Let's raise it to the power of 2? SIMON PAMPENA: Correct, so that's multiplying it by itself. And then what do you get in the middle? BRADY HARAN: You're going to get 2, I'd bet. SIMON PAMPENA: That's right. So now you've got 2 in there, so what are you going to do? BRADY HARAN: Subtract 2. SIMON PAMPENA: Yes! So look at that. So you've just taken an irrational number, and with this game you've brought it down to 0. How about the square root of negative 1? We've gone from numbers that you know and love to fractions, OK, to irrational. This is irrational numbers. Now we've gone into what they call complex, or some people call imaginary, which is a terrible name for it. OK, so what can you do to this one here to try and get it down to 0? BRADY HARAN: Well, I'm just going to square it and add 1. SIMON PAMPENA: There you go. No flies on you, mate. So there you go. So we've been able to play this game with three or four very different types of numbers, quite special. But what about something else? What about the square root of 2 plus the square root of 3? What can you do with that? So, let's see. The square root of 2 plus the square root of 3. Now we're gonna square it. OK, so this is a little bit of high school maths. 2 plus 2 times this by this, which is 2 the square root of 2 times square root of 3 plus this squared. So that's 3. So this reduces down to 5 plus 2 by the square root of 2 by the square root of 3. OK, so this is what we've done. We've done that there, but look what's popped out. A number that we can use, a whole number. So what we'll do is on this side, we'll go 5 plus 2 by the square root of 2 by the square root of 3, and now we'll take away 5. So we'll end up getting 2 by the square root of 2 by the square root of 3. And this is good, because there's no plus sign in the middle. What can we do next? Well, we're going to square all of that. So 2 squared is 4, and the square root of 2 squared is 2. And the square root of 3 squared is 3. OK, so that dot is another way of saying times. And so that one is 2/8, two 4's are eight, eight 3's 24, done. So if we go 24, take 24, boom, we get down to 0. What I wanted to show you, the reason why I wanted to show you this is because all these different numbers look very complicated, unrelated, but let me show you. Now, let's replace all the numbers we put in with x. x take 10 is 0, 4x take 3 is 0, x squared take 2 is 0, x squared plus 1 is 0, and this one is x squared take 5, all squared, take 24 equals 0, which if we expand out, so look. These all look like algebra problems. So what we did was in our game, we picked numbers, and we tried to get them to 0. But the opposite could have been here, let's solve for x. Now, this is the stuff that you get taught in school. This is algebra, and it so happens that the family that all these numbers belong to, even the square root of negative 1, is algebraic numbers. So we've actually found a home for some of the biggest stars of maths, the numbers that have caused huge problems and schisms, what is the square root of negative 1? Square root of 2 from the ancient times, the Pythagorean times. People died because of this number. But somehow we've found a family for these numbers, algebraic numbers. OK. So next, we're going to need another sheet of paper. We've chosen some numbers. What about a special number? What about e? Now this number here-- if you're not familiar with it-- this number is a fantastic number for maths. And what it is is that if it's a function, a function of e to the x, e to any number that you raise it to, OK? On the graph, when you graph it like so, the y value is also at the slope of the tangent at that point. So it's really, really important to natural growth. It's like a really fantastic number. It means a lot to life, really, but it's actually a super crazy number. Super, super crazy. One of the expressions I can show you for it is actually an infinite sum. So I'm going to blow you away. It's 1 plus 1 on 1 plus 1 on 2 plus 1 on 6 plus 1 on 20-- anyway, it keeps going forever and ever. But can we play the game with this number? Can we bring this number down to 0 using the rules of our game? BRADY HARAN: Can we do it with algebra? SIMON PAMPENA: Can we do it with algebra? That's right. BRADY HARAN: All right. Can we? SIMON PAMPENA: Well, for ages and ages and ages, e's been around for about 400 years. Nobody really knew. I mean, this number is really, really important, and no one knew. It so happens, we can't. BRADY HARAN: It can't be done. SIMON PAMPENA: It can't be done, and I'll show you why. Well, I'll kind of try and show you why, because it's actually really tricky. But it was a guy called Charles Hermite, and he basically showed-- right, so I'm going to use these symbols here, because I don't know what the formula will be. He basically showed if you try and play the game, right, bringing e to any power that you want, whole power and timesing it by any whole number. So if you claim that there does exist some bit of algebra that you can bring it down to 0, he showed that you'll lead to a contradiction. Basically, he showed that there was a number, a whole number that existed between 0 and 1. Obviously, there's not. Obviously, there's not. But this is what you do in maths, is that if you want to show you something is impossible, you kind of assume that it's true, and then you show that it creates a contradiction. So this is amazing. So this is what Hermite discovered, and this is really, really a fantastic-- I mean, everyone should be excited by this, because e is not algebraic. So what number is it? Well, it somehow transcends what we're capable of doing. The thing with algebra is that's how we build numbers. Like, that's our world is built with algebra. Like, any number that you kind of deal with in your everyday life has a lot to do with algebra. You're just adding, taking, dividing, things to the power, but e is not. So it somehow transcends maths. So that's what they called it. e is transcendental. It's actually [INAUDIBLE] show you other than e. You know why, is because-- well, this is the interesting thing. e wasn't the first transcendental number. They discovered a transcendental number, Liouville, I think his name is, discovered a transcendental number quite a long way before this, 30 years before this. But it was like, through construction. So he was actually trying to find a number based on the rules of the game that didn't fit. What's special is that e was already, it's already a superstar of maths, e. Like, people knew about it. So this was an extra piece of information. But then people asked this question. What about pi? BRADY HARAN: Superstar. SIMON PAMPENA: The superstar. This is the superstar of math. 2,000 years old. What is pi? Is pi algebraic or transcendental? So you've got to imagine as a mathematician, OK, you love pi. Like, it comes with the territory. You cannot not like pi. So this is the thing, is that you could actually add to the knowledge of pi. You could add something new, which is incredible. I mean, I would die a happy man if I could do that. So this question came up, what is pi? Is it algebraic or transcendental? And so it was about, probably 1880s that a guy called Lindemann actually came up with the answer. He showed, and again, this is a very tricky thing that he showed, he showed e raised to any algebraic number is transcendental. So for example, e to the 1, e. That's a good thing, because e should be transcendental, because it's already been proven. Because 1 is algebraic. Your favorite number, Brady, e to the 10. That's transcendental, right? e to the square root of 2, e to the i. Right? What about pi? So how could you use this fact here, e to the a, so a is any algebraic number, is transcendental? How can you use that fact to show that pi is transcendental? OK, so this is the thing. Again, it's a proof by contradiction. So, this is what he did. He said, let's assume pi is algebraic. So pi is algebraic. That means there's a formula for it. OK, what's that formula? Who knows? Because it doesn't exist. But as an example, if you're an engineer, you'd say, oh, yeah, pi, 22 on 7. All right? OK, cool. So that means pi times 7 take 22 equals 0. Right? As an example. That's not true, by the way. There's no way I'm claiming that to be true. Don't you dare cut it and say Simon thinks that's true. It's not true. Pi, 22 on 7. Pi, 22 on 7. I know pi to quite a few decimal places, and that's obviously not true. And an actual fact, just so I can tell you, another really nice approximation of pi is the cube root of 31. It's actually pretty close. So that could be another formula. So that means if we cube that and take away 31, that equals 0. OK, so we've got like these phony equations. This is the big kicker. This is the big kicker. We're going to use another superstar equation, OK? e to the i pi equals negative 1. So this is Euler's identity. It's a famous one, isn't it? But look at it. Look what it says. e raised to the i pi is negative 1. Now, i pi, OK, if we assume pi is algebraic, that means i pi must be algebraic. So e to an algebraic number has to be transcendental. But is negative 1 transcendental? It's not, because we can play the game, and we can get it to 0. So by using another increase piece of maths in your formula, imagine this is like you're making a film, like you're doing a maths film, and you've just got the biggest Hollywood star in the world to start in it. In your proof. Starring in your proof. So this here, e to the i pi is negative 1. If indeed this was algebraic, this would have to be transcendental, so that means i pi cannot be algebraic. And who's the culprit? Well, it's not the square root of negative 1. It's pi. So pi cannot be algebraic, which means pi must be transcendental. So there's something really tricky going on, and that's why I like it. Because the tricky stuff is where all the awesome maths is. In maths, perfection is important. But then, anyone who uses maths-- for physics or chemistry, or whatever you want to do-- then they can kind of use approximations. I'm not interested in approximations.