字幕表 動画を再生する 英語字幕をプリント 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.