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  • 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?