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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--

• 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.

• 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?

• 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

• 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?

• 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.

• 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

• 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

• 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.

• superstar of maths, e.

• Like, people knew about it.

• So this was an extra piece of information.

• But then people asked this question.