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