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.
