We're gonna do all the numbers. On Numberphile
we've done a lot of the numbers, some would say, but we haven't done all the numbers. Originally
we did the whole numbers. And these are the classics, my goodness. We've done 11, that was an early one.
We've done 3435, seventeen - right, all the whole numbers sit in here.
But then there are other types of numbers. If we go one step out, the rational numbers the ones that are ratios,
these are - you know, you get a seventeenth, you get I don't know things over twelve, you get all sorts of...
So now you've got all the rational numbers. We've gone beyond that though. And the rationals technically include the whole numbers,
they're a subset, but I'm doing this as what people call Venn diagram, which is wrong,
it's an Euler diagram. Because I'm not showing every single possible combination. - (Brady: Are negative numbers whole numbers?)
You know, I
put in a twelfth because I thought I was being hilarious and then I immediately thought, urgh I wasn't gonna put negatives on this diagram.
So I regret that, for two reasons. Opening the negative can and the expression on your face.
So I'm gonna make that a plus, there we are. In fact, this whole sheet of paper is just gonna be the
reals.
Positive re- you know, it works for negative reals, what am I saying? Have the negatives, it's fine.
But the sheet is - are all the reals and
inside here I've put whole numbers and then I've put rational numbers. If you - you can obviously get complex numbers coming up,
we're not gonna do that,
we're gonna stay down here. And I'm gonna gradually work our way out until we get a greater distance out
than Numberphile has ever gone before, right? We're going for the new Numberphile record: how far out into the weird reals
can we go? But we've done rational. Next one up the constructables. And these often aren't mentioned,
you don't have to add this in as a category, but I quite like constructables.
More importantly what people tend to go for, the next one out are the algebraic. Okay
so Simon did a fantastic video about algebraic numbers, and when you go outside,
transcendental numbers.
"I mean this number is really, really important and no one knew." - And so a lot of the number categories refer to in or out
of these different sets. So rational numbers are everything inside the blue line,
irrational numbers are everything outside the blue line.
You've got constructable numbers are anything inside the purple line, unconstructable numbers are outside this. And this light blue line out here:
algebraic numbers are everything inside there, and
transcendental numbers are everything
outside of there. And so constructable numbers are things that you can construct with a pencil and a compass and a ruler.
So Phi, you can do that, the golden ratio because you can do root 5, so you can get Phi.
You can do root 2, that lives in here,
that's kind of fun.
Algebraic numbers are the solution to an algebraic equation. If it's a square root or lower you can put it in constructable,
you can draw it. If it's higher than that
you can't. So the cube root of 2 is an algebraic number. And then outside algebraic you've got...beyond that, right?
So you've got things like pi. Pi lives out here, pi is transcendental. e is
transcendental. The natural log of 2, that's out there. Things that aren't a nice, neat solution to an algebraic equation.
And they're out there, right? This is how people tend to categorise all the numbers.
This is the fringe of, kind of, what we understand in mathematics and what we've done on Numberphile.
So e was the first number that was proven to be transcendental. And so that was proven in
1873 - so reasonably recent, given that some of these are thousands of years old. We knew e existed but we didn't know where it went.
Pi, we didn't know where that went. Pi was proven to be out here in
1882. So e to the power of pi was proven to be out here in only
1934, so that was a more recent one we managed to prove is out there. And there are loads
which we don't know. Pi to the e: we don't know. e to the e: we don't know. Pi to the pi:
we don't know, right? These are all on the cusp. We know that one of e times pi or
e
plus pi; one or more of those are transcendental,
we don't know which. Or both, right? But we know at least one of them is. Most numbers
that we know are algebraic sit around here, and we don't know if they're definitely
transcendental or if they're still algebraic. For the most part
we haven't got a clue, right. If you look at the entry for transcendental numbers on Wikipedia or MathWorld,
there's a list. Here's the only ones we know and that's it,
right. Some of our favourite big ol' numbers: Graham's number - in here. Googolplex - in here, right.
Whole numbers, doesn't matter how big it is, it's in there.
There are infinitely many whole numbers but there are also infinitely many rational numbers, same infinity.
There are infinitely many constructable numbers, infinitely many algebraic numbers;
but all countable infinity, is the smallest infinity possib-ly many of
these numbers in here. And there is one more circuit out. You know what? Should we chuck on the last one?
I can add another loop and it cleans up all of these and it puts them all in a neat bow.
This is the collection of
computable numbers.
Computable numbers? What's that- well this means we can compute them. So we can compute e,
we can compute pi, right?
It's not the solution to a nice equation, but I can write down a system by which you will get the decimal places,
right? And so we do this, we print them out on a very long bit of paper, roll 'em out on a runway,
it's hilarious, right?
"We're here on a runway because, for some reason, Brady has printed out the first 1 million digits of pi."
So this came down to Alan Turing in
1936. And everyone remembers that Turing invented the computer, with the Turing machine and that was in his paper on computable numbers.
He was looking at if you can compute all numbers, and he showed there are numbers that exist out here
but we just don't know what they are. I call them the dark numbers,
right, all these numbers that we know they exist - Turing showed us,
but they're, they're so hard to grasp. And occasionally we see them but not often.
Well hang on, why isn't pi out here? We can compute pi. Well, it takes forever,
like I could write down a
infinite series which gives you pi. And I can give you the rules for writing out the infinite series, and I could write them on
a postcard, or some finite amount of space and go: here are the rules for calculating pi.
You're going to have to do them forever, but the rules are finite. So for everything else in here
I can write a description of how to get all the digits, out here
they're only definable by writing out all the digits. And no - and that's actually most numbers. In here,
this is the countable infinity land right? There's, there's, there's infinitely many of these, but the smallest
infinitely many. Out here is a bigger infinitely many. So the vast majority,
for the strongest definition of vast majority you can come up with of numbers, are not computable. So we in this nice little
island of numbers that make sense, and then outside is this vast, vast world of all the reals
which are only definable by writing out their digits. And we've spotted a few! So there's one called the Chai-lin?
constant? I'll have to double check I got that right. Chaitin. Vaguely speaking, it's the probability,
for a certain way of writing a computer program,
if you generate a computer program at random, if it will run and come to a stop.
Right, and that probability is a naive way of describing it, and it depends on how you write a program.
So, in fact, there are lots of these constants, but we know they're all out here. The only way to get them is to
work out every single digit individually, there's no equation, no algorithm that spits it out. It's an uncomputable number. And they're so
mysterious and hard to understand at fringes of our comprehension of numbers,
but they're out there and the scary thing is most numbers are out there.
(Brady: How can we even know one of them? It seem- it feels like an unknowable unknown.)
It is insane that we even know a couple of them because when I described the numbers out here,
I'm super hand-wavy because I don't understand them, right. I've read - I like I've tried. I read the paper,
I'm like, man,
this is beyond me. Like, the maths to try and grapple with the numbers out here is insane.
But what's incredible is we have done it, right?
So if you look up uncomputable numbers, there are a few examples of ones out here. Although,
interestingly,
when you get these weird numbers you can, you can just - you can kind of make artificial ones.
Okay, so I'm now gonna kind of ruin my lovely neat diagram by putting on a whole new category.
This is the category of what are called normal numbers, which is a bit of a silly name.
It just means that every possible
sub-grouping of digits is equally likely to be in there. And a lot of people say like, for example pi. Everyone goes: your phone
number's somewhere in pi, your name is somewhere in pi, the complete works of William Shakespeare if tuned into digits are somewhere in pi.
We don't know that. We could move pi into here- - (Brady: Which we may yet do?)
We may yet do! It may be in here, everyone seems to think it is. All the digits we've checked imply
it's in here, but we not yet managed to prove that. We don't know if pi is in here.
We don't know if e is in here,
we don't know if root 2's in here. Even though for all of these, if you look through the digits,
you can find any string of digits you want. I found my name in all of them,
right, because they've all got a lot of digits that are
suitably random, and you can find sub strings in there. But we've not managed to prove any of those are normal numbers.
Would you like to see one number we have managed to prove? So this, I love this number. It's called Champernowne's constant.
Is normal. It's one of the few numbers we know is normal.
(Brady: Con-stat.) - Consta- that's an n! Look at it, it's just climbing under the a. - (Ah yeah)
And so Champernowne's constant is one of the few numbers we know is normal, and it goes zero point one two
Three - I've memorised it - four five six seven eight nine,
well, 10, one zero, 11, 12 - and it's just all
the whole numbers. 14,
15, 16 and so on. - (Brady: That's cheati-)
one eight, one nine, two zero
I've got them all,
and so on.
(Brady: That's like brute-forcing the problem)
It really is! It really is. I call this an artificial number because
Champernowne just went: can I find a number which is normal? And he came up with this procedure.
He just went, right you just put all the numbers in order, all the integers in base 10,
and you done. And it's true,
right? Because whatever you want to find - it's in there eventually, because it's just all the whole numbers listed out and then
turned into digits.
But that's all numbers are right? Whole bunch of digits, perfectly valid number, right? But - and it is computable, right.
So actually - - (Brady: It's like the least efficient way of doing it.) - It is. There's a slightly more efficient one. The Copeland-Erdős
number is the same idea but only the primes. So it's 2, 3,
5, 7, 1 1, 1 3, so on - 1 7 etc. So exactly the same idea. Slightly more efficient,
and it's also normal. So we didn't start with these numbers and go
I wonder if they're normal? These people sat down and went: I'm gonna make a normal number. How can I do that?
And they generated these. Now, Champernowne's number is transcendental. So this is it in base 10, obviously
you can do it in different bases, and it is computable. It is normal and it is transcendental.
So transcendental means it's outside algebraic, it's there.
And not the speed of light - Champernowne's constant sits in this part of the diagram.
So the only normal numbers we know for certain are artificial ones that are being constructed for that purpose.
We have never started with a number and then discovered it is normal,
we have no test. There is no process for taking a number and proving
it's normal. Like, mathematics - like hopefully one day we'll have a test,
currently we don't. Which means all the normal numbers we have are artificially generated,
we are yet to take anything from here and show that it's allowed to move over this line.
(Brady: There's an elephant on your diagram there.) - Is it over here? - (Yeah) - Well, very interesting
you should say that, Brady. This section is empty. And this is the only
properly empty section of the
diagram. Now up here we had that one number whose name I couldn't remember properly- - Chaitin.
All - this set of numbers deal with whether or not a program will halt, we've got numbers in this category, right?
So this section we've got numbers. This is
completely empty. And that's because the only normal numbers we know of are ones that we made for that purpose.
And the fact that we made them for that purpose means we have a rule for
generating them, which means they must be computable. To have an uncomputable
normal number would be incredible. But this is currently empty, but we have managed to prove one thing.
We've managed to prove that most numbers are normal, and most numbers are uncomputable.
So actually this is the biggest section. This is numbers, right?
This is a trivial blip, right, in the world, not- this is where numbers are right?
And we had none! So in the main category of numbers, where all the numbers are,
apart from a few trivial side-effects, right? We know zero of them. You know, as mathematicians
we think we're getting somewhere, but up until now we have found none of the numbers.
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Chaitin.