One of the things I find most fascinating about maths is the fact that it is made up of ideas.

It's just this bunch of ideas and concepts and structures and things that kind of the way I see it just it's our floating out there in the universe, waiting for someone to come and discover them and play with them and do something interesting with on what this means.

Is that the same mathematical idea?

The same piece of maths could be picked up and played with by many different mathematicians all over the world and throughout time as well.

An example of this is this piece of matter here, which is from China in 13 03 and you can see there are numbers in there.

If I show you what those numbers are, you can see this is a triangle of numbers that are arranged with a pattern.

So each number is the sum of the two numbers above it.

You've got four is three plus one and six is three plus three.

And you might see this.

You know, you might have seen this before, potentially at school But you may not know this as the young wee triangle you might think of this is being called Pascal's triangle on.

That's because that's what it's called.

But Pascal wasn't born until the 17th century, so this triangle actually existed several 100 years before Pascal and in fact, even existed before that.

This is the exact same structure from three year 900 on.

In fact, this kind of thing happens all the time.

Mathematical ideas get kind of picked up by different people in different places.

And what I'd like to do is tell you the story of one particular piece of maths that goes from a very long time in the past, all the way through to the modern day.

Andi is based on a simple idea.

So here I have some cans and I could use these cans if I wanted to kind of create a display for my supermarket, where I sell cans to create a nice little display like this little kind of triangular shape like that on dhe.

A reasonable question you might ask yourself if you're building this kind of shape is how many cans do I need to build different sizes of stack.

You know, if I've got four rows here, I've got one plus two plus three plus four cans in general.

What do these numbers didn't What's the pattern in these numbers?

And you might reasonably see that this is, you know, one plus two plus three plus four.

And you could calculate this way.

So if I've got, you know, different numbers of cans in my pyramid, this is how I could calculate them.

But you can also see that if I take the one can from the top row and add it to the four cans from the bottom row, I get five.

And if I take the two and add it to the three, I also get five.

And this is not a coincidence.

This happens with any size of stack that you use.

If you've got numbers the increased like this, you take the top one and add it to the bottom one.

You get the same thing as if you add the seconds atop one to the seconds about one because one of them's gone up from the other one's gone down, so you get the same total, so you can do this all the way down to the middle, and it means that you compare up all the numbers.

So you have pairs of numbers, each of which adds up to one more than the biggest number, and you have the number of pairs is half the number of layers in your stack.

So in this case, I've got four layers.

So I've got four divided by two pairs.

That's too.

I've got four plus one is five in each pair, two times five is 10 camps.

And I knew that because these are my cans.

But if I had a stack that was, say, 20 cans high, I would have 20 plus 1 21 cans in each pair on 20 divided by two is 10 pairs of can't.

So I'd have 210 cans all together, and that means I've got a patent for this.

I know that the number of cans divided by two times the number of cans plus one will always give me the total number of cats.

And this is the way that you see this formula often written.

And if I wanted to do this for something more interesting, so not just a flat triangle of cans.

If I wanted to make kind of a three d pyramid, I could do this is well, so I have one count on the top underneath that I have four cans arranged in a square underneath that I have nine cans arranged in a square on so on and so forth.

And there is a general pattern for this as well.

If I want to know how many cans or use altogether, it's this, and it's a bit more complicated than the 1st 1 but it still gives you a general rule on if I wanted to go on further than this because, of course, mathematicians always do.

I could have one count on top.

I don't know why I would do this, but I could have a two by two by two cube of cans underneath that and then a cube that's three by three by three cans under the fat.

This is now just the sum of the cube numbers, and there is a general formula for this as well.

And it looks like this, and these kinds of ideas have been thought about for a very long time.

People have played with this particular bit of maths since a very, very long time ago.

So, for instance, Pythagoras you may have you may be familiar with Pythagoras is greatest hits.

But the followers of Pythagoras were known to arrange pebbles in triangles on the floor to try and work out how many you could get in a in a triangle shaped like this.

So they were looking at these kinds of questions.

This is a mathematician from around about 1000 years later from India on Daria Butters, widely considered one of the first physicists.

And he knew that pie was 3.1416 which is pretty good for the time on aerial battle, also considered thes formula, the sums and the sum of squares in the sum of cubes on dhe.

This is a mathematician called al Carajo, who is based in Baghdad again and in the 500 years later also worked on this particular type of mathematics and also discovered Pascal's triangle 700 years before.

Pascal was quite a popular thing to do, but at the time, maths was done in a quite different way to the way that it's done now, so they wouldn't necessarily have had these formula in the same form that we're familiar with.

So this is from Mario Butters writing on the topic.

Andi, it's it's quite dense.

There's a lot going on here, but I'll put up the equation as well.

So you can see what's going on here.

The sixth part, That's the six on the bottom, off the product of three quantities, consisting of the number of terms that's n the number of terms plus one that's N plus one and twice the number of terms plus one is the sum of the square.

So this is kind of a word version of that same mathematical formula.

And at the time, this was the way that people did mathematics.

We didn't have this kind of algebraic notation that we use today.

So I guess the next person to come into this story is gonna be the first person to write down these formula properly.

I guess in a in a numerical form on that was a mathematician called Thomas Harriet, who was also a map maker and a navigator, and a translator in an ethnographer on DDE.

One of the things that he did was write down these equations So you have some of numbers That sum of squares that some of cubes on the one in the bottom corner There is this some of fourth powers s o the patent carries on on dhe.

The next person who comes along is this mathematician Johannes Foul Harbor, who was from Germany.

And he wrote down formula for the 1st 17 powers, which sort of feels like maybe there's something kind of general here.

Maybe we could actually get a formula for any power.

If I If I choose how many dimensions my supermarket display is gonna be in, I can then work out how many cans I'm gonna need.

Onda, the person that is the next person that kind of comes along in this story is a mathematician called Yakoub Annuity.

What the newly realized was that these formula you can be rewritten so that they all look very similar.

So instead of saying and over two times n plus one, you can spread it out so that you've got the some off first powers.

I guess some of numbers is 1/2 n plus 1/2 end square.

This is just the same equation, but rewritten on dhe If I write the sum of the squares, I get a similar thing.

I've got a fraction in front of an and a fraction in front of an end squared in a fraction in front of an end.

Cute on.

I can do this for all of them.

You just end up with more and more powers of end.

In this case, there's no fraction in front of the end because it zero but you get a sequence of numbers and these numbers essentially tell you what these formulas are.

And the newly discovered a way to find these numbers A pattern in these numbers.

That meant that you could create these formula on what the newly found was a set of numbers, which we call the Bernoulli numbers.

That's the next bunch there.

Quite a lot of them are something over six that we don't know why eso these air the Bernoulli numbers and these are a set of fractions again.

A lot of them are zero.

There are patents in these as well, but we knew.

We realize that if you take these numbers and combine them with some numbers from Pascal's triangle, it turns out you can generate those fractions, which means you can create these formula for any power that you like.

And you can also generate the Bernoulli numbers.

And in order to create the binary numbers, you need the previous binary number.

So you got the 1st 1 which is one you can then work out the 2nd 1 You can then use those to to work out the 3rd 1 and so on.

Now we call the new was very smug about this discovery.

He said, With the help of this table, it took me less than half of 1/4 of an hour to find that the 10th powers of the 1st 1000 numbers being added together will yield the sum.

This which is quite impressive.

You know, this is definitely a step forward, and it's a brilliant discovery, although potentially not necessarily, but knew he was the first person.

Discover this as always.

This is a mathematician from Japan, Seki Tucker, cause you discovered these numbers and published about a year before Benue read it.

But for some reason, they're called the binary numbers.

So who knows?

Anyway, this could be, in some sense, the end of the story because we have an answer to this question.

In general, if I want to build any number of stacks in any number of dimensions, how many cans do I need?

I can work out the binary numbers and then I can work out the formula using them.

But it's not quite the end on the next person who comes along with this particular bit of maths is a person called Ada Lovelies on Dhe.

She's a very, very famous example of someone who's connected potentially.

If you've heard of it, you might have heard of her in the context of computing because Ada Lovelace worked with this machine on I should qualify.

This is not the entire machine.

This is one small part of the machine on the whole thing was never actually built.

So technically, Ada Lovelace worked with an imaginary computer that never really existed.

This is just kind of one small bit of it on.

It was designed by Charles Babbage.

Andi Ada was brought in to help with some of the writing up of this, so she understood how this machine worked.

She was translating some papers on wrote a big appendix on the end just to show what this machine could do.

And it was again, a hypothetical thing.

But you could program this machine to do anything, So you kind of control that using a punch card system.

So a bit like the jacquard looms of the time you punched some holes in a thing, feed it into the machine on it would run a program, and it could calculate, in theory, anything you wanted it to.

Aunt Ada wanted to show off what this machine was capable of doing, and she wrote this program and the title of There is a diagram for the computation by the engine off the numbers of canoeing.

So this was the thing that she did.

This was the piece of mass that she chose to show off using what could potentially have been the very first ever computing machine on.

Do you know this has got in it the binary numbers.

There's a little B zero on a B one popping out there on the right and a B two, and you can see that it is cold for generating the numbers.

And I'm incredibly impressed by this because this was a computer program that was written for a computer that didn't actually exist.

You can't debug this right.

This is just This is so impressive on.

But it happened that she picked this particular bit of Mass.

It was something that she had been playing around with herself, to try and calculate the numbers to actually show what this could do.

And I don't think this is the end of the story, either, because the newly numbers connected various of the bits of Massa's well.

So there's a piece of mass called the Riemann Hypothesis, which is currently one of the most famous outstanding problems in mathematics.

If you can solve the Riemann hypothesis, there is a prize of mental a $1,000,000 available to the mathematician that first solves it, which makes it quite an exciting prospect.

But it does connect to this same piece of maths, So if someone were to make progress with that, they might potentially be the next person in this story as well.

But I think what this has shown me, I don't normally talk about mathematicians this much.

I mostly just talked about maths, but I thought it was interesting to think about the people who did this because these mathematical ideas that float around in the universe, just waiting for someone to come along without mathematicians.

That's all they are.

They're just ideas.

And until someone comes in and grabs it and plays with it and uses it to do something interesting or useful it along, the ever just be an idea, Thank you very much.