# Multiplying monkeys and parabolic primes

so today we've got a very very special guest who is going to perform an incredible feat for you to introduce the special guests abroad my multiplying monkeys so he is one of those multiplying monkeys it's a very very old

Toy patented in 1916 and actually got one of those original toys here so it comes in this box and out comes educated monkey and so what the monkey does is it multiplies and it does this by moving around the feet I've got a bit of a

Movie here to show this to you ok so at the moment two feet at 2 and 11 and so the window shows 22 and I will move the legs or out of it so now we're out 2 times 9 is 18 and now we move 2 feet again this time yeah now it's 5 times 9

Which is 45 now you can't actually move 2 feet all the way together to 5 so to do squares you need some special trick so for squares we actually move things out all the way to here to the special symbol so the that foot goes out to the

Special symbol and then we've got 25 happening at the moment and to get the other squares we just move that leg there okay so we move this leg 36 49 64 81 and 100 you know so very nice it actually comes with a little bit of a

Card here so you can you can slide this card behind and so that will replace all the numbers and when you use it with this card you can also do addition so quite nice I've also got two modern versions of this you can actually buy

This so this is an exact replica of what I've got there the original toy and this is kind of a variation so under there quite readily available if you want one of those things okay and now we're ready for the for the main act the main act is

Going to be performed by our friend x squared so this is just the graph of x squared and we're going to show you how x squared can multiply and do all kinds of other really amazing things okay so let's just put the X's in X's are in how

Do we multiply with the parabola okay I mean we take a number three and you see it's a bit stranger because I don't have negative numbers on that side right so it's most positive numbers on that side and on that side anyway we'll take a

Three and we want to multiply by four and so to multiply what we do is we just make this connection here between this point and that point and then we get this intersection here with their y-axis and that's the product three times four

Is 12 now obviously that could be a coincidence so let's just move things around a little bit so here we've got four times five is 20 seems to work mmm okay well let's just think about you know a couple of scenarios where it's

Clear that it has to work okay and those are obviously the squares so if we go up to five times five then obviously up here we are 25 on the other side we're 25 so if we draw the horizontal across we have to cross the

25 pretty obvious all right and for all the other squares to and so 3 times 3 2 times 2 1 times 1 and you know just just to check it out a little bit before we even think about what's really happening here well we're with 1 1 times anything

Should be the same number right so 1 times 1 is 1 here 1 times 2 is 2 1 times 3 is 3 1 times 4 is 4 1 times 5 is 5 what are the special numbers do we have 0 okay 0 times 5 is 0 works very well ok now just in general if you take any

Number here and any number there what you get there is really a times B ok and it's actually fairly easy to calculate it I'll just give you a hint how to do it and maybe somebody can do it in the comments ok so I mean the simplest way

That I've been able to come up with this well you just realize that up here we add a squared ok and so we just have to add two ace this distance here and this distance is really easy to calculate if you know the

Slope of this red line okay because then you just multiply here this distance which is a by the slope and you get that so at a squared two whatever that is and you're done and how do you get the slope also very easy that triangle here we

Know everything we know that this one here is B squared minus a squared and we know that this is a plus B okay so maybe somebody can do it in the comments huh okay so this really works now what else can this do okay now let's have a look

At again 4 times 5 is 20 and now I'm going to count down from here to zero okay so we're going to move that one here so we're going down to 4 obviously result is okay down to 3 down to 2 down to 1 down to 0 what comes next

What comes next minus 1 but there's no minus 1 but let's move it anywhere in the right sort of direction here oh we lost something yeah okay well let's move it up a bit and there actually it is you know 4 times minus 1 is minus 4 great so

This works what's the next thing you should try well what I want to do is I'm going to move this 4 here towards the right and eventually I want to end up in the negatives there okay but something interesting is going to happen along the

Way and you can kind of see or it can you see what's going to happen along the way no okay well just imagine that that square wandering over there eventually it's going to come into incidence with the one that's already there and there

Something interesting is going to happen okay let's just do this so going down now zoom in okay so at the moment we're still okay minus 1 times 3 is minus 3 good now at the moment we've got basically this line

Connecting these two points right now when the squares come together the points also come together and basically I mean if you've done calculus you know exactly what happens then this what's called a secant line as lined up

It's across the parabola turns into a tangent line the line that touches the parabola at that point there okay and so we're going down – – and there we have the tangent line tells us that 1 times minus 1 is minus 1 perfect

And actually before we keep on going I mean that actually gives you what basically the derivative of x squared or if you ever want to like construct a tangent to any of the points here and this actually suggests how we can do it

Okay so let's just choose any point here for example that point is hovering above well 2 – 2 that's what corresponds to it 2 times minus 2 is minus 4 so let's just put that point in here and now we just connect this point with that point and

That should be the tangent yeah perfect very nice or I mean if you want to skip all this and just want a recipe well what you should do if you've got a point up here the parabola you just measure this distance and then you put that same

Distance down here and then you make the connection that gives you the tangent line right obviously you can also do it on the other side same thing okay now back to kind of moving that number into the negatives this way alright so here

We go so 3 goes down again now the whole thing turns into tangent now we're going to multiply by 0 that's fine and now we're going to go into the negatives there and now we actually have negative times negative is positive

How nice is that right minus 1 times minus 1 is is plus 1 okay but it can do even more right you can do it more we don't have to stop there so next trick we're going to do is I mean its whole multiplying business

Works for any numbers so it could have like PI down there and square root of 2 and you can multiply them and it works all right now at the moment what we want to do is want to focus on on integers ok we're going to multiply integers and

Actually what I've done here is I've noted the points that chorus on the parabola to correspond to 2 to 3 to 4 on both sides okay now what I want to do is I want to connect all these points up in all possible ways all right let's just

Do it so the first connection is here which corresponds to two times two so obviously at four and this time we're actually going to cancel out whatever we get here as product again so four is

Gone next connection is between two and three so it gives the product of six or six goals right and now I'm just going to kind of keep on going and getting rid of the products that we get there so let's keep on going so I'm just one more

And well can we see something already just kind of reading up from the bottom so there's one there's two there's three there's five so what is that all the prime numbers exactly so 3 5 7 11 13 14 14 14 14 sub-prime so why why what's

Happening here we ran out of numbers if you actually drawn the whole parabola and just done the whole thing like put in all connections we would actually get all the all the prime numbers so it is kind of instant Prime's instant prize

Draw a parabola put all those thoughts on make our connections whoosh you got all the all the prime numbers how nice is that all of them plus one day there was a reason of course why we didn't start

This with the point that corresponds to one one because if you multiply by one and would get rid of everything right all right another reason why one shouldn't be a prime okay now it's actually not the end of it but I'll just

Give you some hints and you can kind of explore those yourself so these are a couple of things they can explore so for example I mean we can multiply with this thing but you can also divide so the symmetry of the

Parabola has something to do with multiplication being commutative eight times B is equal to B times a you can actually do square roots and a couple of different ways in two different ways there's something very deep within

Mathematics and if you know a little bit more you might know about multiplication addition on elliptic curves so see whether you can find some connections between what we've done here and that and then how does it work for complex

Numbers maybe how would you do other operations or something like this actually I might just show your division quickly okay and maybe the symmetry business and then that's it for today okay

So division how do you do it we want to divide 15 by something that's divided by three okay so the only thing you have to do is connect these two points up and just extend the lines up to there and we go down that's the result 15 divided by

3 is 5 well pretty obvious okay what about the symmetry business well 3 times 5 is 15 ok and now 5 times 3 is 15 obviously and you can really see you know that's got a lot to do with the symmetry of the parabola it's very nice

Ok and that's well me taking a bow for the parabola because parabola has a bit of a problem doing this