We've talked before in a video about

Pandigital numbers. And a Pandigital number was

a number that used all the digits one

to nine. and we... [Brady]: What about zero?

[James]: It didn't always use zero, I think sometimes

we excluded 0 and sometimes we included 0

and you get different numbers with

different properties and we had

a nice time looking at Pandigital numbers

That was a bit of fun

[Brady]: Yeah, it is there. People can click on it.

[James]: There?

[Brady]: Uh... Yeah! About there. Alright.

[James]: Now I want to show you a Pandigital Formula

So it's gonna be a formula, that uses all the digits

1 to 9. And it's really nice actually

Let me right down the formula.

(1+9 to the power, so I'm gonna use -4

the power 6 times 7 is all in the

bracket ), and then I take that to a power

which is going to be 3 to the power 2 which

is to the power of 85. So this is the formula,

So let's just check if it's a pandigital formula

let's check all the digits are there

We've got 123456789

Yes. I've used all the digits from 1 to 9.

But what's so special about that formula?

It's really cute.

This formula is equal to 2.718281828

459...

This formula is actually... e

It's approximately e. How approximately though?

How good is it?

This is correct to 18 trillion trillion (18 times 10^24)

digits, decimal places. [Brady]: NO! Go away!

[James]: Yeah, I know it's great. This lovely

little formula is incredibly accurate for e

[Brady]: How did that happen?

[James]: It's beautiful! I was telling you how that happened.

So this formula, it's not

not a very old one. It was found in 2004

by a guy called Richard Sabey, and he

did something very clever to get this

formula because the definition of e,

It's the limit as n tends to infinity of

1+ (1 over n), to the power n. Now,

he did something clever. He found a

number which I'm gonna capital N.

And it's gonna be a big number. Actually I'm gonna let it

be this thing we've got here.

This power. so he had a number 3 to the power

2 to the power 85 and he did this clever thing

I'm just going to mess around with this

a little bit, and I hope you're happy with

how powers work because I'm going to use

some properties of how powers work, and

this is what I'm going to do. I'm going

to take one of these- 85 here- and i'm

going to use it to square the 3. So it

becomes 9 now to the power 2 to power 84

and now i'm going to another clever

thing if you have you have powers work

this is also equal to 9. I'm going to

double the 2 there so it becomes a 4. And 84

gets halfed, so it becomes a 42 this is all

properties of powers and i'm going to

split up the 42. This becomes 9 to the power

4 to the power 6 times 7, 'cause that's 42.

If I stick a minus sign in there, it will

become 1 over N. So, what it's got now is

a formula in this shape- 1 plus (1 over N)

to the power n- but this N is massive

this is a massive number, and it's so big

that makes it an incredibly accurate

approximation of e. It's not infinity, it's not

tending off to infinity and being e

exactly. It's big enough to make

something that is correct to 18 trillion

trillion digits. There was a challenge- it

some sort of challenge on the

Internet- as a challenge you

think about this for the same

competition he actually came up with a

few formulas and he was going through

the classic mathematical constants so he

did e and e's the best one and I love e

so much. But he also did pi (π), and others.

The π one, i'll show you- I don't like it

as much I don't think it's good but it's

clever that he's

done it with a pandigital formula, but

I'll show you why it's not so good

So he's got a formula for pi (π), and it's

going to be approximately equal to, and

this is it, it's going to be: 2 to the power 5 to the power .4

- .6, - and in brackets here, we got (.3 to the power 9

/ 7). And then take that bracket

we're going to take it to the power

.8, to the power .1

[Brady]: Yeah, lot's of points.

[James]: Lot's of points, and instead of stepping 0.6

and 0.1 he's used . He's cheating

maybe I don't know by not using the

zeros, I'm not as satisfied with it.

This is accurate to pi(π) up to 10 decimal

places. It's a lot less impressive

so you've got a pandigital formula for

pi(π), for 10 decimal places with a bit of a cheat

I don't know depends on how you feel

about that- using points instead of zero point something

I don't like that formula as much.

[OUTRO]

[James from "Pandigital Numbers"]: And then there's a number that can be

divided by 10 as well and that's also

unique so that would be a unique

pandigital number including 0