[James]: Yeah this is really nice.
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