Subtitles section Play video Print subtitles [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

A2 formula power james brady clever decimal Incredible Formula - Numberphile 2 0 林宜悉 posted on 2020/04/05 More Share Save Report Video vocabulary