Question

I know that pi has no repeating sequence of digits. But is it possible to say whether there is a more complicated pattern based on some sequence of digits?

I don't know if irrationality could imply that it's impossible, because I think you could create an irrational number by defining a pattern of digits that don't repeat, which would make it irrational and generated by a pattern...

Answer 1

Some numbers are irrational because they are proven to be so. The sequence of digits are completely (you could say) set in stone and have no way to generalise it except the original formula used to derive the irrational number.

Answer 2

But there are connections between things unknown to the people who were originally just looking at the circumference of a circle, right?

For example, in the number e, there is a very interesting pattern when you look at its continued fraction expansion, and that can be used to generate the number, which has nothing to do with what people originally found the number from.

So I'm not so sure you can say that it can only be derived from the original formula.

Answer 3

Yes. The other ways are actually the formula in disguise. e for example, have one equation, that can be formulated in a different way, giving a new equation, that was actually the same as the last.

Answer 4

The original definition of e something to do with that fraction, which can be formulated into a Riemann sum and into an integral. The fraction was the original formula.

Answer 5

I don't know what you mean. The different ways of generating a number generate the same number. So they're "the same" in that sense. But their methods and where they came from can be extremely different and have no connection to what originally someone thought it would be. So because pi has some connection to circles, it doesn't mean it can't have other connections to things we don't know about yet. And I wonder if it's possible some of those things might be some very convoluted pattern in the digits. I don't know if it's possible or not though.

Answer 6

Well, I'll give you an example of irrational numbers. Tell me a random string of numbers. Given a super computer, I will find those particular numbers in the trailing digits of e, pi or any irrational numbers. This is how strange they are.

Answer 7

The methods are actually quite 'same' in a sense. It's like multiplication and addition. You derive multiplication from addition.

Answer 8

yes I know they're strange. That's not under dispute. What I wonder is, could it be possible (or is it proven impossible) that no very strange way of generating them from a pattern could be found?

Working from the other direction, could it be possible to define a pattern that generates an irrational number?

I think maybe 0.123456789101112131415... is irrational. But it's clearly generated by a pattern.

Answer 9

Well, as said, the pattern you seek is the formula in which the irrational number they are formed.

Answer 10

Could you restate that? I didn't understand the sentence.

Answer 11

well, for example, the pattern of e is that fraction with the limit.

Answer 12

But for pi. Is it possible to have a pattern of a form where you are given some sequence of digits, and then some rule to construct the sequence of digits in pi, involving permutations and other sorts of operations on the sequence? no calculations.

Or is there some theorem which says it's impossible?

I know that it can't have an infinite sequence of repeating digits, or else it's rational. But does it extend into more elaborate rules for permuting some sequence etc?

Answer 13

That's what I'm really asking.

I know there are ways to generate pi. But they require calculation.

I wonder if it could be at all possible to have some way, simple in operation, elaborate in directions, to generate pi.

Answer 14

I can tell you this: Everything and nothing is simple in maths. To mathematicians, the way to generate pi is probably the most beautiful and simple thing they ever saw.

Answer 15

That doesn't tell me much. And you haven't justified saying "*the* way"...

Answer 16

Hmm...What I meant was that formula you used to derive pi is the most simple, easiest way in maths. There is no other way.

Answer 17

I never mentioned any specific formula to derive pi...

And there are several ways to derive pi.

Answer 18

Oh sorry about that lol. Yup, all leading to the same answer, but actually all of them can be derived from themselves...

Answer 19

All you're saying is that they end up with the same number. There are methods that are completely different. I'm sorry but, I think you're talking nonsense. You can't say that all of the numerical techniques to calculate pi are somehow derive-able from any of the analytical techniques...

They end up with the same number. That's all. And you haven't said anything about my question except that "pi is nice, and pi generating algorithms generate pi." I'm tired and you're talking nonsense not helpful to the question.

Answer 20

Sorry I 'm unable to help.

Answer 21

As long as no infinite repetition of any sort, then that number is irrational?
All rational numbers can be expressed as non-terminating REPEATING decimals.

Answer 22

@Shadowys where in the decimal representation of pi (or e) is the sequence

...000...