Repeating Digits of Pi

a1call · 2017-10-02, 02:39

There is this thread in another forum.

I am interested in members of this board's input.

Thanks in advance.

https://forum.cosmoquest.org/forumdi...and-Technology

ETA:
I made correction to my post on the other board:

11 changed to 1010.

a1call · 2017-10-03, 17:13

The conversation has taken an interesting turn.
If an ideal random number generator is guaranteed to output any given number, given infinite trials, can we conclude that any given finite sequence of digits will occur somewhere in the pi's decimal expansion?

R. Gerbicz · 2017-10-03, 20:27

Quote:

Originally Posted by a1call

There is this thread in another forum.
https://forum.cosmoquest.org/forumdi...and-Technology

Some part already solved in the "Does pi contain pi?" thread. Obviously we omit the trivial (identity) representation, and then it is impossible to find pi if you skip finitely many digits, because as it is pointed out pi would be rational.

The interesting (not solved there) part is that: can you get x if you delete infinitely many digits of a general x number?
If x is rational then the answer is yes (Why?).
If x is irrational (like pi) then you can still get x: let H={d: 0<=d<10 and d appears in x infinitely many times}. It is clear that there exists k, that after the k-th decimal digit of x you see only digits from H. The strategy is: keep the first k digits of x, then skip a digit, you can still find all digits (in order) of x, because these digits are in H, and hence these appears infinitely many times in x. After you have found the (k+1)-th digit at the t-th position then skip the (t+1)-th digit and use induction from here. Note that this is also a (harder) proof for the easier case, when x is rational, and ofcourse |H|>=2 if x is irrational (though it is not used).

a1call · 2017-10-05, 01:15

As pointed out on the other board, I am having a lot of problems with the normal numbers:
https://en.wikipedia.org/wiki/Normal_number

Does anyone else see that the definition is based on the false assumption that a number with infinite digital expansion in base 10 will never have terminating digits in any other base or that it can't have a bias in the density of digits in some bases and not in others?

Thanks in advance for any clarifications and/or inputs.

science_man_88 · 2017-10-05, 02:17

Quote:

Originally Posted by a1call

As pointed out on the other board, I am having a lot of problems with the normal numbers:
https://en.wikipedia.org/wiki/Normal_number

Does anyone else see that the definition is based on the false assumption that a number with infinite digital expansion in base 10 will never have terminating digits in any other base or that it can't have a bias in the density of digits in some bases and not in others?

Thanks in advance for any clarifications and/or inputs.

if it's non-repeating and non-terminating it would be irrational and hence not terminate in any rational base ... also your complaints show, you haven't read the wikipedia article in full where they declare things b-normal.

a1call · 2017-10-05, 02:41

Quote:

Originally Posted by science_man_88

if it's non-repeating and non-terminating it would be irrational and hence not terminate in any rational base ... also your complaints show, you haven't read the wikipedia article in full where they declare things b-normal.

Quote:

Intuitively this means that no digit, or (finite) combination of digits, occurs more frequently than any other, and this is true whether the number is written in base 10, binary, or any other base.

Rational base is your introduction. It is not present in the Wikipedia article.

science_man_88 · 2017-10-05, 02:43

Quote:

Originally Posted by a1call

Rational base is your introduction. It is not present in the Wikipedia article.

well the integer bases are all rational so if you believe they are talking about intger bases the statement still stands.

a1call · 2017-10-05, 02:46

The phrase "integer base" does not exist in the article either.

axn · 2017-10-05, 04:20

Quote:

Originally Posted by a1call

The phrase "integer base" does not exist in the article either.

Quote:

Originally Posted by TFA

In mathematics, a normal number is a real number whose infinite sequence of digits in every base b[1]

From the first line of wikipedia article, if you hover over the [1], you'll see this "The only bases considered here are natural numbers greater than 1"

a1call · 2017-10-05, 05:58

Thank you axn.
Will check that out next time I am on a PC.

Dr Sardonicus · 2017-10-05, 15:58

I don't know whether the number pi is normal to base ten (or to any other base). I default to the "assumption of ignorance" that it is normal, because I am ignorant of any reason to think it is not normal. (Also, the decimal digits calculated to date seem to pass the "randomness tests" thrown at them.)

However, the "output of a random number generator" idea for the digits might come to grief, at least in hexadecimal. I might well be wrong here, but if the digits can reasonably be viewed as sequentially put out by a random number generator, then it seems to me there should be no way to calculate the n^th digit without calculating all the previous digits first. And at least in hexadecimal, there is a method for finding the n^th digit without calculating all the previous digits first.

2017-10-02, 02:39	#1
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2⁶·31 Posts	Repeating Digits of Pi There is this thread in another forum. I am interested in members of this board's input. Thanks in advance. https://forum.cosmoquest.org/forumdi...and-Technology ETA: I made correction to my post on the other board: 11 changed to 1010. Last fiddled with by a1call on 2017-10-02 at 03:08

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2017-10-03, 17:13	#2
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 11111000000₂ Posts	The conversation has taken an interesting turn. If an ideal random number generator is guaranteed to output any given number, given infinite trials, can we conclude that any given finite sequence of digits will occur somewhere in the pi's decimal expansion?

2017-10-05, 01:15	#4
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2⁶·31 Posts	As pointed out on the other board, I am having a lot of problems with the normal numbers: https://en.wikipedia.org/wiki/Normal_number Does anyone else see that the definition is based on the false assumption that a number with infinite digital expansion in base 10 will never have terminating digits in any other base or that it can't have a bias in the density of digits in some bases and not in others? Thanks in advance for any clarifications and/or inputs.

2017-10-05, 02:46	#8
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2⁶·31 Posts	The phrase "integer base" does not exist in the article either.

2017-10-05, 05:58	#10
a1call "Rashid Naimi" Oct 2015 Remote to Here/There 2⁶×31 Posts	Thank you axn. Will check that out next time I am on a PC.

2017-10-05, 15:58	#11
Dr Sardonicus Feb 2017 Nowhere 2×3×719 Posts	I don't know whether the number pi is normal to base ten (or to any other base). I default to the "assumption of ignorance" that it is normal, because I am ignorant of any reason to think it is not normal. (Also, the decimal digits calculated to date seem to pass the "randomness tests" thrown at them.) However, the "output of a random number generator" idea for the digits might come to grief, at least in hexadecimal. I might well be wrong here, but if the digits can reasonably be viewed as sequentially put out by a random number generator, then it seems to me there should be no way to calculate the n^th digit without calculating all the previous digits first. And at least in hexadecimal, there is a method for finding the n^th digit without calculating all the previous digits first.