Is Pi Infinite and Nonrepetitive? Exploring the Possibility of Pi Inside Pi

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In summary, Pi is infinite and nonrepetitive, and it is believed to contain every finite sequence of digits somewhere in its decimal expansion. However, it is not known for certain if this is true, and it is also not periodic. Additionally, there is no proof that every finite number combination is in pi, as pi would have to be a rational number for this to be true. Therefore, pi cannot contain itself in a non-trivial sense.
  • #1
Edi
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If pi is infinite and nonrepetive and every number combination is in pi, somewhere, does that mean pi itself is in pi somewhere.. ? (that would make it periodic)
 
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  • #2
Yes, it is, starting at the first digit :)
 
  • #3
Edi said:
If pi is infinite and nonrepetive and every number combination is in pi, somewhere, does that mean pi itself is in pi somewhere.. ? (that would make it periodic)

If pi is "normal in base ten" then that would mean that every _finite_ sequence of decimal digits occurs somewhere in the decimal expansion of pi.

There can only be countably many non-terminating decimal expansions found in consecutive digits in the decimal expansion of pi -- the one starting at the first digit, the one starting at the second digit, the one starting at the third digit, etc.

Since there are uncountably many non-terminating decimal expansions, it is certain that not all of them appear.
 
  • #4
If pi is infinite and nonrepetive and every number combination is in pi, somewhere, does that mean pi itself is in pi somewhere.. ? (that would make it periodic)
Is there any proof that every (finite) number combination is in pi?
 
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  • #6
Dead Boss said:
Is there any proof that every (finite) number combination is in pi?

It is not known whether pi is normal in base 10.
 
  • #7
[itex]\pi[/itex] is not periodic. If it were it would be in [itex]\mathbb{Q}[/itex] which it isn't.
 
  • #8
telecomguy said:
The string 31415926 occurs at position 50,366,472 counting from the first digit after the decimal point.

http://www.angio.net/pi/bigpi.cgi
Interesting but does not answer the question which was about the entire countable string. On the other hand, Edi seems to be under the impression that we could have entire string, then additional digits which is not possible.

Edi, it is NOT known whether "every number combination is in pi" is true or not.
 
  • #9
You are looking for a number n such that 10^n*pi-pi is an integer. Call this integer q.
Then (10^n-1)*pi = q
pi = q/(10^n-1)
To find such an n, pi would have to be a rational number, which is isn't.
So, no, pi cannot repeat or contain itself.
 
  • #10
Khashishi said:
You are looking for a number n such that 10^n*pi-pi is an integer. Call this integer q.

As has been pointed out, n=0, q=0 satisfies this criterion. Pi contains itself -- in a trivial sense.

Then (10^n-1)*pi = q

Yes
pi = q/(10^n-1)

pi = 0/0 ?
 
  • #11
you know what I meant.
 
  • #12
Khashishi said:
You are looking for a number n > 0 such that 10^n*pi-pi is an integer.

Happy?
 
  • #13
Well, I could be even more picky and take n = 0.1200175... and q = 1.
 
  • #14
jbriggs444 said:
Well, I could be even more picky and take n = 0.1200175... and q = 1.

Please stop hijacking this thread. We all know what Khashishi meant.
 

1. What is "Pi inside pi inside pi"?

"Pi inside pi inside pi" refers to a mathematical concept in which the digits of the number pi (3.14159...) are found within the digits of pi itself, creating a seemingly infinite pattern.

2. How is this concept possible?

This concept is possible because pi is an irrational number, meaning it has an infinite number of non-repeating digits. Therefore, any sequence of numbers can be found within pi, including itself.

3. Is this concept purely theoretical or does it have practical applications?

The concept of "pi inside pi inside pi" is primarily a theoretical concept, but it has been used to test and develop algorithms for computing large numbers of digits of pi. It also serves as an interesting example of the infinite nature of pi and irrational numbers in general.

4. Are there any limitations to this concept?

As pi is an infinite and non-repeating number, theoretically any sequence of numbers can be found within it. However, as the sequence gets longer, the likelihood of its occurrence decreases exponentially.

5. Can this concept be applied to other irrational numbers?

Yes, this concept can be applied to other irrational numbers with infinite and non-repeating digits, such as the square root of 2 or Euler's number. However, the occurrence of a specific sequence within these numbers is also unlikely due to their infinite nature.

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