Pi and trascendental numbers - no repeating sequence of digits

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Discussion Overview

The discussion revolves around the nature of the digits of pi, particularly whether it can be considered to contain a repeating sequence of infinite length, and what that implies about its classification as a decimal. Participants explore the implications of infinite sequences and the concept of randomness in relation to pi.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants question whether an infinite sequence can contain a repeating sequence of infinite length, suggesting that such a concept may not make sense.
  • One participant proposes that if there is an infinite sequence of numbers, then every possible combination, including repeating patterns, must occur within that sequence.
  • Another participant expresses uncertainty about the concept of a "random infinite sequence," pondering whether it could eventually start repeating.
  • A later reply clarifies that a repeating sequence of infinite length cannot exist, as there is no endpoint to allow for repetition, contrasting this with rational numbers that have finite repeating sequences.
  • Participants discuss the randomness property of pi and the notion of normality, noting that it remains unproven whether pi exhibits such properties.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of infinite sequences and their potential for repetition. Multiple competing views are presented regarding the implications of infinite sequences and the characteristics of pi.

Contextual Notes

Limitations in the discussion include a lack of formal definitions for terms like "repeating sequence of infinite length" and "random infinite sequence," as well as unresolved questions about the nature of pi and its digits.

naes213
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Just pondering...

If pi continues without ending and can be considered to "contain an infinite number of digits" then isn't there a repeating sequence of infinite length contained in pi, thus making it a repeating decimal?

Obviously not...but what is the reason?



Thanks!
 
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How can an infinite sequence repeat? Please define that for me.
 
First let me qualify this statement by saying that I'm certainly not an expert and am just supposing based on very minimal mathematical background.

In my mind...which again may be seriously flawed...if there is an infinite sequence of numbers then each and every possible combination of numbers must occur in that sequence, including some type of repeating pattern.
 
well I'm not much of a maths expert at all but i can't see a flaw in what you've said,
in fact you could say almost anthing you want about "an infinite sequence of numbers" if you're asked to proove it, you probably cant, but you probably can't be proven wrong either.

but i like your thinking. makes me wonder if the phrase "random infinite sequence" is really a sound idea! maybe when you get to infinity it starts repeating! in fact i can't think of a single good reason why it doesn't start repeating at decimal digit (84^986^123^4848)-1 ??/ lol
 
naes213 said:
Just pondering...

If pi continues without ending and can be considered to "contain an infinite number of digits" then isn't there a repeating sequence of infinite length contained in pi, thus making it a repeating decimal?

Obviously not...but what is the reason?



Thanks!

The reason is simply enough that the concept of a repeating sequence of infinite length doesn't make any sense unless you define what you mean and then it would most likely not be a repeating sequence of infinite length anymore, but some sort of ramdomness property like normality, (which indeed can neither be proved or disproved for pi at the moment)
 
naes213 said:
Just pondering...

If pi continues without ending and can be considered to "contain an infinite number of digits" then isn't there a repeating sequence of infinite length contained in pi, thus making it a repeating decimal?

Obviously not...but what is the reason?



Thanks!

What do you mean by "a repeating sequence of infinite length"? As said above, a "sequence of infinite length" can't repeat- there is no "end" to the sequence so that it can repeat! I imagine you are thinking of the fact that any rational number is "eventually repeating". In that case the "repeating sequence" is not of infinite length. In 1/3= .33333..., the "repeating sequence" is just "3"- and the entire rest of the number is just that- no room for any other digits.

A simpler example is 0.101001000100001... which obviously, though it is of infinite length, never "repeats"- there is always one more 0 between 2 1s.
 
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