Pi and trascendental numbers - no repeating sequence of digits

[naes213]

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!

 

[daniel_i_l]

How can an infinite sequence repeat? Please define that for me.

 

[naes213]

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.

 

[phlegmy]

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

 

[Pere Callahan]

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)

 

[HallsofIvy]

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.