B Can an infinite decimal expansion of a number repeat once?

[docnet]

TL;DR Summary

It's commonly known the decimal expansion of an irrational number never repeats or terminates. The word 'never' made me think what about the case for rational numbers. Are there rational numbers with infinite decimal expansions that repeats just once, or any finite number of times?

It seems like there is no number whose infinite decimal expansion has a finite number of repeated segments, since if the repeated segment is finite in length, it must be repeated an infinite number of times for the repetition to continue indefinitely. And if the repeated sequence is infinite in length, then there cannot be a repeat since the segment itself goes on indefinitely.

 

[Orodruin]

TL;DR Summary: It's commonly known the decimal expansion of an irrational number never repeats or terminates.

Is it? Those seem like contradictory statements unless you have access to an infinite amount of digits. For example, 1 repeats in pi pretty fast: 3.1415…

What is true is that an irrational number cannot terminate or end with an infinite periodic repetition of just the same digits in the same order.

The word 'never' made me think what about the case for rational numbers. Are there rational numbers with infinite decimal expansions that repeats just once, or any finite number of times?

Unclear what you mean by this. Does $0.111\overline{2}$ qualify as repeating 1 thrice?

 

[PeroK]

TL;DR Summary: It's commonly known the decimal expansion of an irrational number never repeats or terminates. The word 'never' made me think what about the case for rational numbers. Are there rational numbers with infinite decimal expansions that repeats just once, or any finite number of times?

It seems like there is no number whose infinite decimal expansion has a finite number of repeated segments, since if the repeated segment is finite in length, it must be repeated an infinite number of times for the repetition to continue indefinitely. And if the repeated sequence is infinite in length, then there cannot be a repeat since the segment itself goes on indefinitely.

I guess the idea is that the digits never repeat until halfway to infinity, then they repeat for the second half of the decimal expansion? From halfway to infinity until infinity. Unfortunately, there is no point halfway to infinity!

 

[gmax137]

Unfortunately, there is no point halfway to infinity!

Or maybe this is fortunate?

 

[docnet]

Is it? Those seem like contradictory statements unless you have access to an infinite amount of digits. For example, 1 repeats in pi pretty fast: 3.1415…

What is true is that an irrational number cannot terminate or end with an infinite periodic repetition of just the same digits in the same order.

Sorry for my vague language! I now know that I was trying to say a decimal expansion of an irrational number doesn't terminate or end with an infinite periodic repetition of just the same digits in the same order.

Unclear what you mean by this. Does $0.111\bar{2}$ qualify as repeating 1 thrice?

Yes, but my apologies for using unclear language. It was a question of whether it's possible for some rational number that has an infinite decimal expansion to have just one repetition of just the same infinite number of digits in the same order (Perok told me no, it isn't possible).

 

[WWGD]

Sorry for my vague language! I now know that I was trying to say a decimal expansion of an irrational number doesn't terminate or end with an infinite periodic repetition of just the same digits in the same order.

Yes, but my apologies for using unclear language. It was a question of whether it's possible for some rational number that has an infinite decimal expansion to have just one repetition of just the same infinite number of digits in the same order (Perok told me no, it isn't possible).

Well, you may want to be more precise, as each of the digits $\{0,1,..,9\}$ will be repeated infinitely often. Being just slightly facetious here.

 

[docnet]

Well, you may want to be more precise, as each of the digits $\{0,1,..,9\}$ will be repeated infinitely often. Being just slightly facetious here.

?

 

[DaveC426913]

It seems to be a poorly-defined question. How long does a sequence of digits have to be to qualify as a repeating sequence?

Does 0.232357... count?
Does 0.22357... count?
Does 0.11111... count?
(Not that these are decimal expansions, (except the last one), but suppose they were?)

If it's an irrational number that never ends, how can you know some given sequence doesn't repeat?

 

[docnet]

It seems to be a poorly-defined question. How long does a sequence of digits have to be to qualify as a repeating sequence?

Does 0.232357... count?
Does 0.22357... count?
Does 0.11111... count?
(Not that these are decimal expansions, (except the last one), but suppose they were?)

@Orodruin already cleared this up and I acknowledged my vague language and the question I was trying to ask.

Does 0.232357... count?

No, because we are talking about cases where the repetition continues to infinity. In a vague way, I wanted to know if it's possible to find a bijective map that inverts the sequence of numbers $1,2,3,...$ about the mean of the set $\{1,2,3,...\}$ which @PeroK implied can't exist since there is no halfway to infinity. Hypothetically, if in another mathematical universe there is such a map, it would be possible to have an infinite decimal expansion of a rational number that has two adjacent instances of just the same digits in the same order. As @Orodruin helped me understand this is an impossible scenario that would lead to contradictions, and the question came from my poor understanding of these numbers. The point is I'm asking questions to help improve my understanding of the math system.

Does 0.22357... count?

No.

Does 0.11111... count?

Yes $0.\bar{1}$ counts. We know that a segment of $1$s of length $n$ repeats infinitely often in that expansion, with each segment having the same digits in the same order. We know that such a segment of $1$s can't repeat a finite number of times, because the decimal place that is $\frac{1}{2}$ of the way to infinity (or is any nonzero fraction $\frac{1}{n}$ of the way to infinity) isn't definable.

 

[Vanadium 50]

as each of the digits will be repeated infinitely often.

I believe .12112111211112.... is irrational and does not have all ten digits repeating infinitely often. Or at all in 8 of 10 cases.

 

[docnet]

I just remembered that
$$\sum_{i=1}^\infty n=-\frac{1}{12}$$
which makes the mean of the set $\{1,2,3,...\}$ equal to $-\frac{1}{24}$. I'm not sure if it would be useful in providing any clarity though.

 

[docnet]

^^On second thought, I'm not sure if my statement about the mean is correct at all. Also, my use of the term in a previous post was not precise to begin with.

 

[gmax137]

I just remembered that
$$\sum_{i=1}^\infty n=-\frac{1}{12}$$

I think you meant to say
$$\sum_{n=1}^\infty n=-\frac{1}{12}$$

I watched the numberphile video but I still don't think this is correct. PS, not a mathematician so what do I know.
()

 

[Orodruin]

I just remembered that
$$\sum_{i=1}^\infty n=-\frac{1}{12}$$
which makes the mean of the set $\{1,2,3,...\}$ equal to $-\frac{1}{24}$. I'm not sure if it would be useful in providing any clarity though.

Are you saying that there are two integers?

The mean of a set is the sum of the elements divided by the number of elements.

 

[docnet]

Are you saying that there are two integers?

The mean of a set is the sum of the elements divided by the number of elements.

Yes! you're right. Yikes I think I need to take a break because my brain just broke.

 

[fresh_42]

I just remembered that
$$\sum_{i=1}^\infty n=-\frac{1}{12}$$
which makes the mean of the set $\{1,2,3,...\}$ equal to $-\frac{1}{24}$. I'm not sure if it would be useful in providing any clarity though.

If you really want to open that barrel then please study ...
https://www.physicsforums.com/insights/the-extended-riemann-hypothesis-and-ramanujans-sum/
... first and the proofs therein.

However, this is a completely new subject and should be discussed in a new thread, hopefully after the participants will have read that article.

This thread is closed now.