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My question is similar to (but different from) the one here.

I came across this sentence on Wikipedia: "The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over."

Is it possible for an unbounded sequence of digits to repeat in the decimal expansion of an irrational number? Is the concept of an infinitely long sequence of digits compatible with the concept of that sequence repeating and, if so, what discipline in math addresses such a thing?

bblohowiak
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  • Where would the repeat start if the first occurrence is infinite? – John Douma Oct 29 '18 at 20:32
  • My understanding of the definition of an irrational number is an infinite non-repeating decimal and hence cannot be expressed as a ratio of two integers. – Phil H Oct 29 '18 at 20:51

2 Answers2

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The only way for an infinite sequence of digits to repeat is if the second occurrence is a shift of the first occurrence, with offset $n$ say. But then the sequence repeats with period at most $n$.

TonyK
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Your proposal: "Is it possible for an unbounded sequence of digits to repeat in the decimal expansion of an irrational number?"

makes no sense. How would you ever determine if an unbounded sequence repeats?

David G. Stork
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