Maximum length of repeating digits in a decimal

Question

Total math novice here. I'm wondering if (as I think should be the case) there is no maximum length of a repeating decimal period, for example:

- 0.33333... (period 1)
- 0.252525... (period 2)
- 0.142857142857... (period 6) 
- etc.

The reason I am curious about this is that I have learned that the decimal expansions of irrational numbers don't terminate or repeat. But if there is no upper limit to the period that a repeating decimal can have, then where is the difference between the two? I am guessing that although the repeting pattern can grow without bound it is always finite (I think?) whereas with an irrational number the pattern never restarts again after a finite period. But it still makes my head hurt :-) so I am wondering if there are any explanations for it that I might be able to understand as a non-practitioner.

Thanks for any help!

Answer 1

Yep, you're right. Rational decimal expansions repeat eventually. You may have to go for millions or billions of digits, but they'll repeat - you can always find some number $n$ so after some point in the string of digits every $n$th digit will be the same. Irrational ones don't repeat at all.

Answer 2

To answer the first part, yes these period can get arbitrarily large. Take a look at the number: $$ \frac{1}{10^n-1}$$

For $n=1$, this gives $\frac{1}{9}=0.1111...$

For $n=2$, this gives $\frac {1}{99}=0.010101...$

For $n=3$, this gives $\frac {1}{999}=0.001001001...$

The decimal expansion of each number has period $n$ (the decimal expasion is just the repeating pattern of $n-1$ zeros followed by a $1$), and we can make $n$ as big as we would like.

For the second part of your question, the difference is that irrational numbers don't have a period. It's not that they have an infinitely large period after the decimal place; they just don't have a period whatsoever. They never get to a point where they start repeating. If the number ever continues to infinitely repeat itself, then that number is a rational number.

The difference is that every single individual rational number has a finite period. There are rational number with whatever finite period size we want, but there is never a rational number with an infinite period size.