Could we have a repeating rational decimal number with more than 10 repeating digits (something like $0.0123456789801234567898...$) after the decimal point?
What is the maximum number of repeating digits after the decimal point in a number?
Could the answer be generalized to state that we could / couldn’t have a repeating rational number in base $b$ with more than $b$ repeating digits?
Answer
The period of a periodic sequence of digits can be as large as you like. To see this, multiply the number by $10^T$, where $T$ is the period, and then subtract the original number. Since this is definitely a whole number $n$ – the repeating parts of the sequence cancel out – the original must have been a rational number, specifically $n/(10^T-1)$.
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