Side Topic: 'It's Not a Toom-ah'

[Xyzzy]

Originally Posted by manfred4

A fraction can at most have as many repeating digits as the denominator is as its number. And you can easily see, if you missed some digits for the repeat - on division by hand for example you see a different reminder on that step. So you know, its not really the repeating digit.

Originally Posted by Nick

Yes - for any positive integers a and b, the decimal expansion of the fraction must start repeating at the bth digit after the decimal point at the latest. You can calculate the decimal expansion by doing long division of a by b, and each step of the calculation ends with a remainder. Once you have passed the decimal point, if you ever get remainder 0 then you have finished, and written the fraction as a decimal with only finitely many digits. Otherwise, each step ends with a remainder which is an integer in the range 1 to b-1 inclusive, so the first b remainders cannot all be different. And once the same remainder occurs again, the steps in the calculation will be exactly the same as the previous time, generating the same digits over and over again.

Try as an example.

Okay, that works!

So, to summarize:

• If we do long division (a/b) and we get a 0 remainder somewhere along the process, the job is done and the decimal expansion is exact.
• If we do long division (a/b) and we get a repeating decimal, that repeated part has to be fewer digits than the (absolute?) value of b. Every interim step to get there will have a unique remainder until the repeating starts. We will know when the repeating starts because we will have a rerun remainder.
• Because we are doing long division, which is the definition (?) of a rational number, any job we do has to end with an exact answer (remainder = 0) or be a repeating decimal.

Thanks!

[Xyzzy]

https://en.wikipedia.org/wiki/0.999....m_in_education

[R.D. Silverman]

Originally Posted by manfred4

A fraction can at most have as many repeating digits as the denominator is as its number. And you can easily see, if you missed some digits for the repeat - on division by hand for example you see a different reminder on that step. So you know, its not really the repeating digit.

We will get to this subject in the number theory discussion, but not until later.
There is a purely number-theoretic explanation for the period length of a decimal fraction.