Rational if and only if Decimal Expansion Repeats

[jinydu]

It is a well-known fact that a real number is rational (i.e. can be expressed as the ratio of two integers) if and only if its decimal expansion eventually consists of an infinitely repeating string. But I've actually never learned a full proof.

I have learned one direction, which I can illustrate using an example

Let x = 0.142857142857142857...
Multiply by an appropriate power of ten so that the decimal point occurs right after the first appearance of the string: 1000000x = 142857.142857142857
Subtract to cancel out everything after the decimal point: 999999x = 142857
Divide by the coefficient of x to get 0.142857142857142857... = 142857/999999

It's easy to see that this argument will work in general.

But I'm not so sure about the other direction... Showing that the decimal expansion of a rational number eventually repeats itself indefinitely.

[wblipp]

Do they still teach long division by hand?

There are only a finite number of remainders, so they have to repeat eventually.

[jinydu]

Sure I learned long division. Not sure what you're trying to get at though.

[Mini-Geek]

Originally Posted by jinydu

Sure I learned long division. Not sure what you're trying to get at though.

Imagine you're trying to divide 1 by 7 with long division. Eventually you'll get the same answer with the same remainder i.e. it's repeating.
In a shorter example than 1/7, say 1/3,
1. you'd try to divide 1 into 3 and it's 0 times with 1 left over (0.), then
2. divide 10 into 3 and see that it's 3 times with one left over (0.3), then
3. divide 10 into 3 and see that it's 3 times with one left over (0.33),
... and so on forever (i.e. it's repeating exactly).

[wblipp]

Originally Posted by jinydu

Sure I learned long division. Not sure what you're trying to get at though.

They must not drill it much, though. Mini-Geek explained it - but I'll provide another set of words in case you are having difficulty still.

After you have used up all the digits in the dividend (the "a" value if we are calculating the decimal expansion of a/b), each step will be

1. bring down a zero
2. determine the next digit of the quotient
3. multiply the digit by the divisor (the "b" value)
4. subtract to get a new remainder.

If you ever get to a remainder that you have seen before, you will get the same digit in the quotient, and same product, and the same next-remainder as the last time - everything, including the quotient digits - will repeat the previous result.

But the remainder is always less than "b", so there are only b possible values - so you MUST see a remainder you have seen before in no more than b+1 steps.

A few thousand hand done long divisions - standard elementary school drill in the era before electronic calculators - would have made this all obvious. Except in Yorkshire, of course, where they didn't have hands so they to hold the pencils with their toes. At least the lucky ones that had toes.