20081001, 01:13  #1 
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Rational if and only if Decimal Expansion Repeats
It is a wellknown 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. Last fiddled with by jinydu on 20081001 at 01:13 
20081001, 01:38  #2 
"William"
May 2003
New Haven
936_{16} Posts 
Do they still teach long division by hand?
There are only a finite number of remainders, so they have to repeat eventually. 
20081001, 01:48  #3 
Dec 2003
Hopefully Near M48
1758_{10} Posts 
Sure I learned long division. Not sure what you're trying to get at though.

20081001, 02:12  #4  
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
10AB_{16} Posts 
Quote:
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). Last fiddled with by MiniGeek on 20081001 at 02:13 

20081001, 07:45  #5  
"William"
May 2003
New Haven
2·3^{2}·131 Posts 
Quote:
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 nextremainder 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. 

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