Long division is a collapsed version of the Euclidean Division algorithm. First, let’s remind ourselves what the Euclidean Algorithm is:
|Euclidean Division Algorithm: If m and n are integers and if n > 0, then there are unique integers q and r such that m = nq + r and|
Another way to write this is to say that
Let’s see how you use this to find the decimal representation of any rational number. I will illustrate the process by finding the decimal representation of
This first application gives us 7, the integer part of this decimal. Each succeeding application uses 10 times the remainder from the previous step. Here are the first 6 lines, which yield a quotient of 7.238095 and a remainder of
There are only 20 possible non-zero remainders when dividing by 21 so the cycle of the quotients that begins 238095… must repeat after at most 20 steps. In fact, since , the cycle repeats after 6 steps.
Now compare each line with the steps of the calculation from Long Division below. The final remainder 5 is equal to a remainder six steps earlier, so the cycle of quotients, 238095, will be repeated if the long division is continued.