As I write this, in 2003 A.D., students in the third or fourth grades in school are still being taught long division, although in the higher grades most students obtain quotients using calculators.
In teaching students long division the teacher would do a few examples and turn the student over to the computer which would drill the student in how long division works - but not why it works.
There is a reason why teachers or computers seldom explain why long division works. You see, long division is really the Division Algorithm of the ancient Greeks. That is, given two integers a and d, where d IS POSITIVE, there exist unique integers q and r such that
a = qd + r, 0≤ r < d. To show the student why long division works, you could use the calculator to verify the results. But it still does not constitute proof. One proves that long division works by showing that the result it obtains is absolutely the correct value for every possible dividend-divisor pair.
This is done using the Division Algorithm formula. Now, it is not difficult to convince a grade 4 student about the Division Algorithm formula.
However, even adults with a few years of university mathematics training would have to strain to write down a proof of it.
What is long division? It is a division of one number by another in the long way that "brings down" each digit of the dividend.
For instance, 12345/12 is accomplished thus:
In the first step above, we asked for the largest multiple of 12 that would go into 1. The answer is 0. We place 0 in the quotient row. And place 12×0 = 0 under the digit 1 of the dividend.
We subtract 0 from 1 and "bring down" the next digit which is 2. We ask for the largest multiple of 12 again, and so on.
The long division above gives the answer
12345 = 12(1028) + 9
or 12345/12 = 1028 + 9/12
This is a particular example of long division. Now we consider the general procedure of long division.
Suppose we wish to divide an integer a by a positive integer d.
Let a = anan-1...a1a0 in decimal representation.
Since a has n+1 digits, the long division will have n+1 steps as each digit of the dividend generates exactly one step and gives exactly one digit of the quotient.
At step i, we do this:
Look for the largest integer qi so that d × qi does not exceed Bi, where Bi is defined as follows:
After we reach i = 0, what remains is r, the remainder.
Bn = an and
Bi = 10(Bi+1 - d × qi+1) + ai, 0 ≤ i < n.
Write qi to the right of qi+1in the quotient row.
In the particular example of 12345/12, we stopped with the remainder which was 9. However, we could continue, by writing 12345 as 12345.0000... Then we bring down the digit 0, place a decimal point in the quotient row, and then look for the largest multiple of 12 that will go into 90, and so on.
Why Long Division works
In the long division procedure, the dividend must equal the sum of the remainder and all the numbers that have been subtracted.
But the numbers subracted are d×qi with place value 10i. So
a = (d × qn)10n +(d × qn-1)10n-1 + ... + (d × q1)101 + (d × q0) + r, where 0 ≤ r < d
= d × (qn10n + qn-110n-1 + ... + q1101 + q0) + r
= d × qnqn-1...q1q0 + r
= d × q + r, 0 ≤ r < d.
Enter the Division Algorithm!
According to it, q and r must be unique. That is, the q we have found in the long division is indeed the one and only value possible, namely the quotient of a when divided by d.
Now that we know why long division works, it is easy to extend to dividends that are not integers.
Suppose a = 758.9 and d =5. Then a/5 = (1/10)(7589/5) so that we carry out the long division involving two integers and then divide the answer by 10 which is accomplished by moving the decimal point left.
Finally, noting that the Division Algorithm is valid in any base, we can extend these arguments to any base just as well for base 10.
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Last updated - November 8, 2003