Why Does Long Division Work?

Date: 12/22/2006 at 07:03:29
From: Peter
Subject: long division re-invention

Hi.  I've been looking everywhere and can't find a real explanation of 
the "long division" algorithm.  I can of course use this algorithm (I 
learned it looooong ago in elementary school) but what I'm asking you
is to "re-invent" this algorithm.  I find it very confusing to know
that it works but not know why.

Date: 12/22/2006 at 10:37:57
From: Doctor Peterson
Subject: Re: long division re-invention

Hi, Peter.

Here is an attempt (I'm not sure how successful) to explain why long 
division works to a kid:

  Long Division
    http://mathforum.org/library/drmath/view/58499.html 

I'll try to repeat the same example using algebraic notation rather 
than manipulatives (bundles of sticks).  Here's our problem:

      __18_
    4 ) 73
        4
        -
        33
        32
        --
         1

We're dividing 73, which is 7*10 + 3, by 4.  We first divide 7 by 4, 
giving

  73 = 7*10 + 3

     = (4*1 + 3)*10 + 3

Here I've used the basic idea of division, which is

  dividend = divisor * quotient + remainder
      7    =    4    *    1     +     3

Now we rearrange this so that we have the first digit of our quotient:

     = 4*(1*10) + (3*10 + 3)

Note that the remainder of that first division, the 3, is actually a 
number of tens, so I've combined it with the 3 in the dividend that 
we haven't touched yet.  That whole bundle still has to be divided by 
4; so do that, finding that 33 = 4*8 + 1. (The 8 is the quotient, and 
the 1 is the remainder.)  Put that into what we've already done:

     = 4*(1*10) + 4*8 + 1

We're almost done.  We just have to combine our two quotients:

     = 4*(1*10 + 8) + 1

There we have it: the quotient is 1*10 + 8 = 18, and the remainder is 
1.

If you have any further questions, feel free to write back.  I'm sure 
I can fill in details or extend this to more complicated problems if 
necessary.  Incidentally, if you've taken algebra you've probably seen 
long division of polynomials, which uses the same general algorithm, 
but without the need for "borrowing" between digits, so it's more 
straightforward.  Some of the details are easier to understand in that 
context.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/