Long Division

AllPages RecentChanges Links to this page Edit this page Search Entry portal Advice For New Users

• Is the product of 4 consecutive
  positive integers always one
  less than a square?

• Yes.
  If n is the second integer, the square root
  is $n^2+n-1.$ Interestingly, I used the algebraic
  version of square root by long division.
• Show me your work will you? I don't think
  I know this algebraic square root business.

So let's start with the division problem, computing 999999 divided by 7.

Modern teaching doesn't use long division any more, apparently. Instead they use a thing called "chunking". In fact, "chunking" is nothing more that long division done with more guesses and less formality.

1 +------------- 7 | 9 9 9 9 9 9 7 --- 2 9 9 9 9 9

So we start by ignoring everything except the most significant part. We ask how many 7s we can take out from the first digit. The answer is just one. So we have

    999999 = 700000 + 299999

That table then says that

 9999999 / 7 = 100000 + 299999/7 

We now iterated. We can remove 4 chunks of 7 from 29, so that's removing 40000 lots of 7 from 299999. Again, we lay that out in the tabular fashion:

1 4 +------------- 7 | 9 9 9 9 9 9 7 --- 2 9 9 9 9 9 2 8 ----- 1 9 9 9 9

So we are, at each stage, deciding what the next digit should be, multiply by 7, subtract from the remaining amount, and then repeat. This is laid out in a formal, tabular fashion. At each stage we have an intermediate sum that makes sense.

        1 4 2 8 . .
      +-------------
    7 | 9 9 9 9 9 9
        7
       ---
        2 9 9 9 9 9
        2 8
       -----
          1 9 9 9 9
          1 4
         -----
            5 9 9 9
            5 6
           -----
              3 9 9

This can be used for polynomials too. So, for example:

                           x^2 - 2x + 2
            +---------------------------------
x^2 + x - 3 | x^4 - x^3 - 3x^2 + 8x - 5
              x^4 + x^3 - 3x^2
             ------------------
                  -2x^3        + 8x
                  -2x^3 - 2x^2 + 6x
                 -------------------
                        + 2x^2 + 2x - 5
                          2x^2 + 2x - 6
                     -------------------
                                      1

So we can perform long division with polynomials in exactly the same way as we can perform long division with numbers, and in effect, it's just a formalised way of doing "chunking."

So, on to Square Roots By Long Division ...