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Long Division, Egyptian Division, Guessing

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Date: 06/13/2001 at 17:21:28
Subject: Different method for two-digit division

Is there another method for two-digit division? I don't care how
complicated it is (really). I can do fractions, decimals, negative
numbers a little. I just have a problem doing two-digit division. The
method we're taught in school is as follows:

28/180
1) Guess (STUPID) I guess 5
2) Check it    28x5 = 140
3) Too low, guess again. I guess 7 28x7 = 196
4) Too high, guess again. I guess 6   28x6 = 168
5) Find the remainder   180-168 = 12

Is there a better method that doesn't involve so much guessing?

Kiran
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Date: 06/14/2001 at 08:30:07
From: Doctor Peterson
Subject: Re: Different method for two-digit division

Hi, Kiran.

I don't know of a method (other than using a calculator, or doing it
in binary) that doesn't involve guessing. The best thing to do is to
improve your guessing and correcting skills.

But first, I should point you to the Egyptian method, which is
actually a form of binary division despite its ancient origins, and,
though slow and complicated, doesn't require any guessing:

Egyptian Division
http://mathforum.org/dr.math/problems/fogg6.23.98.html

1  28      180     68     4
2  56*   - 112*  - 56*  + 2
4 112*   -----   ----   ---
68     12     6
(rem) (quo)

What I did here was to double 28 repeatedly, keeping track in the left
column of what multiple of 28 each doubling represents. Then, starting
at the bottom of the list, I repeatedly subtracted each doubling that
fits, first 112 and then 56, and marked them (*) in the list. When the
remainder was less than 28, that was the remainder for the whole
problem, and the sum of the multipliers (4 and 2) corresponding to the
numbers I subtracted (112 and 56) was the quotient. No guessing,
little thinking - and a lot of work, especially for larger numbers
than this.

You can find many explanations of long division in our archives that

Three- and Four-Digit Long Division
http://mathforum.org/dr.math/problems/candace03.15.99.html

Now let's look at your example and see how we can make long division
work better:

>    28/180
> 1) Guess (STUPID) I guess 5
> 2) Check it    28x5 = 140
> 3) Too low, guess again. I guess 7 28x7 = 196
> 4) Too high, guess again. I guess 6   28x6 = 168
> 5) Find the remainder   180-168 = 12

First, we can try to guess better the first time. Did you just guess
randomly based on the general sizes of the numbers? In this case, I
would round 28 up to 30, which would go into 180 6 times. Since I
rounded the divisor up, I know the correct quotient will be at least
6; I doubt it will be more. So I would try 6 first. As you see, I
would be right.

It does take some skill and experience to get it right; and in fact
that is probably the greatest benefit of teaching you long division in
a world of calculators! It's good to get a feel for estimation,
because that's our best defense against blind trust of machines that
may be used incorrectly. The part of this that you hate, the
"guessing," is actually the most useful part, once you are no longer
forced to do long division on paper, but still need to make quick
checks occasionally.

I don't always make the right guess the first time myself; learning to
correct a wrong guess efficiently is an important part of the process.
Usually I am off by no more than one, so my next guess is almost sure
to be right. And in fact your guess turned out to be off by one.
Increasing it by 2 suggests that you have very little confidence in
your estimation skills (or are just trying to make the method look as

But could we have seen ahead of time that your next guess should be 6?
Yes. When you found that 5 was too low, it was because 180-140 = 40,
which is larger than 28. But how much larger is it? Since 40-28 = 12,
if we add only 1 to the quotient, the new remainder will be less than
the divisor. Do you see that? The remainder left by your wrong guess
tells you exactly how much to increase the guess by. So once you have
made a wrong guess, you know enough to make a right guess the next
time.

Let me go through that again. You tried 5, and found that 180-5*28
gives a remainder of 40. Now we divide that by 28, and see that this
gives 1 with a remainder of 12. (We don't really have to do another
whole division. Since the answer will almost always be 1, you just try
subtracting 1 and see if that works.) This means we can add 1 to the
quotient, and the new remainder will be 12.

Now suppose you had first guessed too high, say 8. Then 8*28 = 224,
which is too big. How much too big? 224-180 = 44. Subtract 28 from
this and we still have 16. This means that if I had guessed one less,
7, I still would have been too big by 16; so I have to subtract 2 from
my guess. I try 6, and find it's right. I'm not surprised, because I
knew it would work.

Put all this together, and you find a more positive way of looking at
long division. Don't think of it as a series of wrong guesses, but as
a series of successive approximations. Each attempt you make leads you
step doesn't have to be a guess. It's not a hunt-and-peck system, but
a guidance system, leading you unerringly toward the goal. Making
better guesses (I should say estimates) just means you take fewer
steps.

By the way, the Egyptian method takes a lot more steps; by using your
mind (intelligent "guessing") rather than a purely mechanical method,
you save time. That's how math usually works, and it's why long
division took over from earlier methods.

I hope this helps a bit. I've actually made a couple of discoveries in
writing up how I think, because I'd never quite put it this way!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
Elementary Division

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