Long Division, Egyptian Division, Guessing

Date: 06/13/2001 at 17:21:28
From: Kiran Sonnad
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
6) Answer    168R12

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

Kiran

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   

Ignore the part about fractions. Here's your problem worked this way:

    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 
may help you; here's one:

   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
> 6) Answer    168R12

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 
bad as possible).

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 
closer to the final answer, giving you more information so the next 
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/