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/
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