Division, Not Long Division

Division is the most powerful arithmetic operation. It makes comparisons. When the numerator and denominator have the same units, the comparison makes a dimensionless number, the only kind that the universe cares about. Long division, however, is something else entirely. In my post “Dump algebra,” many commentators objected to my loathing of long division. But long division is not division! Long division is just one way to do the computation, and is far from the most useful way.

I’ll illustrate with an actual example of division. For my environmental-protection lawsuit, now in the Massachusetts Supreme Court, I needed to divide 142,500 by 4655. Here is the long-division calculation, my first use of the method in 30 years:

The calculation took me a few minutes with paper and pencil, some of the time to reconstruct the algorithm details and to get the bookkeeping straight — even though I already knew the answer quite accurately.

I knew the answer because I had already applied a more enjoyable method: skillful lying. I turned the numerator 142,500 into the nearby and convenient 150,000; and turned the denominator 4655 into the nearby and convenient 5000. Dividing 150,000 by 5000 gives 30. It’s likely to be an accurate estimate, because the two errors (increasing the numerator and increasing the denominator) partly compensate.

The next correction is not too hard, and comes from estimating the two errors. The actual numerator of 142,500 is 7500 less than 150,000, a decrease of 5 percent. To fix this error, decrease the estimate by 5 percent. The actual denominator of 4655 is 345 less than 5000, a decrease of, let’s say, 7 percent (7 percent of 5000 is 350). To fix this error, increase the estimate by approximately 7 percent. The two fixes together require increasing the estimate by 2 percent (7 percent minus 5 percent). So, 30 becomes 30.60—which is very close to the actual quotient of 30.6122…

This whole calculation took about 10 seconds in my head. There’s no need for long division, and I hope that I live another 30 years without using it again.

For students, learning long division mostly means learning like a parrot. A classic example is from the National Assessment of Educational Progress (NAEP) results reported in 1983 (Carpenter, T., et. al., “Results of the Third NAEP Mathematics Assessment: Secondary School,” The Mathematics Teacher, 76:652-659). Thirteen-year-olds across the country were asked:

An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed?

70 percent of the students did the long division correctly (the result of 1128/36 is 31 and 1/3). From doing the division correctly, the most popular answer, chosen by 29 percent, was the meaningless 31 R 12 (31 with a remainder of 12) buses. Another 23 percent chose 31 buses, leaving 12 soldiers stranded. Only 18 percent chose the correct answer of 32 buses. Even then it’s not clear how many of the 18 percent were sure of their answer or were just guessing between 31 and 32.

Here is a flow diagram illustrating the answer distribution:

It’s easy to learn long division yet understand little.

 

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  1. Charles says:

    I taught math in junior and senior high for 32 years, retiring in 1997. I taught basic math up to and including Calculus in that time.

    It’s my opinion that the number one thing which has destroyed mathematics education is the calculator. Theoretically, it can be useful, but it’s not used the right way… all it’s used for is to do simple arithmetic that should be learned in elementary school. Too many kids, even in calculus, can’t do simple arithmetic, but rely on the almight crutch – the calculator.

    I taught my calculus students without a calculator; the other teachers taught the calculator. EVERY year, on the departmental final exam, my students averaged 10 points higher than the other classes. The only difference was the absence of calculators. The other teachers were equally capable with the material, if not more capable.

    Take the calculator out of school…. learn the tables at an early age… learn the algorithms.. KNOW HOW to get the answer by yourself…

    Give a class a basic multiplication of two 5-digit whole numbers using the four-function calculator and most will get it wrong… virtually all of them. If they learned the multiplication table and multiplication algorithm, they’d do much better.

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    • Sanjoy Mahajan says:

      I agree about calculators. I never used one in school myself. My favorite solution these days is the QAMA calculator (see my Freakonomics post about it), which forces the user to estimate first before it will give an answer!

      -Sanjoy

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  2. Phillip Bryan says:

    Look up the topic of ‘Euclidean Domain’ (an object of study in advanced algebra.) You will learn that the ‘loathed’ long division algorithm is a form of the ‘Euclidean Algorithm’ the existence of which gives the name to the object of study, and results in a rich body of interesting and useful results, both in Mathematics, itself, and in applications.

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  3. Phillip Bryan says:

    Have you tried to program your method? Long division is easy to program.

    After one example of what happens when one neglects a remainder, the students will know to include them for the next problems. This error also provides good motivation for introducing the greatest integer function, so we can make a ‘silk purse’ out of this ‘cow’s ear.’

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