# 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.

## ajb8

If you really want kids to learn you can't get bogged down with cheap tricks, you have to teach the true foundations of mathematics. Replacing long division with this "division by estimation" is an insult to pure math.

"We have to reinvent the wheel every once in a while, not because we need a lot of wheels; but because we need a lot of inventors."

## jagiela

Years ago there was this nifty invention called a calculator that would do all these computations for you. Then years later, they came up with web access for your cell phone and now your phone can do all this for you. You don't even need to guess anymore.

There is just so much worthless education out there: algebra, multiplication tables, cursive writing. Its all so silly but it keeps teachers employed

## Drloko

So you discovered estimation? Great. Next, let's just set pi to 3 and the square root of two to 1.5. The point of long division is to provide a method to get the exact answer. Because to some people it's important to have the correct answer, even if that means keeping track of all if those tedious decimal places.

## Not A Math Nerd

Wasn't that same "rough guess" stuff used on one of the landers send to Mars a few years ago - you know, one of the ones that went SPLAT. Well, I guess not - that was due to unclear communications and inaccuracies in units. However, it is that type of rounding/estimations that causes errors that can create damaging situations.

I can appreciate your use of rounding and estimations, and how they can be more accurate than leaving 12 soldiers with an extra 20 mile hike to reach their training site. But the students do need to learn long division - along with logic to use that long division wisely.

Please do not bemoan the use of a tool, however tedious (and long division can be tedious) because folks haven't been taught the logic of HOW to use it properly.

## Sanjoy Mahajan

The Mars Climate Orbiter crashed because the engineers spoke in different languages (units): "The peer review preliminary findings indicate that one team used English units (e.g., inches, feet and pounds) while the other used metric units for a key spacecraft operation. This information was critical to the maneuvers required to place the spacecraft in the proper Mars orbit. " See http://mars.jpl.nasa.gov/msp98/news/mco990930.html

Extra accuracy would't have fixed this problem. But if the engineers had been really lucky, a really inaccurate calculation could have canceled out the error from ignoring the differing units!

## Gwen Nicodemus

I love seeing articles that explain different ways to do things. My son is dyslexic. He can't work out a 2-digit multiplication or division problem on paper, but if I ask him verbally, he generally gets the answer. Or really close. He has his own "cheats" like this, but he can't quite articulate them to me.

As as far as the 10-second thing, I can believe that. I've watched my kid do it. I've watched my husband do it. Actually, I can do it too.

## Holli

Long division becomes useful later when learning how to divide polynomials, especially ones that have remainders. You have to learn the process in the simpler realm of numbers first. Besides that...how many of these kids can do percents, much less in their heads? Or know how to compensate for errors? Kids are more concrete thinkers, not abstract. They need to do it on paper, not in their heads! And the example you give about the buses is not an error in long division, but applying it logically. It's more understanding the concept, than the actual computation. Even using your method, they might not have gotten it right...

## Charles

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.

## Sanjoy Mahajan

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

## Phillip Bryan

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.

## Phillip Bryan

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.'