Long Division

You probably remember how to do long division. Maybe you’re a little vague about it because, after all, you rarely have to solve a long division problem and, anyway, if you had to do one now you’d use a calculator. But if you completed your elementary school education you were taught how to do long division using a pencil and a piece of paper. Most likely you weren’t taught why it worked. Besides, it was hard enough to memorize the method.

Of course you know how to do long division. But just to refresh your memory, here’s how:

First, you write the number you want to divide and then you draw a horizontal line close above it and an a vertical line (somewhat curved so you won’t mistake it for the numeral 1) down from the left end of the horizontal line. Now the big number is snug inside there, under the roof. Next, you write the number you want to divide by to the left of that last line. The layout should look like this.

After you’ve arranged the numbers that way, you do a complicated sequence of calculations which combine guesswork with multiplication and subtraction, and you keep jotting down the result of those calculations in a specific location beneath the number you’re dividing. That sequence of moves is called an algorithm.

In primary school you were taught the algorithm, told to memorize it, and you did. But it’s very unlikely you were told why this odd rigmarole method of calculating numbers and writing them in precise locations produces the right answer. Furthermore, you weren’t taught long division until you had learned how to add or subtract or multiply big numbers, and when you did those calculations you always began on the right-hand side and worked toward the left. But in long division you were told to begin on the left and work toward the right. And no one said why.

When you divide a big number by a smaller number, you’re slicing the bigger number into pieces that are exactly the same size as the smaller number. The answer to any long division problem is how many of those identical small pieces it takes to make the big number.

Now, the simplest way to find how many of the smaller quantity are in a bigger one is to subtract the smaller number from the bigger one, and keep on subtracting again and again until the bigger one is gone. Then you can count how many times you did the subtraction and that’s the answer. It’s a simple brute-force method, guaranteed to be accurate but lethal to your brain. You’d have to make over two hundred and fifty subtractions to solve the problem we have here. Fortunately, Henry Briggs invented his clever long-division algorithm and gave it to us, free for nothing. Here’s what he looked like in formal attire.

Briggs’ method uses something similar to the brute-force strategy mentioned earlier, he subtracts the smaller quantity from the larger one until there’s nothing left of it. But instead of subtracting one small quantity over and over again, he goes to the side of the number with the largest mathematical value –- the left side –- and subtracts a big chunk of small quantities in one big bight from that side. In fact, using the Briggs algorithm it takes only three subtractions to reduce the big 3514 to zero. We all owe Briggs a big Thank You for saving us a huge amount of time and bother.

The reason you have to write down your division problem in this odd way — the divisor to the left of the big number and the calculations moving from left to right under the dividend –- the reason is because the number system we use assigns values to the numerals according to the position they take in line. For example, in the number 1775 the numeral 7 which occupies the second place from the right has the value of 70, while the 7 which occupies the third position from the right has a value of 700. Another way of describing this is to say that the place on the right-hand end is for ones and the next place on the left is for the tens, and to the left of that is the place for the hundreds, and so it goes to infinity as the number grows leftward.

In our example, the number we’re dividing into small identical pieces is composed of four parts. The first and biggest part is that huge group of 3 thousands at the left end. The next big bunch are those 5 hundreds, and after that comes just 1 ten and a four. Now when you look at 3,514 you can see it as three thousand, five hundred and fourteen, or you can see it as thirty-five hundred and fourteen. Most of us feel daunted when trying to guess how many times 14 goes into 3,514, but by using the algorithm we break the big number into its parts and everything goes easier and faster.

We begin with those 3 thousands, and take only the numeral 3 and forget about the zeros which are merely place holders for the hundreds and tens and that little four which we can deal with later. Of course, 14 can not go into 3, so we give up on the 3 thousands and add them to the hundreds so we’re dealing with all 3500 of them. Again, we can forget about those zeros and just look at the 35.

By the way, the image below is our hard-working Henry Briggs in less formal garb.

Probably your elementary school teacher didn’t tell you these things when teaching you how to do long division. Instead, you were told simply to try dividing the first numeral in the dividend by the divisor, and if it didn’t go, take the first two numbers of the dividend and again figure how many times the divisor could go into that. If you found a good number that worked, then you wrote it on the horizontal line precisely above the second numeral in the dividend.

All right, obeying your teacher and following the algorithm, we figure that 14 can go into 35 twice with something left over, so we write the numeral 2 on the line above the 5, multiply 14 by 2 and write the answer, 28, beneath the 35. And, privately, we know that the 35 stands for 35 hundreds and 28 stands for 28 hundreds. Then we subtract 28 from 35 and get 7. In other words, we’ve made our first subtraction and only 700 are left of the 3500 we started with. This is amazing progress!

Yes, there are some leftovers, the 35 hundreds also had 14 glued to the right end, which is to say it had a 10 and a four. So we copy and 1 and write it to the right of the 7, making 71. Then with a little thought we figure that 5 times 14 fits into 71, so we write the numeral 5 on the line above the big number, do the multiplication and write 70 beneath the 71.  Unlike those young kids doing this for the first time, we know that we’re subtracting 700 from 710. That leaves 10 and the solitary 4. That’s, yes, 14, so we write 1 on the top line and now have 251. That’s how many sets of 14 you can get in a group of 3,514.

Now you know how the long-division algorithm works and why it gives you the right answer. And you also know why elementary school teachers don’t try to explain it to their pupils.

At this point you might find some relief in the songs and satires by Tom Lehrer, a mathematician with a talent at comedy. In particular, you might be interested in his song “The New Math” which goes crazy explaining how to subtract by using the conventional subtraction algorithm that we all learned around third or fourth grade. Lehrer wrote the song at a time when some primary schools experimented with trying to teach kids not only how to do mathematics, but also how the number system works.

Like the Briggs, the 17 th century mathematician who gave us his long-division algorithm free, for nothing, Tom Lehrer gave away his copyrights to all his songs and lyrics and they’re available, like Briggs’ algorithm, free for nothing. Here it is: