It's Long Division

Arithmetic seems to terrorize a significant number us, students both young and adult. And nothing is more emblematic of that terror as long division.

The alternative rock band Fugazi captures the sense:

It's a long time coming,

It's a long way down,

It's long division,

Crack and divide.

There is nothing more daunting to students we have tutored than long division. It seems a special torture invented in Hell for their personal torment.  But it need not be so.

Arithmetic was invented largely to keep track of human trade and commerce. How else would one keep things straight? Purchased 200 wheels of cheese, sold 192, lost 8 to vermin, made a modest profit. Without arithmetic, it would be impossible to track.

These early traders weren’t schooled mathematicians, but they weren’t stupid. They learned how to mark sticks or make imprints on clay tablets to record numbers. They added and subtracted to determine sales and inventories and profits and losses. Arithmetic was not an abstraction but the means to a very important end.

At some point they realized that when a customer ordered five crates of clay pots, each crate containing 10 pots, there might be a quicker way to determine the total of pots needed than to add 10 together 5 times. Hence was eventually born multiplication, with a variety of  creative solutions: the Ancient Egyptian method, the Russian Peasant method, the Gelosia method. Human ingenuity knows no bounds when trying to avoid tedious labor. These methods required a few brain cells but were better than adding a column of the same number, over and over and over.

But, then, technology forced us to take many steps back.

Imagine in the 1940s, the slowest, most unwitting arithmetician among us. That would be an electromechanical computer. The computing machines of Alan Turing’s day (of “Imitation Game” fame) were notably obtuse. Huge, hot, electric sparking and fumes, they could add a column of numbers not that much faster than you could. They had but two fantastic advantages: absolute accuracy and immunity to fatigue.

Oddly enough, though, these electrical behemoths didn’t know how to subtract. Or multiply. Or divide. All they knew how to do was add.

So how did the machines of Turing’s era perform subtraction and multiplication and division when all they knew was addition?

Addition is a basic, primitive operation. If you are given two apples, then three more, you will have a total of five apples.

                2 + 3 = 5

Even the Common Core methods will agree with that.

But say you are given five apples, then two are taken away. We know the answer is three – we did the subtraction in our heads. But Turing’s machine can’t subtract. So, brilliantly, we instruct it to add five and the negative of two:

                5 + (-2)= 3

Still using only the basic operation of addition, the machine has succeeded in subtracting by simply adding the negative of the subtrahend (the number being subtracted). We can now say that the machine knows how to subtract:

                5 – 2 = 3

…but in fact, we know that it is adding the negative to do so.

How about multiplication? That is simple for the sparking behemoth as well. After all, it is tireless, and simply replicates the feat of early humans  before the many multiplication methods were invented. It merely adds, repeatedly. Five cases of ten clay pots is calculated thusly:

                10 + 10 +10 + 10 + 10 = 50

Multiplication is just repetitive addition. (With some fussing with decimal points, but we can ignore that for the time being).

Now for the promised key to long division. If multiplication is repeated addition, might you suppose that division is repeated subtraction? Yes, you might, and you would be absolutely correct.

Let’s try to divide 49 by 12 given this method. Keep track of how many times we can subtract 12 and stop only when the balance is not greater than 12.

                49 / 12

                49 – 12 = 37         (1^st subtraction)

                37 – 12 = 25         (2^nd subtraction)

                25 – 12 = 13         (3^rd subtraction)

                13 – 12 = 1           (4^th subtraction)

                1 left over

The answer is 4 with a remainder of 1. That’s division, using only subtraction. And remember that subtraction is just the addition of negatives. This is how Turing’s useful idiot, the sparking monster that could only add, was able to subtract and multiply and divide as well.

The moral of this story is that arithmetic is ancient. It is rooted in the real world, not an abstract implement of student torture.  And, hopefully, by realizing that a quite stupid computer can do this, you will have the confidence that you can, too.

Now, to those of you who found this all very simplistic, it’s time to volunteer as a tutor. Some yearning student needs your insights. This is how you give back.