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.

War and Secrets

Allied freighter attacked and sunk by German U-Boat

From the German point of view, it was the Second Happy Time. During 1942, the Axis sank 609 Allied ships totaling over 3 million tons at a cost of only 22 U-boats. It was, indeed, a happy time for the Germans as they focused their attention on the east coast of North America. Lack of preparation, no blackout of American cities, and an underequipped Atlantic U.S. Navy led to enormous German success.

(The First Happy Time was in 1940 and 1941, as Germany took on the unprepared British Royal Navy and the convoys they protected).

There was one other key to German dominance of the shipping lanes: Engima. Enigma was a German cipher machine used to encrypt messages. Enemy military commanders were able to gather intelligence and send orders with little chance of eavesdropping by the Allies. Convoys were discovered and submarines ordered to intercept in complete secrecy. The toll on Allied shipping was appalling.

The British established a top-secret effort at Bletchley Park to take on the challenge of breaking Enigma. To complicate matters, there were several variations of the machine in varying degree of sophistication. With the able assistance of Polish mathematicians, the British began to gain some success in decoding Luftwaffe and German Army messages. But the German Naval Enigma was a substantially different machine and defied cryptanalysis. Our convoys were sitting ducks.

And then fortune smiled. On October 30, 1942, the British Air Force spotted the German submarine U-559 on the surface off the coast of Egypt.  The airplane summoned the destroyer HMS Hero which closed on the submarine and forced it to submerge. Other destroyers joined the hunt and U-559 was severely damaged by depth charges. Losing trim, she came to the surface and was boarded by British sailors who retrieved her Enigma machine and code books.

Meanwhile, brilliant mathematician Alan Turing, who had been working on code breaking at Bletchley Park, turned his attention to the Naval Enigma. (Alan Turing is the father of our digital computer – thank him as you use your laptop computer or iPhone). Turing was finally able to break the Naval Enigma code using his deep experience at Bletchley and the newly obtained U-559 materials. The Second Happy Time came to an abrupt end for the Germans, and the sealift of men and materiel accelerated as we prepared for the invasion of the continent.

The Bletchley Park operation was very complex. It involved spies in the field, the best mathematical minds, radio listening stations, and science-fiction room-sized calculating machines like Colossus. And all of this was top secret and continued to be so long after the war had ended. All who were involved were required to sign the Official Secrets Act and vow to remain silent. And this secrecy was vital to ensure that the Germans did not know that their codes had been broken. The lives of millions and the very outcome of the war were at stake.

In thanks, the British recognized Turing by awarding him the Order of the British Empire (OBE) in 1945, then convicted him in 1952 of the criminal act of homosexuality. Turing kept his silence and committed suicide two years later. (Last month, over 60 years too late, the Queen issued a Royal Pardon “in fitting tribute to an exceptional man”.)

Which is why it is so incongruous that Edward Snowden has hundreds of thousands if not millions of fans.

Snowden revealed far more than the NSA collection of telephone metadata. A Pentagon report just sent to Congress asserts that most of the documents Snowden took relate to military operations. According to House Intelligence Chairman Mike Rogers, "The vast majority of the material was related to the Defense Department, and our military services," not NSA operations.

You might think, “So what? We’re not in a real war.” And you would be wrong. Our struggle against the virus of radical Islamic jihad is no less grave than those earlier anti-submarine patrols on the dark, storm-tossed Atlantic.

The lives of our citizens and the lives of our troops are now less secure thanks to Mr. Snowden. You can be a fan if you like. But Alan Turing, who sacrificed much and contributed even more, would most likely disagree.