Devlin's Angle

Base Considerations

As a child growing up in England in the 1950s, a large part of my early schooling was spent learning various archaic systems of measurement. Not content with a monetary system that mixed base 12 and base 20 arithmetic, my ancestors had bequeathed me and my compatriots all manner of number systems for measuring land area, liquids, grain, and whatever. Among those number systems was a delightful one from the 13th Century English wine and brewing trade. Largely on account of the wonderfully evocative terminology, I can remember it to this day. It goes like this.

2 gills = 1 chopin, 2 chopins = 1 pint, 2 pints = 1 quart,
2 quarts = 1 pottle, 2 pottles = 1 gallon, 2 gallons = 1 peck,
2 pecks = 1 demibushel, 2 demibushels = 1 bushel or firkin,
2 firkins = 1 kilderkin, 2 kilderkins = 1 barrel, 2 barrels = 1 hogshead,
2 hogsheads = 1 pipe, 2 pipes = 1 tun.

Of course, the only hard part about learning this system was remembering the order of the words. The number system itself is good old binary arithmetic, these days found all over the world etched into wafers of silicon.

We are so used to computers nowadays that it seems obvious that computer arithmetic should be performed in a binary fashion. After all, this is the most natural form for a computer, which is ultimately, a "two-state" machine (the current in a circuit may be either on or off, an electrical "gate" may be either open or closed, etc.). But this was not always the case.

When the first American high-speed (Hah! That was then.) electric computers were developed in the early 1940s, they used decimal arithmetic, just as did their inventors. But in 1946, John von Neumann suggested that it would be better to use the binary system of arithmetic, since which time binary computers have been the norm. (Not that this was the first time that calculating machines made use of the binary system. Some French machines using binary arithmetic were developed during the early 1930s, as did some early electric computers designed in the United States by John Atanasoff and by George Stibitz, and in Germany by Konrad Zuse.)

There is, of course, nothing special about the decimal number system we use every day. Certainly it was convenient in the days when people performed calculations using their fingers. Assuming a full complement of same, it is essential that there is a "carry" when we get to ten. And that's all there is to it. (As the satirical performer and mathematician Tom Lehrer once remarked, base-8 arithmetic is just like ordinary arithmetic-if you are missing two fingers.)

Though binary arithmetic is ideally suited to the electronic computer, other kinds of computing device have used other methods. For example, anyone old enough to remember those mechanical calculating machines that used to fill offices in the 1950s and 1960s will recall that they worked in a finite decimal arithmetic. Though such machines performed subtraction, all numbers represented on the calculator were 'positive'. If you subtracted 1 from 0, for example, the machine would display an entire row of 9's. This is because, as far as the machine was concerned, this really was -1. If you added 1 to a full string of 9's, you would get a carry all the way along the number, and off the left hand end, leaving zero; i.e. (-1) + 1 = 0.

A particularly intriguing possibility is to use a negative base in the number representation. For instance, you could represent your numbers in the negadecimal system, where the base is -10. In this system, the number 211 is equal to the decimal number

1 + (1 x -10) + (2 x -10 x -10),

i.e. 191. Again, 35 in negadecimal is the same as the decimal number

5 + (3 x -10) = -25.

So in negadecimal, negative numbers (like -25) can be represented without negative signs being necessary. In fact, any number can be written out in negadecimal notation, and regardless of whether the decimal number is positive or negative, no sign is necessary in negadecimal.

Playing with arithmetic having negative bases is an amusing classroom exercise, but, strange as it may seem, a computer was once built that used "-2" base arithmetic. It was the UMC-1, a Polish-made computer of the late 1950s and early 1960s, of which several dozen were made and installed.

These days, of course, it seems that only the banks have computers that use negative base arithmetic-or so it appears from the statements my bank keeps sending me. But that is another story.

All of which makes an excellent setting for the following puzzle. Fill in the next two members of the following sequence: 10, 11, 12, 13, 14, 15, 16, 17, 20, 22, 24, 31, 100, -, -.

-Keith Devlin

Devlin's Angle is updated at the start of each month. The above article is adapted from Devlin's book All The Math That's Fit To Print, published by the MAA in 1994.