A Swarm of Bases

Looking at Mayan ways of herding numbers, where base-20 numeration (the vigesimal) is dominant, with one exception, I was reminded that ancient cultures, going far back into the BC regions, also liked working with numbers other than the decimal. The Mayans used positional notation for their numbers but, unlike us, who move from right to left as numbers increase, they moved from top to bottom. At each position they showed multiples of 20 except in the third place. There they used an 18-base structure (octodemical). Their numeration came from the construction of calendars. The first position (topmost) included units, the second weeks (20 days), and the third years. If they had been consistently vigesimal, they would have derived a year by multiplying 1 week of 20 days by 20. That would have yielded a 400-day year—a little too long for comfort. Therefore they used 18-base numeration in that position to get a year of 360 days. Thereafter, each successive period was 20 times the earlier. The Mayans also had a symbol for zero; it had the shape of an eye.

During Babylonian times (around 1800-1600 BC), base-60 numeration (sexagesimal) was common—whereas the Egyptians used the decimal system. Use of the sexagesimal seems to have been motivated by the problem of fractions, known to plague all elementary grade children in decimal eras. Sixty is evenly divisible into halves, thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, twentieths, and thirtieths fractions, whereas 10 is divisible cleanly only into halves and fifths.

Now, of course, we also use a sexagesimal system daily—if not hourly—without having our eyes glaze up: the sixty seconds to the minute, 60 minutes to the hour, the 360 degrees of the circle, which is 6 x 60. And as we know instinctively that 15 minutes are a quarter hour, so we also know that 90° are a quarter of a circle. Our longer measurements of time are also derivable from 60—as in hour, and day—but when it comes to a year, Mother Nature shakes off our precision and insists on pesky fractions.

The Babylonians, unlike the Mayans, applied their numbers to geometry and mathematics, not to calendars. They stuck to a lunar calendar which is something of a nightmare to describe.

Modern people all over the world also daily use other bases without ever thinking about them. Two that are hidden—but the nerds know—are base-2 (binary), the core language of computers, and built on top of it the base-16 (hexadecimal), of which a half is the 8-bit byte. Some folks, stuck with very old computers, may also be served by 12-base (duodecimal) numbers.

The duodecimal is under our linear measurements (12 inches to the foot); hexadecimal hides in weights (a dram is 1/16 of an ounce, an ounce 1/16 of a pound), and in volumes (an ounce is 1/16 of a pint, the quart and gallon are multiples of pints). And so on. Note that bases other than 10 all have more clean fraction than the decimal. Sexagesimal wins, with ten such. The 12-, 18-, and 20-base have four, the 16-base has three.

So what is the point of all this? The point is that, before modern calculators, there was motivation (to use a word the nerds are fond of) for using all kinds of bases for ease of calculation—and habit has preserved those practices in ordinary life. And the final point is to praise the decimal point—which has freed us of calculating awkward fractions in the old way in an age that is becoming ever more fractionalized.

—————————

My source here is A History of Mathematics by Carl B. Boyer, revised by Uta C. Merzbach, John Wiley & Sons, on the Babylonians p. 25, on the Mayans p. 213.

The Scale’s the Thing

In a way it pleases me that some nations drive on the left, some on the right, that some homes hang the toilet roll so that the sheet comes from the back, others so it unfurls up front. It pleases me that the metric hasn’t conquered all and that the inch still survives. Vive la difference.

Yesterday I quoted a bit of verse using the word milliard; Brigitte’s comment pointed out that we don’t use that number here. It is European. She wondered about the difference.

Well, according to the French, who claim to have invented both ways of naming numbers, it has to do with the scale that you use. There is a long scale and a short. The long scale is used in most of Europe, the short scale in  the United States, Britain, and most English-Speaking countries, but the British made the switch from long to short relatively recently (1974), when Prime Minister Harold Wilson told parliament that in government statistics billion, henceforth, would mean 10⁹, as is common in our usage, rather than 10¹² as had been the case in Britain theretofore.

The long scale is the older of the two and was formulated by the French mathematician Nicolas Chuchet (approx. 1445-1488). The short scale was developed in the seventeenth century by  French mathematicians bent on reform and became very influential by the nineteenth. In the long scale, as shown below, actual names are attached, after the million, only to numbers that are multiple of a million (6 zeroes)—the only exception being the first in the series. The short scale has names for multiples for each thousand of millions. Here the two schemes.

Long Scale	Short Scale	No. of Zeroes
million	million	6
milliard	billion	9
billion	trillion	12
thousand billion	quadrillion	15
trillion	quintillion	18
thousand trillion	sextillion	21
quadrillion	septillion	24
thousand quadrillion	octillion	27
quintillion	nonillion	30
thousand quintillion	decillion	33
sextillion	undecillion	36
thousand sextillion	duodecillion	39
septillion	tredecillion	42

The long scale is using the Latin for two, three, four, etc. to designate multiplications by millions; thus a million x million is a billion, a million x million x million is a trillion, and so on. The names thus indicate how many times to multiply a million’s six zeroes to get the size of the number: billion is 2 x 6, trillion is 3 x 6 zeroes. Numbers that fall between these don’t have names of their own. The short scale advances by multiplying the previous number in the scale by 1,000. The names, in effect, indicate how far down the short scale that you are. A way to use them is to exploit the multiplicand, 1,000, with its three zeroes. We use the name to multiply 3—then add 3 to get the number of zeroes. Thus, sextillion: 6 x 3 = 18 + 3 = 21 zeroes or 10²¹.

The long scale is odd only because its first number after million (milliard) is only a multiple of 1,000; all others are multiples of millions. The short scale is more consistent, always advancing by three zeroes, but, unfortunately, it uses the old names from the long scale. Thus octillion, in that system, has 27 zeroes, and what exactly does that have to do with eight?

Thirteen

Numerology, particularly the meaning of the number 13 and its rotation (31), expansions (49, 94, 58, 85, 67, 76—numbers the digits of which you must add), and its multiples (26, 39, etc.) give me the same innocent pleasure as do meaningful coincidences, sometimes called serendipities. A genuinely original and insightful modern parapsychologist, J.E. Kennedy, has suggested that the ultimate purpose of the phenomena he studies may serve as reminders to us that something transcending the obvious may simply be there. I’m putting this in rather too simple a summary. My object is not to discuss the insight, which is more sophisticated. To learn more about Kennedy’s thoughts, I recommend perusal of his papers, accessible here. What I want to talk about, lightly, as it were, is the number 13 itself. Contrarian as it is—being viewed as unlucky—in my family it has been a portent of good fortune, in the last generation as in mine, and seemingly also in the next one down in time.

This is an innocent preoccupation because like everyone else I’m a complex being and fully possess the kind of rationality which can bluntly, brusquely, and dismissively say: “Give me any other number, and if I know enough details about your life, I could construct the same castles-in-air around that number too! In your life or anyone else’s.” Having put that in, just to stay on the safe ground of modern meaninglessness, let me enjoy my 13 for a moment.

I was born on July the 31st which is obviously very meaningful—after all my birth date, turned around, is 13! We’re presently residing in the 13th Congressional District of Michigan—to be sure I don’t know the numbers of the others where we’ve lived. But there’s a whole string of thirteens that mark our coming to America. The Certificate of Eligibility issued to us in Germany permitting our entry to the United States was dated January 13, 1949. The 49 is significant too. And in my case 113 (thus January 13) is doubly meaningful because my name, ARSEN, decoded numerologically is 5, and 113, magically, collapses into five. And 1949, if you drop the first nine but not the second, also results in 113. Get it? Good. Let us move on. We spent precisely 13 days at sea making the passage from Bremerhaven to New Orleans. Our eyes beheld the American coast on June 13 of 1951 (ignore the year). Then, years later, when Brigitte, Barbara, Monique, and I came to America together, by air this time, we landed in New York on January 13, 1961. Now, finally, to the intriguing image that illustrates this post. It is the shoulder patch, incorporating the coat of arms, of the 13th Infantry Regiment—the military unit in which I served my first several months in the Army. We were “First at Vicksburg,” one of the bloodiest and nastiest events in the history of the Civil War, alas. That number was not quite so lucky for the many who died there. But then a deeper lesson of numerology—hear me, Kabbalists, hear me moderns—is that reality is greater than can be inferred from number, unless that number is 1.