Saturday, December 24, 2011

The amazing world of numbers

The other day, I was trying to get straight the words for the numbers in Italian. While our counting system maintains a strict pattern, the words representing them are irregular. The exceptions to the regularity got me thinking about how number systems developed from primitive times. As a good scientist, I will first propose my hypothesis without looking at the history books, though I admit having some knowledge on the topic.

Consider the necessity of counting to the Shepard while letting his flock out into the field and needing to know if any are missing at the end of the day. The Sheppard added a stone to a pile for each sheep going out into the field and then removed a stone for each returning one. If a stone remained, the Shepard knew that a sheep had gone missing. In addition to being a practical counting technique, this procedure established the one-two-one correspondence between two sets of objects with the same number of elements. From a physicists point of view, one could say that this also established the principle of conservation of stones and conservation of sheep. Their numbers did not change, they just moved about from one location to another.

To solve the issue with proliferating sheep and shortages of stones, the Shepard recognized that he could use a different type of stone to represent, let's say 10, sheep. Thus, after the ninth stone was placed on the pile, the tenth sheep would be represented by the single 10-sheep stone while removing the previous nine. An extra stone would again represent each additional sheep until the 20th, at which point the next 10-sheep stone was used. Later, it became apparent that one need not use a special stone representing 10 sheep. Rather, one could place a stone in a different spot. Thus evolved the base 10 system, with separate symbols, or numerals representing one to ten, with these same numerals representing tens when placed in the tens spot, etc.

The "base" 10 number system most likely originated because of the ready availability of 10 fingers. I find it interesting that the word for finger and number shares the word "digit."

Other bases are also possible. When we were in Italy, I was at first puzzled at the system of Roman numerals engraved into the ancient buildings, which differ from what we commonly refer to as roman numerals today. For example, the modern form IV was represented as IIII and what we would recognizes as IX was written as VIIII. This even carried over to larger numbers, such as CD which they wrote as CCCC. Clearly, the original form of the Roman numeral system seemed to suggest base 5, as one would expect from counting on the fingers of one hand.

Now back to Italian. The pattern from zero to nine is as one would expect for base ten, with unique names corresponding to each numeral. Above dieci (ten), the system becomes schizophrenic. Undici (for eleven) seems to be saying one and ten, but quindici (for fifteen) is irregular in the sense that it is not a compound form of dieci and cinque (five). After Sedici (sixteen), the pattern reverses to diciassette, diciotto, diciannove, etc. Interestingly, the naming pattern for 20 and above continues along a strict convention without exception. Happily, 50 is cinquanta not quinquanta as I would have expected given the expression for 15. This pattern suggests an original base 16, or perhaps 15 with origins in the Roman base 5 system, which later got fixed to be consistent with base 10 convention. Whatever the case, the words hint at a mixture of systems.

Words and grammar can carry secret messages from the past. Ideas that were once imbedded in peoples' minds crept into language and became firmly rooted once it was formalized into written form. Thus, what remains today in any language language provides a snapshot of common usage from the past, which reflects the understanding of those times.

I was excited the morning that these ideas raced through my mind. I then thought about numbers in different languages. The words for the numbers in English are distinct until thirteen, which takes the form of three and ten, etc. Could this show the remnant of a base 12?. This seemed plausible, given the fact that some units of time (12 hours representing half a day and 12 months making a year) seem to have a preference for 12. And don't forget 12 inches to a foot.

I then rattled off the numbers in Ukrainian. It was purely base ten. I had taken French in high school and some in college, but my French got totally erased when I started to learn Italian (except for the time that French got mixed in with my Italian when I said to a French colleague while in Italy "qualchechosa," a hybrid of "qualcosa" (Italian) and "quelque chose" (French)). So, I asked my wife to remind me of how to count to twenty, and the pattern turns out to be the same as in Italian!

Stones may correspond to sheep, but how does one deal with fluids? This is an important quantity when bartering in liquids (as in ordering a beer). At some point, humans must have recognized that liquids are conserved. In other words, they can be moved around and split into smaller amounts, but when recombined, the quantity fills the original container in the same amount.

In the case of liquids and weights, there appears to be a preference for base 2, which makes sense given the ease with which we can split liquids in half over and over again. There are 2 cups to a pint, 2 pints to a quart, 2 quarts to a half gallon, and two half gallons to the gallon. Also, there are 8 ounces to a cup and a pint is 16 ounces (base 16!). And don't forget 16 ounces to a pound. The relationship between weight (or more precisely mass) and volume is clear. 16 ounces of water or beer weighs 16 ounces, or one pint weighs a pound. No wonder!

The birth of the decimal Metric System (base ten) coincided with the French Revolution, when two platinum standards representing the meter and the kilogram were deposited in the Archives de la République in Paris, on 22 June 1799. This was the first step in the development of the present International System of Units, in which the basic units of distance, mass, time and current are the meter, kilogram, second and ampere, respectively.

For convenience, modern day computers use binary, or base two: ones and zeros are easily representable with the gate of a transistor being on (1) or off (0). For programing convenience, the bits (binary digits) are combined into groups of 4 leading to a hexadecimal representation (base 16), where two hexadecimal "digits" can represent the numbers 0 through 255. As such, pairs of hexadecimal numerals are used to describe the alphabet (upper and lower case) the ten base 10 numerals, as well a bunch of special symbols.

The simple act of counting eventually led to the transformation of primitive society into one that can understand the mysteries of nature, which since antiquity had been thought incomprehensible in the absence of deities. Our language, on the other hand, provides clues of the thought processes that went into the development of counting, which forms the basis of mathematics, physics, the sciences, engineering and technology - historically following approximately that order.

After writing this post, I searched Wikipedia for additional information on various bases used by various civilizations in various eras. In the third millennium BCE, the Summarians used a base 60 numeral system, called the sexagesimal system. The symbols used are shown to the right. Incidentally, our system of 60 seconds to the minute and sixty minutes are -- you got it, base 60! But so are angular measurements. There are 60 arc minutes in a degree, and sixty arc seconds in an arc minute. This connection makes sense given that the timing of the sun's apparent motion in the sky is measured as a change in angle over a change in time period.

It is obvious how the base 5, 10, and 20 systems follow from counting with fingers and toes. Thus while the sexagesimal system is base 60, the symbols follow a base 10 pattern. However, an advantage of base 60 is that 60 has a large number of factors (1,2,3,4,5,... you can determine the rest) so that fractions are easier to represent.

According to the Wikipedia article on base 10, peoples using other bases are as follows:

  • Pre-Columbian Mesoamerican cultures such as the Maya used a base-20 system (using all twenty fingers and toes).
  • The Babylonians used a combination of decimal with base 60.
  • Many or all of the Chumashan languages originally used a base-4 counting system, in which the names for numbers were structured according to multiples of 4 and 16.
  • Many languages use quinary number systems, including Gumatj, NunggubuyuKuurn Kopan Noot and Saraveca. Of these, Gumatj is the only true 5–25 language known, in which 25 is the higher group of 5.
  • Some Nigerians use base-12 systems
  • The Huli language of Papua New Guinea is reported to have base-15 numbers. Ngui means 15, ngui ki means 15×2 = 30, and ngui ngui means 15×15 = 225.
  • Umbu-Ungu, also known as Kakoli, is reported to have base-24 numbers. Tokapu means 24, tokapu talu means 24×2 = 48, and tokapu tokapu means 24×24 = 576.
  • Ngiti is reported to have a base-32 number system with base-4 cycles.

So, what became an innocent language lesson on the numbers led to a train of thought that occupied my mind for a morning and gave me the satisfaction of understanding something that was new to me. The fact that I had not learned something new to the world did not bother me a bit.

With all this numerology and talk of ancient number systems, am I worried about 2012? What do you think?

I am finishing this post after our traditional zillion-course Ukrainian Christmas dinner, followed by a glass of eggnog and countless chocolate-covered pretzels, so errors in my logic are undoubtedly plentiful. No apologies!

I wish all of you the best that the holiday season has to offer. Given the international nature of my handful of regular readers, I would be interested in hearing about how you form the words for the numbers in your native languages, and at what point if any, they are irregular. In the meantime, I will be sipping on another eggnog. Good night!

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