## PRELIMINARY TERMS

Let us use "digit" (meaning finger, or toe) to mean a unique symbol for some particular "number" (a value or count) and "numeral" to mean the particular combination of digits that represent a specific number. In numbers represented by numerals of more than one digit, the position of the digit in the numeral may or may not indicate something about its value. In the position dependent systems (such as the ordinary decimal system: dollars > dimes > pennies) it is necessary to have a digit to hold the position in the absence of a count for that position value. For example, 102 is one dollar, no dimes and two pennies, or 1 x 100 + 0 x 10 + 2 x 1, or in exponent notation, 1 x 10^2 + 0 x 10^1 + 2 x 10^0 (recall that any number raised to the zero power is one).

The "base" of a number system using position dependent notation is the number of digits including the zero symbol, otherwise we might denote it by its fundamental multiple (plus number of digits).

## MESOPOTAMIAN NUMBER SYSTEM

This is significant today for the base 60 (as in seconds and minutes) and 6 x 60=360 (as in degrees of a circle). It is especially significant because no scholars* have been found who consider that the true base of digital (finger) counting was derived by the Mesopotamians (Greek for "mid-river" and generic for all of the early civilizations arising between the Tigris and Euphrates rivers, especially Sumerian and Babylonian.

*see George E. Owen, The Universe of the Mind (A History of Mathematics and Physics), The Johns Hopkins Press, Baltimore, 1971; Derek Price, Science Since Babylon, Yale Press, 1961, and Marshall Clagett, Greek Science in Antiquity, Scholar's Bookshelf, Princeton, NJ, 1988.

You have no doubt heard that the base 10 number system (decimal) is because man has ten fingers (five on each of two hands!) That is a facile - if not logical - explanation. The decimal system has ten numerals but one of which is zero ( 0 ). As we have ten fingers we should have ten numerals corresponding to each finger excluding zero or eleven numerals including zero which would give rise to a base 11 number system. A notation of 123 would mean 1 x 11^2 + 2 x 11^1 + 3 x 11^0 = 121 + 22 + 3 = 146 (in decimal value).

It is my belief (suggestion, rationalization, or proposal) that the Mesopotamians counted off on one hand and rather than use the fingers of the other to count sets of five, they would carry the count to one finger of the second hand as 6 by raising one finger and clenching off the five raised fingers of the first hand. Thus using both hands to count sets of 6 they could reach 60. Note that there is implied a zero numeral represented by the fist (no fingers raised). The simple multiples of 2 x 6 =12 and 5 x 12 = 60 no doubt gave rise to the hours of the day (sunrise to sunset) and the near correspondence of the 12 lunar cycles (moonths) in one solar cycle (year). Note that the value of 7 (days in week) is simply the integer number of days that distinguish the obvious four phases of the moon. Thus using this method of finger counting you can hold (in your hands) a count of 35 = 5 (left fingers each worth x 6) + 5 (right fingers each worth x 1). Not quite a count of 1000 we are looking for.

## IONIC GREEK NUMBER SYSTEM

Ionia was to the east of the Aegean Sea from Greece on the land mass connecting to Mesopotamia: present day Turkey - Iraq. Apparently adapted from a Phoenician (later than Babylonian) system, it consisted of twenty-seven digits (three more than in the Greek alphabet) as three sets of nine numerals without position dependence. The three additional symbols were also Phoenician: two of which (F and Q) became letters in the Roman alphabet while the third which did not is represented here by a variation of the Greek Sigma (S).

The system clearly allows number values to be associated with words (ciphers) and opens many possibilities for numerology of all kinds! The first example attributed to the Gnostics for a deity of astronomical significance around the beginning of the current era and the second somewhat more contemporary and a lot more ridiculous.

## ROMAN NUMBER SYSTEM

Only seven numerals: smaller numerals to the left subtract from the follower; others add serially. This gives rise to order dependence but not place dependence. Note that the symbols are powers of ten times one and five as in finger and hand! Both Greek and Roman notation for very large numbers had additional rules.

MDCLXVI = 1000+500+100+50+10+5+1 = 1666: A significant year: Newton (age 24) discovered his three great contributions in optics, calculus and gravitation (to be published over the following 20 years).
MCMXCVIII = 1000+900+90+8 = 1998. Will 1999 = MIM?

## ARABIC NUMERALS

Following the decline of the Roman empire and the survival of learning in the near East in Persia and the subsequent Arabic states, the position (place) dependent decimal number system was developed with the invention of zero (as a symbol for empty set place holder - not as a concept for a quantity of none). It ought to be noted that modern Arabic numbers show no recognizable relation to (European) Arabic numerals.

## COMPUTER NUMBER SYSTEM

As an electronic device the computer only recognizes two states: current flowing or not flowing, only two (binary) symbols are required: 0 and 1. To count in a place dependent system, the successive powers of two must be represented by the Binary digITs (hence the term BIT). The value of the bits in a binary system in increasing order following the (U. S. ounce) volume units as powers of 2: ounce < (liquor)"shot" < gill < cup < pint < quart < pottle (actually British but not a misprint!) < gallon < peck < (grocery)"bag" < bushel; where all but two have legitimate names and those two have practical names. We've cheated a little by switching from liquid to dry measure for the three or four largest. In ounces the succession of containers is: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048. Had we measured in tablespoons (1/2 ounces) we could have doubled these values to 4096 for the thirteenth digit, or a total count of 8191.

If we were content with ten digits we could count to over 1000 using our fingers as binary digits (raised = 1, folded = 0) with the right thumb having a value of 1 (the least significant bit: LSB) and the left pinkie having a value of 512 (the most significant bit: MSB). Two thumbs up is 33, two pistols is 99, and double high fives is 1023. Which is what was promised in the first place.

Let's note on closing that the computer has evolved using eight bits at a time and the name for an eight bit number is BYTE (count from 0 to 255). Two bytes = WORD (16 bits) with a count capability of 65,535 or 64 times 1028 =1 K) or 64 K. Each new generation of microprocessor has increased the "word" size in multiples of bytes, currently at 4 bytes or 32 bits.

Copyright 1997 by P. E. Field
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