Community Standard Sexagesimal
Written Sexagesimal Numerals
See also: Sexagesimal nomenclature and Planck Metric-60 Units
Sexagesimal simply means base 60. It is a numerical system that employs sixty unique numerals and a positional notation using powers of sixty to express larger numbers. In contrast, binary employs two numerals (0 and 1) and a positional notation using powers of two to express larger numbers. Our more familiar decimal (base 10) system employs ten numerals and a positional notation using powers of ten to express larger numbers.
As in decimal, there is a unique numerical symbol for each sexagesimal digit. In decimal (base 10), these symbols are 0 through 9. In base 60, they are 0 through x, as shown in Figure 1 below.
| +0 | +10 | +20 | +30 | +40 | +50 | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| STEM: |  |  |  |  |  | ||||||||
| TOP: | |||||||||||||
| 0 |  | 0 | (0) | A | (10) | K | (20) | U | (30) | e | (40) | o | (50) |
| 1 |  | 1 | (1) | B | (11) | L | (21) | V | (31) | f | (41) | p | (51) |
| 2 |  | 2 | (2) | C | (12) | M | (22) | W | (32) | g | (42) | q | (52) |
| 3 |  | 3 | (3) | D | (13) | N | (23) | X | (33) | h | (43) | r | (53) |
| 4 |  | 4 | (4) | E | (14) | O | (24) | Y | (34) | i | (44) | s | (54) |
| 5 |  | 5 | (5) | F | (15) | P | (25) | Z | (35) | j | (45) | t | (55) |
| 6 |  | 6 | (6) | G | (16) | Q | (26) | a | (36) | k | (46) | u | (56) |
| 7 |  | 7 | (7) | H | (17) | R | (27) | b | (37) | l | (47) | v | (57) |
| 8 |  | 8 | (8) | I | (18) | S | (28) | c | (38) | m | (48) | w | (58) |
| 9 |  | 9 | (9) | J | (19) | T | (29) | d | (39) | n | (49) | x | (59) |
Base 60 symbols consist of two parts, which are joined to create a sexagesimal numeral. The upper portion represents 0-9, the lower portion a additive value (+10, +20, up through +50). The uppermost row and leftmost column of the grid show each partial component, while the rest of the grid shows how those components are combined to create the numerals 0-x (0-59).
Advantages of Sexagesimal
Convenient Divisors
It is believed that the Sumerians and Babylonians chose a sexagesimal numerical system because of its multitude of convenient divisors. Base 60 allows for even division by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, in contrast to decimal, which only has the related pair 2 and 5. This allow numerous fractions that are difficult to represent in decimal to be written very simply. For example, in sexagesimal, the fraction 1/3 is represented with exactly one digit after the sexagesimal radix marker, like so:
0.K
which representes (0).(20), or 0+20/60
In contrast, decimal notation would write 1/3 as 0.333333333… (where 3 is recurring, i.e. it repeats forever).
NOTE: Although the US and UK use a “.” (point) as a radix marker, much of the rest of the world uses a “,” (comma), and would write 1/3 like so: 0,K In this document, the American and British style decimal and sexagesimal point is used.
Fewer Digits Required to Express Larger Numbers
Humans are generally only able to retain 5 or 6 digits in their short term, immediate memory (many of us can only manage 4). Base 60 allows humans to work with and retain much larger numbers than base 10, without having to resort to mnemonic tricks, or memorization (which accesses long-term memory). Furthermore, when long-term memory is used, vastly larger numbers can be memorized and retained longer.
To illustrate this, consider the following representations of the same number:
binary: 101110101110010100110110
decimal: 12248374
sexagesimal: ugJY
Anyone who has studied programming knows how unwieldy binary numbers quickly become. Anything above thirty-two requires too many digits to be useful. Decimal doesn’t explode quite so fast, but as we see above, a fairly common range of numbers does require more digits than the human mind is comfortable with. In contrast, only four sexagesimal digits are required to express the same number. Imagine how much easier it would be to memorize four or five digit phone numbers, rather than seven or ten digit numbers. Basic arithmetic becomes easier as well.
(Fiction) The Autonomous Community couldn’t increase the computational capacity of the biological brains that limited them when operating in the Physical, so they adopted a base 60 numerical standard as an easy way to enhance their biological intelligence while in the Physical, at least with respect to numbers and arithmetic. Work on other bio-mnemonic optimizations, in everything from language to memory retrieval, was suspended when the Community’s worst crisis came to a head.
History of Sexagesimal
Factual History
The Sumerians used a sexagesimal numerical system, which the Babylonians refined with a “positional” notation (”Positional Notation” is explained in the “Primer: Positional and Integer Base Notation.” The system all of us have grown up with is a decimal, or base-10, positional notation system). We still retain some residual use of their system, in degrees of angle (360 degrees in a circle) and in the number of seconds in a minute, and minutes in an hour (the number of hours in a day, twenty-four, comes from another culture which used a different numerical system). Unlike the notation adopted by the Autonomous Community, the Sumerian and Babylonian numerals were cumbersome. Digits were written in cuneiform characters reminiscent of a combination of different tally marks (modern efforts to use decimal numbers separated by commas or colons, and a period or semi-colon as a radix marker aren’t much better, and all fail to exploit the mnemonic advantages of a completely native sexagesimal notation).
The early Babylonians and Sumarians did not have a standard symbol for “0″ (though some used a space as an empty placeholder, and later Babylonians used a dot to represent zero), nor did they have the concept of a radix (decimal or sexagesimal) marker. However, the Babylonians did have a Place-Value notation for fractions of decreasing powers of sixty, a refinement lacking only a radix marker to unify their fractional and integer notations.
Future (fictional) History
The Autonomous Community adopted an advanced sexagesimal system that included the advantages of the positional notation of our decimal system (including a radix marker separating positive and negative powers of 60), unique single-character representations of each digit 0-59, a nomenclature for naming numbers based on powers of 60, and a metric-60 nomenclature for unit prefixes. They designed the system to be concise, coherent, and above all, fairly simple for minds trapped in biological flesh to understand and use, even without the aid of comprehensive knowledge engrams.
Related Topics: Sexagesimal nomenclature and Planck Metric-60 Units
© 2007 Jean-Michel Smith. The contents of this page may be freely used and redistributed under the terms of the Creative Commons Attribution License, Version 2.5 or later.
Autonomy sexagesimal