autonomy - author's canon

sexagesimal

Community Standard Sexagesimal

What is Sexagesimal?

Sexagesimal simply means "base 60". 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" Section below. 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 did use a dot to represent zero), nor did they have the concept of a radix (decimal or sexagesimal) marker. The Babylonians did have a Place-Value notation for fractions of decreasing powers of sixty as well, a refinement lacking only a radix marker to unify their fractional and integer notations.

It is believed that the Sumerians and Babylonians chose base 60 because of its multitude of convenient divisors. Sexagesimal 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 (in this document we represent the radix marker in the British and American style, with a "." or "point." Elsewhere in Europe and much of the world the radix marker is written with a comma, ","), like so:

. (USA/UK, and in this document)

, (elsewhere)

(never mind the funny looking symbols; they are explained fully in the section entitled "Written Sexagesimal Numerals" below). In decimal notation, 1/3 is written 0.3333... (3 recurring, i.e. 3 repeats forever).

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.

Primer: Positional (Place-Value) and Integer Base Notation

So, how exactly does a base 60 numbering system work? First, it is important to understand how our positional base 10 system works (those already familiar with numerical systems of different bases can skip forward to the next section, "Written Sexagesimal Numerals").

Positional notation is so intuitive that we grasp a fairly complex arithmetic equation with just a glance. We immediately understand 195.02 without having to add 5 to 90 to 100, and finally include the fraction 2/100. Positional sexagesimal is just as intuitive, with a little practice.

As you have probably guessed, "positional notation" means the position of each digit relative to the radix marker determines the magnitude of its value. The first place to the left of the radix marker represents the digit times the numerical base taken to the zeroth power (N0 always equals one), the second the digit times the numerical base to the first power, the third the digit times the numerical base squared, and so on. Similarly, the first position to the right of the decimal represents the digit times the numerical base taken to the negative one power, the second position the digit times the numerical base taken to the negative two power, and so on. The results are all added together, giving us the total value of the number. This is true of base 10, base 60, or any other numerical base you care to use.

Stated most generally, any number D can be represented in any base N (N>0) by a series of digits d as follows:

D = djNj + dj-1Nj-1 + ... + d1N1+d0N0 . d-1N-1 + d-2N-2 + ... + d-kN-k

Which, in standard positional notation, is written:

D = (djdj-1 ... d1d0.d-1d-2 ...)N,

where all digits d satisfy the constraint 0≤di<N.

Take, for example, the decimal (base 10) number "103.95":

1 x 102 + 0 x 101 + 3 x 100 + 9 x 10-1 + 5 x 10-2 = 100 + 0 + 3 + 9/10 + 5/100

Which of course, gives us one hundred three and ninety-five hundredths, written in the familiar form of "103.95" in positional base 10.

Similarly, in sexagesimal, each position to the left of the radix marker represents increasing powers of 60, while those to the right of the radix marker represent increasing negative powers of 60. Consider:

(48)(0)(1).(15)(39)

Each symbol represents a number between zero and fifty nine (again, these symbols are defined and explained in detail in the next section). Just like in decimal notation, the position of the symbol defines the magnitude of its value. In the number above, (48) represents "48", (0) is "0", (1) is "1", "." is the sexagesimal (radix) marker, (15) is "15", and (39) is "39".

Sexagesimal positional notation yields:

(48) x 602 + (0) x 601 + (1) x 600 + (15) x 60-1 + (39) x 60-2,

or, using numerals we are more familiar with:

48 x 602 + 0 x 601 + 1 x 600 + 15 x 60-1 + 39 x 60-2, which gives us:

172,800 + 0 + 1 + 15/60 + 39/3600 = 3648 939/3600, or the decimal value:

172,801.2608333... (3 recurring).

Notice how a three digit numeral in base 60 can represent a six digit base 10 number? The difference between decimal and sexagesimal notation grows dramatically with each digit:

(1)(0)(0),(0)(0)(0) represents 777,600,000, while

(59)(59)(59),(59)(59)(59) equals 46,655,999,999!

Which would you find easier to remember, a six digit number or an eleven digit number?

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. 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.

Written Sexagesimal Numerals

As in decimal, there is a unique numerical symbol for each sexagesimal digit. In base 10, these symbols are 0 through 9. In base 60, they are (0) through (59), as shown in Figure 1.

+0+10+20+30+40+50
STEM:
TOP:
0(0)(10)(20)(30)(40)(50)
1(1)(11)(21)(31)(41)(51)
2(2)(12)(22)(32)(42)(52)
3(3)(13)(23)(33)(43)(53)
4(4)(14)(24)(34)(44)(54)
5(5)(15)(25)(35)(45)(55)
6(6)(16)(26)(36)(46)(56)
7(7)(17)(27)(37)(47)(57)
8(8)(18)(28)(38)(48)(58)
9(9)(19)(29)(39)(49)(59)
Figure 1. Sexagesimal Numerals

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)-(59) (0-59)

Spoken Numbers and Metric-60 Unit Prefixes

Numerical nomenclature in Community Standard Base-60 is much simpler than base 10 once you get used to it. In base 10 we have an inconsistent way of naming numbers larger than ten. "Fourteen", but not "fiveteen", "thirty" but not "secondy" or "fourthty", "forty" but not "twoty", "threety", or "fivety", and so on. Larger numbers are even worse. "Hundreds", "thousands", and "millions" are alright, but "milliards" (109 European) means nothing to an American, while "billions", "trillions", and so on mean completely different numbers depending on which country you are in, and which language you speak! "One billion" means 109 to an American, and 1012 to a European, "one trillion" 1012 and 1015 respectively, and so on. To avoid confusion, the BBC typically says "million million" rather than "milliard" or "billion". Really huge numbers have no names at all, with the occasional exception arbitrarily thrown in just to sow a little extra confusion (a "googleplex" is 10100, but what do you call 1080 or 10101?).

With Community Standard Sexagesimal no such confusion exists: the nomenclature is well defined, easy to derive, and perfectly regular, both for numbers themselves and their corresponding metric-60 unit prefixes. It is trivial to derive unique, understandable names for numbers, and metric-60 units, ranging from 60-160 (approximately 3 x 10-285) through 59 x 60160 (around 2 x 10286), a vastly wider range than we have available in decimal.

In Standard Nomenclature, the numbers (0) (0) through (59) (59) are spoken (and written longhand) exactly as they are in base 10. The reason for this is to keep things simple for us native base 10 users, rather than making up words for numbers 10 through 59. So (55) (55) is spoken and written longhand as "fifty-five". Purists devised their own terminology for the numbers (7) (7) and (11)-(59) (11-59), finding the multiple syllables of traditional names such as "twenty-seven" to be ungainly. Although Purist Nomenclature was starting to gain popularity at the time the Community's crisis erupted, standard nomenclature remained more widely used.

The Six Rules of Sexagesimal Nomenclature

The names of each sexagesimal number are easily derived from five simple rules, as are the names and prefixes for the corresponding metric-60 unit modifiers (analogous to "milli" and "kilo" in metric-10 nomenclature). The sixth rule defines how metric-60 unit prefix abbreviations are written.

Rule 1: Choose either Standard or Purist Nomenclature. The choice only affects this rule. Standard Nomenclature: Numerals (0)-(59) (0-59) are spoken and written longhand exactly as their decimal equivalents. Purist Nomenclature: Numerals (0)-(6), (8)-(10) (1-6,8-10) are spoken and written exactly as their decimal equivalents, all other numerals are written and spoken according to Purist Nomenclature.

Rule 2: The first consonant defines the absolute value of the power of 60 (see Figure 2). "Absolute Value" simply means that, in applying Rules #2 and #3, we ignore the sign and treat negative exponents exactly as we do positive exponents.

601b"b" is in "bravo"
602d"d" as in "delta"
603f"f" as in "foxtrot"
604g"g" as in "golf"
605h"h" as in "hotel"
606j"j" as in "Juliette"
607l"l" as in "Lima"
608m"m" as in "Mike"
609n"n" as in "November"
6010p"p" as in "papa"
6011q"q" as in "Quebec"
6012r"r" as in "Romeo"
6013s"s" as in "sierra"
6014t"t" as in "tango"
6015v"v" as in "Victor"
6016w"w" as in "whiskey"
6017y"y" as in "Yankee"
6018z"z" as in "Zulu"
6019ch"ch" as in "channel"

6020th"th" as in "thin"
Figure 2. Deriving the base consonant of a sexagesimal numerical prefix

Rule 3: The base vowel modifies the base consonant, adding 20, 40, 60, 80, 100, 120, or 140 to the absolute value of the power of 60 (see Figure 3).

+ +0a"a" as in "calm"
+ +20e"e" as in "shed"
+ +40i"i" as in "sit"
+ +60o"o" as in "show"
+ +80u"u" as in "Zulu"
+ +100y"y" as in "spy"
+ +120æ"a" as in "gate"
+ +140ē"e" as in "sheet"
Figure 3. Deriving the base vowel of a sexagesimal numerical prefix

Rule 4: The prefix "an" is prepended for negative exponents.

Rule 5: Add "ra" as the final syllable if you're talking metric-60 units, or "zend" if you're just talking base 60 numbers.

Rule 6: (metric-60 units) The base consonant and base vowel define the unit modifier abbreviation (analogous to k for kilo, m for milli, etc.) which is separated from the unit name abbreviation by a hyphen. Negative exponents have the letter a (an abbreviation of "an") prepended. For example, meratocks are abbreviated me-t, while anherameters (about the same order of magnitude as a tock in size) are abbreviated ahe-m.

Examples

Example 1: "a bazend":

Consider (1)(0) (60). Rule #1 (Standard Nomenclature) tells us that we pronounce numerals (0)-(59)as we would in decimal. (1)(0) (60) falls outside of this range. Our first nonzero digit is in the second place to the left of the sexagesimal marker (601). Thus the absolute value of our exponent is (1) (1). Applying rule #2 we get "b" as our base consonant. Applying rule #3 "+(0)"+0) gives us "a" as our base vowel. Our exponent is a positive value, so rule #4 does not apply. Since we are talking about a number, and not a metric-10 or metric-60 unit prefix, we append "zend" per rule #5, giving us the word "bazend".

Thus, (1)(0) is written and pronounced "one bazend" or "a bazend", never "sixty".

Example 2: a simple number:

Consider (1)(46). This number is pronounced "one bazend forty-six." See if you can apply the six rules above to come up with that result. If you can, then you're starting to get the hang of Community Standard Sexagesimal.

Example 3: fractions and negative exponents:

Consider (0).(0)(3)(58) circadians (a short subjective period of time, equal to about 1.10185185 millicircadians). We'll stick stick with Standard Nomenclature, so rule #1 tells us we pronounce each digit as we would its decimal equivalent. We note that the first nonzero numeral is two places to the right of the sexagesimal marker, so our exponent is -(2) (-2) . The absolute value of our exponent is therefor (2) (2) (we drop the minus sign). According to rule #2, our base consonant is "d". Applying rule #3, +(0) (+0) gives us our base vowel as "a". Since we are dealing with a negative exponent, Rule #4 tells us to add the prefix "an" to our base, yielding "anda" as our stem. If we wish to state the value in terms of metric-60 units of measure, we have

(3).(58) ada-C, or "three point fifty-eight andaracircadians."

We could also state this value as a pure number, i.e.

(0).(0)(3)(58) circadians, pronounced any of three ways: "zero point zero three fifty-eight circadians", "three andazend fifty-eight anfazend circadians", or (Shortened Form) "three andazend fifty-eight circadians".

Example 4: Really Big Numbers:

Consider a very long distance: a bazend to the bazend nineteen ((60e79)) meters, in decimal written as 6079 meters. How do we say this huge value, (60e79), in Sexagesimal? First, we choose which nomenclature we wish to use (Standard or Purist). We'll keep using Standard Nomenclature. Second, we notice that

(1)(19) = (19) + (1)(0) ( 79 = 19 + 60 )

Using Rule #2, (19) (19) gives us "ch" as our base consonant. Applying rule #3, +(1)(0) (+60) gives us our base vowel as "o". We are dealing with a positive exponent, so Rule #4 doesn't apply, and we are left with "cho" as our base syllable. According to rule #5, we append "zend" if we are naming a number, and "ra" if we are referring to a metric-60 unit prefix. So, we have one chozend meters (6079 meters), or one chorameter, abbreviated (according to Rule #6) as either (60e79) m (6079 m) or (1) cho-m (1 cho-m) respectively. Of course, in actual practice metric-10 and sexagesimal is only mixed like this when converting units. The Autonomous Community uses metric-60 units exclusively.

Resources, References, and Additional Reading

Sexagesimal Resources

References