Centovigesimal number system

**Base 120** (also centovigesimal, twelfty, decanunqual, decadozenal), is an upper mid-scale base, and is often used as a decimal auxiliary or even a primary base due to being divisible by 2,3,4,5, and 8. Centovigesimal digits can be coded as two duodecimal or decimal digits, forming the much more compact and human-usable encodings Base-12-on-10 and Base-10-on-12. The base was used in several old languages because of the previously mentioned divisor benefits, however the user on this wiki Wendy.krieger is still studying it.

## In historical languages

The vocabulary for the earliest Germanic languages include words that would translate as *decimal* in modern English.  This is because five-scores does not equate to a hundred, but rather six-scores.

The names for the decades above ninety have been recorded in many of the languages, are *tenty* and *eleventy*. To avoid confusion, I have modified these to *teenty* and *elefty,* to avoid conflect with *twenty* and *seventy* respectively.

The use of the long numbers (as base 120 was known) faded out during the 14th century.

### English Numbers

The style of writing numbers over a hundred, is not as in the Latin practice of repeating C, (eg 300 = CCC), but to write it as iii C (ie three + hundred).

The parallel-row stone-board was in use, the rows are lettered away from the user as I, X, C, M. This would make the thousands some 1200, rather than 14400 as below.

## Modern Use

The present use of base 120 came as an auxillary system in the study of higher-dimension polytopes. This involves a lot of fractional parts, arising from the difference of the square of radii.

Also needed was the ability to give concise values for the reciprocals of the factorials. Of the bases studied, the only ones capable of doing this to any level of usability, was 6, 18, 30, 80 and 120, The other bases generate long periods all too quickly, Even of this list, 120 is well in front of the remainder.

### Alternating Arithmetic

The discovery of an algorithm to preform long arithmetic in bases like 120, opened up the possibility of using it as a day-to-day base.

The chief obsticle to finding the algorithm lies in the assumptions that one must have digits from 0 to 119, and that one must learn the full set of tables. Once one gets around this distinction, the digital calculations happen at a lower level than base arithmetic, the numbers are changed according to as they are 'high' or 'low'.

Alternating arithmetic eliminates to learn tables such as 57 times 83. As in decimal, these might be done as a two-step multiplication, typically.

## Change of Idiom

That a base has *b* digits, is a decimal idiom, in that ten-sized bases can be handled in this way. In practice, this applies only to a very tiny range. In mathematical theory, this is extended to all numbers, for which one might substitute a calculator for the tables for long arithmetic.

Historically, the bases are best thought of in terms of The Stoneboard or Abacus, where stones represent units, and the place on the board the value of the stone. Unlike the current decimal system, which is done in one row, historical systems like 20 and 60 are done in two rows to the column.

The columns are then a high (tens) cell and a low (units) cell, are named to the left of the units, as hundreds, thousands, centions, and millions. Cention and Million reflect the Greek pattern of 'numbers in the second myriad', are latin forms for *big hundred* and *big thousand*.

The order of columns in the current use reflect the historical order, where divisions were placed high, so that a unit in one column would divide into a divisorable number of tens in the next. This is consistant with dividing the dollar into quarters, of 25c each.

## Features of the system

Like 6 and 28, 120 is of the form (2^n-1)2^(n-1). This means that a binary set of weights (1, 2, 4, 8, 15, 30, 60) can be efficiently used with it, if one runs from 1/8 to 8 of a unit.

120 is the first multiple of six to be surrounded by composites. This means the "alpha" 1,1 and omega 1,-1 periods have useful primes not found in the base. 1.01 = 11\*11 and E9 = 7\*17. One can *cast out* multiples of seven or seventeen in calculation, to see if the arithmetic is correct in modulus arithmetic. In decimal, this is casting out nines.

There are a huge number of regular numbers in the base. These are numbers whose prime divisors are 2, 3, and 5. The number of new regular numbers under 120^n is very close to 49.76617n²-12.95n+4. With so many numbers with terminating reciprocals, it is possible to build a table of reciprocals to replace division. Such was done in sexagesimal.

**Unit fractions 1/2-20 in Base 120**