Base 24

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The base-24 (quadrivigesimal) system is a numeral system with 24 as its base.

There are 24 hours in a day (nychthemeron, day–night cycle), so solar time includes a base-24 component.

See also base 12.

                           Decimal Equivalent
       10  twenty-four      24             24
      100  "               24^2 =         576
    1 000  "               24^3 =      13 824
   10 000  "               24^4 =     331 776
  100 000  "               24^5 =   7 972 624
1 000 000  "               24^6 = 191 102 976

The digits used for numerals ten (10) to twenty-three (23) may be the letters "A" through to "P" ("I" and "O" are skipped to prevent confusion with the digits 1 and 0 in some typefaces).

Fractions[edit]

Quadrivigesimal fractions are usually either very simple

1/2 = 0.C
1/3 = 0.8
1/4 = 0.6
1/6 = 0.4
1/8 = 0.3
1/9 = 0.2G
1/C = 0.2
1/G = 0.1C
1/J = 0.18

or complicated

1/5  = 0.4K4K4K4K... recurring (easily rounded to 0.5 or 0.4K)
1/7  = 0.3A6LDH3A6... recurring
1/A  = 0.29E9E9E9... recurring (rounded to 0.2A)
1/B  = 0.248HAMKF6D248.. recurring (rounded to 0.24)
1/D  = 0.1L795CN3GEJB1L7.. recurring (rounded to 0.1L)
1/P  = 0.11111... recurring (rounded to 0.11)
1/11 = 0.0P0P0P... recurring (rounded to 0.0P) (1/(5*5))

As explained in recurring decimals, whenever a fraction is written in "decimal" notation, in any base, the fraction can be expressed exactly (terminates) if and only if all the prime factors of its denominator are also prime factors of the base. Thus, in base-10 (= 2×5) system, fractions whose denominators are made up solely of multiples of 2 and 5 terminate: ¹⁄8 = ¹⁄(2*2*2), ¹⁄20 = ¹⁄(2×2×5), and ¹⁄500 (2²×5³) can be expressed exactly as 0.125, 0.05, and 0.002 respectively. ¹⁄3 and ¹⁄7, however, recur (0.333... and 0.142857142857...). In the duodecimal (= 2×2×3) system, ¹⁄8 is exact; ¹⁄20 and ¹⁄500 recur because they include 5 as a factor; ¹⁄3 is exact; and ¹⁄7 recurs, just as it does in base 10.

In practical applications, the nuisance of recurring decimals is encountered less often when quadrivigesimal (or duodecimal) notation is used.

However when recurring fractions do occur in quadrivigesimal notation, they sometimes have a very short period when they are numbers containing one or two factors of five, as 5² = 25 is adjacent to 24. The other adjacent number, 23, is a prime number. So certain powers of five are palindromes in the quadrivigesimal notation:

51 =          5
52 =         11
53 =         55
54 =        121
55 =        5A5
56 =       1331
57 =       5FF5
58 =      14641

The multiples of decimal hundred are 44, 88, CC, GG, LL, 110, etc.

Natural languages[edit]

Umbu-Ungu, also known as Kakoli, is reported to have base-24 numerals.[1][2] Tokapu means 24, tokapu talu means 24×2 = 48, and tokapu tokapu means 24×24 = 576.

References[edit]

  1. ^ Gordon, Raymond G., Jr., ed. (2005), "Umbu-Ungu", Ethnologue: Languages of the World (15 ed.), retrieved 2008-03-16 
  2. ^ Bowers, Nancy; Lepi, Pundia (1975), "Kaugel Valley systems of reckoning", Journal of the Polynesian Society 84 (3): 309–324