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Base 24
From Wikipedia, the free encyclopedia
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| List of numeral system topics | |
| Positional systems by base | |
| Decimal (10) | |
| 2, 4, 8, 16, 32, 64 | |
| 3, 9, 12, 24, 30, 36, 60, more… | |
The quadrovigesimal or base-24 system is a numeral system with 24 as its base.
There are 24 hours in a day, so our time keeping system includes a base-24 component. See also base 12.
Decimal Equivalent
10 twenty four 24 24
100 ? 24^2 = 576
1000 ? 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).
Quadrovigesimal 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 quadrovigesimal (or duodecimal) notation is used.
However when recurring fractions do occur in quadrovigesimal 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 powers of five look funny in the quadrovigesimal 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.