Base 24
Encyclopedia
The base- system is a numeral system
with 24 as its base.
There are 24 hour
s in a nychthemeron
(more commonly, a day
), 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).
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 certain powers of five are palindromes
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.
Numeral system
A numeral system is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
with 24 as its base.
There are 24 hour
Hour
The hour is a unit of measurement of time. In modern usage, an hour comprises 60 minutes, or 3,600 seconds...
s in a nychthemeron
Nychthemeron
Nychthemeron , occasionally nycthemeron or nuchthemeron is a period of 24 consecutive hours...
(more commonly, a day
Day
A day is a unit of time, commonly defined as an interval equal to 24 hours. It also can mean that portion of the full day during which a location is illuminated by the light of the sun...
), so solar time
Solar time
Solar time is a reckoning of the passage of time based on the Sun's position in the sky. The fundamental unit of solar time is the day. Two types of solar time are apparent solar time and mean solar time .-Introduction:...
includes a base-24 component.
See also base 12
Duodecimal
The duodecimal system is a positional notation numeral system using twelve as its base. In this system, the number ten may be written as 'A', 'T' or 'X', and the number eleven as 'B' or 'E'...
.
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
Quadrovigesimal fractions are usually either very simple1/2 = 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 certain powers of five are palindromes
Palindromic number
A palindromic number or numeral palindrome is a 'symmetrical' number like 16461, that remains the same when its digits are reversed. The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under reversal of its letters...
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.