It shares with all fixed-radix numeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the characteristics of the representations of rational numbers and irrational numbers. See decimal and binary for a discussion of these properties.
Relation to other positional number systems
* | 1 | 2 | 3 | 10 | 11 | 12 |
1 | 1 | 2 | 3 | 10 | 11 | 12 |
2 | 2 | 10 | 12 | 20 | 22 | 30 |
3 | 3 | 12 | 21 | 30 | 33 | 42 |
10 | 10 | 20 | 30 | 100 | 110 | 120 |
11 | 11 | 22 | 33 | 110 | 121 | 132 |
12 | 12 | 24 | 36 | 120 | 132 | 210 |
Quaternary | 1 | 2 | 3 | 10 | 11 | 12 | 13 | 20 | 21 |
---|---|---|---|---|---|---|---|---|---|
Binary | 1 | 10 | 11 | 100 | 101 | 110 | 111 | 1000 | 1001 |
Decimal | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
Quaternary | 22 | 23 | 30 | 31 | 32 | 33 | 100 | 101 | 102 |
Binary | 1010 | 1011 | 1100 | 1101 | 1110 | 1111 | 10000 | 10001 | 10010 |
Decimal | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 |
Quaternary | 103 | 110 | 111 | 112 | 113 | 120 | 121 | 122 | 123 |
Binary | 10011 | 10100 | 10101 | 10110 | 10111 | 11000 | 11001 | 11010 | 11011 |
Decimal | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 |
Relation to binary
As with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix 4, 8 and 16 is a power of 2, so the conversion to and from binary is implemented by matching each digit with 2, 3 or 4 binary digits, or bits. For example, in base 4,
- 30210_{4} = 11 00 10 01 00_{2}.
Although octal and hexadecimal are widely used in computing and computer programming in the discussion and analysis of binary arithmetic and logic, quaternary does not enjoy the same status.
By analogy with bit, a quaternary digit is sometimes called a crumb.
Occurrence in human languages
Many or all of the Chumashan languages originally used a base 4 counting system, in which the names for numbers were structured according to multiples of 4 and 16 (not 10). There is a surviving list of Ventureño language number words up to 32 written down by a Spanish priest ca. 1819.^{[1]}
The Kharosthi numerals has a partial base 4 counting system from 1 to 10.
Hilbert curves
Quaternary numbers are used in the representation of 2D Hilbert curves. Here a real number between 0 and 1 is converted into the quaternary system. Every single digit now indicates in which of the respective 4 sub-quadrants the number will be projected.
Genetics
Parallels can be drawn between quaternary numerals and the way genetic code is represented by DNA. The four DNA nucleotides in alphabetical order, abbreviated A, C, G and T, can be taken to represent the quaternary digits in numerical order 0, 1, 2, and 3. With this encoding, the complementary digit pairs 0↔3, and 1↔2 (binary 00↔11 and 01↔10) match the complementation of the base pairs: A↔T and C↔G and can be stored as data in DNA sequence.^{[2]}
For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010 (= decimal 9156 or binary 10 00 11 11 00 01 00).
Data transmission
Quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits.
See also
References
- ^ "Chumashan Numerals" by Madison S. Beeler, in Native American Mathematics, edited by Michael P. Closs (1986), ISBN 0-292-75531-7.
- ^ http://2010.igem.org/files/presentation/Hong_Kong-CUHK.pdf
External links
- Quaternary Base Conversion, includes fractional part, from Math Is Fun
- Base42 Proposes unique symbols for Quaternary and Hexadecimal digits
- Graphical imaging of numeral systems