Quaternary is the base-4 numeral system. It uses the digits 0, 1, 2 and 3 to represent any real number.

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

A quaternary multiplication table
* 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
Numbers one to twenty-seven in standard quaternary
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,

302104 = 11 00 10 01 002.

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

  1. ^ "Chumashan Numerals" by Madison S. Beeler, in Native American Mathematics, edited by Michael P. Closs (1986), ISBN 0-292-75531-7.
  2. ^ http://2010.igem.org/files/presentation/Hong_Kong-CUHK.pdf

External links