Quaternary numeral system

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.

[edit] 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 programming in the discussion and analysis of binary arithmetic and logic, quaternary does not enjoy the same status.

[edit] Hilbert curves

Quaternary numbers are however 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.

[edit] 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]

[edit] 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.

For example, the nucleotide sequence GATTACA can be represented by the quaternary number 2033010 (= decimal 9156).

[edit] 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.

[edit] References

1. ^ "Chumashan Numerals" by Madison S. Beeler, in Native American Mathematics, edited by Michael P. Closs (1986), ISBN 0-292-75531-7.

[edit] See also

• Conversion between bases

[edit] External links

• Quaternary Base Conversion, includes fractional part, from Math Is Fun
• Base42 Proposes unique symbols for Quaternary and Hexadecimal digits