# Quaternary numeral system

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Quaternary numeral system

Quaternary is the base-num|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 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.

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.

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

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).

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.

References

ee also

* Conversion between bases

External links

* [http://www.mathsisfun.com/numbers/convert-base.php?to=quaternary Quaternary Base Conversion] , includes fractional part, from Math Is Fun
* [http://hauptmech.com/base42/wiki/index.php?title=Base4 Base42] Proposes unique symbols for Quaternary and Hexadecimal digits

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