# Quaternary numeral system

Quaternary is the base four 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.

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.

Interestingly, 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).

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