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Quaternary numeral system

 

 
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Quaternary numeral system
 
 
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Quaternary is the base
Base (mathematics)

In arithmetic, the base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b....
- numeral system
Numeral system

A numeral system is a writing system for expressing numerals , and a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
. It uses the digit
Numerical digit

In mathematics and computer science, a digit is a symbol used in numerals , to represent numbers, in Positional notation numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e....
s 0, 1, 2 and 3 to represent any real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
.

It shares with all fixed-radix
Radix

In numeral system, the base or radix is usually the number of unique Numerical digit, including zero, that a Positional notation numeral system uses to represent numbers....
 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 number
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
s and irrational number
Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
s. See decimal
Decimal

The decimal numeral system has 10 as its Base . It is the most widely used numeral system....
 and binary
Binary numeral system

The binary numeral system, or notation with a radix of 2. Owing to its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers....
 for a discussion of these properties.

ith the octal
Octal

The octal numeral system, or oct for short, is the radix-8 number system, and uses the digits 0 to 7. Numerals can be made from Binary numeral system numerals by grouping consecutive digits into groups of three ....
 and hexadecimal
Hexadecimal

In mathematics and computer science, hexadecimal is a numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen....
 numeral systems, quaternary has a special relation to the binary numeral system
Binary numeral system

The binary numeral system, or notation with a radix of 2. Owing to its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers....
.






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Quaternary is the base
Base (mathematics)

In arithmetic, the base refers to the number b in an expression of the form bn. The number n is called the exponent and the expression is known formally as exponentiation of b by n or the exponential of n with base b....
- numeral system
Numeral system

A numeral system is a writing system for expressing numerals , and a mathematical notation for representing numbers of a given set, using graphemes or symbols in a consistent manner....
. It uses the digit
Numerical digit

In mathematics and computer science, a digit is a symbol used in numerals , to represent numbers, in Positional notation numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e....
s 0, 1, 2 and 3 to represent any real number
Real number

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits co...
.

It shares with all fixed-radix
Radix

In numeral system, the base or radix is usually the number of unique Numerical digit, including zero, that a Positional notation numeral system uses to represent numbers....
 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 number
Rational number

In mathematics, a rational number is a number which can be expressed as a quotient of two integers. Non-integer rational numbers are usually written as the vulgar fraction , where b is not 0 ....
s and irrational number
Irrational number

In mathematics, an irrational number is any real number that is not a rational number ? that is, it is a number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero....
s. See decimal
Decimal

The decimal numeral system has 10 as its Base . It is the most widely used numeral system....
 and binary
Binary numeral system

The binary numeral system, or notation with a radix of 2. Owing to its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers....
 for a discussion of these properties.

Relation to binary

As with the octal
Octal

The octal numeral system, or oct for short, is the radix-8 number system, and uses the digits 0 to 7. Numerals can be made from Binary numeral system numerals by grouping consecutive digits into groups of three ....
 and hexadecimal
Hexadecimal

In mathematics and computer science, hexadecimal is a numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F to represent values ten to fifteen....
 numeral systems, quaternary has a special relation to the binary numeral system
Binary numeral system

The binary numeral system, or notation with a radix of 2. Owing to its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by all modern computers....
. Each radix
Radix

In numeral system, the base or radix is usually the number of unique Numerical digit, including zero, that a Positional notation numeral system uses to represent numbers....
 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 bit
Bit

A bit is a binary numeral system numerical digit, taking a value of either 0 or 1. Binary digits are a basic unit of information Computer data storage and transmission in digital computing and digital information theory....
s. For example, in base 4,
302104 = 11 00 10 01 002.
Although octal and hexadecimal are widely used in computing
Computing

Computing is usually defined as the activity of using and developing computer technology, computer hardware and computer software. It is the computer-specific part of information technology....
 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
Hilbert curve

A Hilbert curve is a Geometric continuity fractal space-filling curve first described by the German mathematician David Hilbert in 1891....
. 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
Chumashan languages

Chumashan is a family of languages that were spoken on the southern California coast , in neighboring inland regions , and on three nearby islands ....
 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 Ventureņo language number words up to 32 written down by a Spanish priest ca. 1819.

Genetics

Parallels can be drawn between quaternary numerals and the way genetic code
Genetic code

The genetic code is the set of rules by which information encoded in genetic material is Translation into proteins by living cell s. The code defines a mapping between tri-nucleotide sequences, called codons, and amino acids....
 is represented by DNA
DNA

Deoxyribonucleic acid is a nucleic acid that contains the genetics instructions used in the development and functioning of all known living organisms and some viruses....
. The four DNA nucleotide
Nucleotide

Nucleotides are molecules that comprise the structural units of RNA and DNA. Additionally, nucleotides play central roles in metabolism. In that capacity, they serve as sources of chemical energy , participate in cell signaling , and are incorporated into important cofactors of enzymatic reactions ....
s in alphabetical order, abbreviated A
Adenine

Adenine is a nucleobase with a variety of roles in biochemistry including cellular respiration, in the form of both the energy-rich adenosine triphosphate and the cofactor s nicotinamide adenine dinucleotide and flavin adenine dinucleotide , and Protein biosynthesis, as a chemical component of DNA and RNA....
, C
Cytosine

Cytosine is one of the five main bases found in DNA and RNA. It is a pyrimidine derivative, with a heterocyclic aromatic ring and two substituents attached ....
, G
Guanine

Guanine is one of the five main nucleobases found in the nucleic acids DNA and RNA, the others being adenine, cytosine, thymine, and uracil. In DNA, guanine is paired with cytosine....
 and T
Thymine

Thymine is one of the four bases in the nucleic acid of DNA that make up the letters GCAT. The others are adenine, guanine, and cytosine. Thymine always pairs with adenine....
, can be taken to represent the quaternary digits in numerical order
Collation

Collation is the assembly of written information into a standard order. One common type of collation is called alphabetisation, though collation is not limited to ordering letters of the alphabet....
 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 pair
Base pair

In molecular biology, two nucleotides on opposite complementarity DNA or RNA strands that are connected via hydrogen bonds are called a base pair ....
s: A?T and C?G.

For example, the nucleotide sequence GATTACA
Gattaca

Gattaca is a 1997 in film science fiction film drama film written and directed by Andrew Niccol, starring Ethan Hawke, Uma Thurman and Jude Law with supporting roles played by Loren Dean, Gore Vidal and Alan Arkin....
 can be represented by the quaternary number 2033010 (= decimal 9156).

Data transmission

Quaternary line code
Line code

In telecommunication, a line code is a code chosen for use within a communications system for transmission purposes. Line coding is often used for digital data transport....
s have been used for transmission, from the invention of the telegraph
Electrical telegraph

The electrical telegraph is a Telegraphy that uses electric Signal s. The electromagnetic telegraph is a Machine for human-to-human Transmission of coded text messages over wire....
 to the 2B1Q
2B1Q

Two-binary, one-quaternary is a physical layer encoder used for Integrated Services Digital Network basic rate interface. 2B1Q uses four signal levels, which are -450 mV, -150 mV, 150 mV and 450 mV, each equivalent to two bits ....
 code used in modern ISDN circuits.

See also

  • Conversion between bases
    Radix

    In numeral system, the base or radix is usually the number of unique Numerical digit, including zero, that a Positional notation numeral system uses to represent numbers....


External links

  • , includes fractional part, from Math Is Fun
    Math Is Fun

    Math Is Fun is an educational website maintained by Rod Pierce devoted to the concept that mathematics is, indeed, fun.There are several aspects to the website:...
  • Proposes unique symbols for Quaternary and Hexadecimal digits