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Quaternary numeral system

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

Four is the largest number within the subitizing range and one of two numbers that is both a square and a highly composite number (the other being 36), making quaternary a convenient choice for a base at this scale. Despite being twice as large, its radix economy is equal to that of binary. However, it fares no better in the localization of prime numbers (the smallest better base being the primorial base six, senary).

Quaternary 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 systemsEdit

Numbers zero to sixty-four in standard quaternary
Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Quaternary 0 1 2 3 10 11 12 13 20 21 22 23 30 31 32 33
Octal 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17
Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F
Binary 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111
Decimal 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Quaternary 100 101 102 103 110 111 112 113 120 121 122 123 130 131 132 133
Octal 20 21 22 23 24 25 26 27 30 31 32 33 34 35 36 37
Hexadecimal 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F
Binary 10000 10001 10010 10011 10100 10101 10110 10111 11000 11001 11010 11011 11100 11101 11110 11111
Decimal 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47
Quaternary 200 201 202 203 210 211 212 213 220 221 222 223 230 231 232 233
Octal 40 41 42 43 44 45 46 47 50 51 52 53 54 55 56 57
Hexadecimal 20 21 22 23 24 25 26 27 28 29 2A 2B 2C 2D 2E 2F
Binary 100000 100001 100010 100011 100100 100101 100110 100111 101000 101001 101010 101011 101100 101101 101110 101111
Decimal 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
Quaternary 300 301 302 303 310 311 312 313 320 321 322 323 330 331 332 333 1000
Octal 60 61 62 63 64 65 66 67 70 71 72 73 74 75 76 77 100
Hexadecimal 30 31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F 40
Binary 110000 110001 110010 110011 110100 110101 110110 110111 111000 111001 111010 111011 111100 111101 111110 111111 1000000

Relation to binary and hexadecimalEdit

addition
table
+ 1 2 3
1 2 3 10
2 3 10 11
3 10 11 12

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,

2302104 = 10 11 00 10 01 002.

Since 16 is a power of 4, conversion between these bases can be implemented by matching each hexadecimal digit with 2 quaternary digits. In the above example,

23 02 104 = B2416
multiplication
table
× 1 2 3
1  1 2 3
2  2 10 12
3  3 12 21

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.

Although quaternary has limited practical use, it can be helpful if it is ever necessary to perform hexadecimal arithmetic without a calculator. Each hexadecimal digit can be turned into a pair of quaternary digits, and then arithmetic can be performed relatively easily before converting the end result back to hexadecimal. Quaternary is convenient for this purpose, since numbers have only half the digit length compared to binary, while still having very simple multiplication and addition tables with only three unique non-trivial elements.

By analogy with byte and nybble, a quaternary digit is sometimes called a crumb.

FractionsEdit

Due to having only factors of two, many quaternary fractions have repeating digits, although these tend to be fairly simple:

Decimal base
Prime factors of the base: 2, 5
Prime factors of one below the base: 3
Prime factors of one above the base: 11
Other prime factors: 7 13 17 19 23 29 31
Quaternary base
Prime factors of the base: 2
Prime factors of one below the base: 3
Prime factors of one above the base: 11
Other prime factors: 13 23 31 101 103 113 131 133
Fraction Prime factors
of the denominator
Positional representation Positional representation Prime factors
of the denominator
Fraction
1/2 2 0.5 0.2 2 1/2
1/3 3 0.3333... = 0.3 0.1111... = 0.1 3 1/3
1/4 2 0.25 0.1 2 1/10
1/5 5 0.2 0.03 11 1/11
1/6 2, 3 0.16 0.02 2, 3 1/12
1/7 7 0.142857 0.021 13 1/13
1/8 2 0.125 0.02 2 1/20
1/9 3 0.1 0.013 3 1/21
1/10 2, 5 0.1 0.012 2, 11 1/22
1/11 11 0.09 0.01131 23 1/23
1/12 2, 3 0.083 0.01 2, 3 1/30
1/13 13 0.076923 0.010323 31 1/31
1/14 2, 7 0.0714285 0.0102 2, 13 1/32
1/15 3, 5 0.06 0.01 3, 11 1/33
1/16 2 0.0625 0.01 2 1/100
1/17 17 0.0588235294117647 0.0033 101 1/101
1/18 2, 3 0.05 0.0032 2, 3 1/102
1/19 19 0.052631578947368421 0.003113211 103 1/103
1/20 2, 5 0.05 0.003 2, 11 1/110
1/21 3, 7 0.047619 0.003 3, 13 1/111
1/22 2, 11 0.045 0.002322 2, 23 1/112
1/23 23 0.0434782608695652173913 0.00230201121 113 1/113
1/24 2, 3 0.0416 0.002 2, 3 1/120
1/25 5 0.04 0.0022033113 11 1/121
1/26 2, 13 0.0384615 0.0021312 2, 31 1/122
1/27 3 0.037 0.002113231 3 1/123
1/28 2, 7 0.03571428 0.0021 2, 13 1/130
1/29 29 0.0344827586206896551724137931 0.00203103313023 131 1/131
1/30 2, 3, 5 0.03 0.002 2, 3, 11 1/132
1/31 31 0.032258064516129 0.00201 133 1/133
1/32 2 0.03125 0.002 2 1/200
1/33 3, 11 0.03 0.00133 3, 23 1/201
1/34 2, 17 0.02941176470588235 0.00132 2, 101 1/202
1/35 5, 7 0.0285714 0.001311 11, 13 1/203
1/36 2, 3 0.027 0.0013 2, 3 1/210

Occurrence in human languagesEdit

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 have a partial base 4 counting system from 1 to decimal 10.

Hilbert curvesEdit

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.

GeneticsEdit

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 transmissionEdit

Quaternary line codes have been used for transmission, from the invention of the telegraph to the 2B1Q code used in modern ISDN circuits.

ComputingEdit

Some computers have used quaternary floating point arithmetic including the Illinois ILLIAC II (1962)[3] and the Digital Field System DFS IV and DFS V high-resolution site survey systems.[4]

See alsoEdit

ReferencesEdit

  1. ^ Beeler, Madison S. (1986). "Chumashan Numerals". In Closs, Michael P. (ed.). Native American Mathematics. ISBN 0-292-75531-7.
  2. ^ "Bacterial based storage and encryption device" (PDF). iGEM 2010: The Chinese University of Hong Kong. 2010. Archived from the original (PDF) on 2010-12-14. Retrieved 2010-11-27.
  3. ^ Beebe, Nelson H. F. (2017-08-22). "Chapter H. Historical floating-point architectures". The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library (1 ed.). Salt Lake City, UT, USA: Springer International Publishing AG. p. 948. doi:10.1007/978-3-319-64110-2. ISBN 978-3-319-64109-6. LCCN 2017947446.
  4. ^ Parkinson, Roger (2000-12-07). "Chapter 2 - High resolution digital site survey systems - Chapter 2.1 - Digital field recording systems". High Resolution Site Surveys (1 ed.). CRC Press. p. 24. ISBN 978-0-20318604-6. ISBN 0-20318604-4. Retrieved 2019-08-18. […] Systems such as the [Digital Field System] DFS IV and DFS V were quaternary floating-point systems and used gain steps of 12 dB. […] (256 pages)

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