Quaternary numeral system

'''Quaternary''' {{IPAc-en|k|w|ə|ˈ|t|ɜr|n|ər|i}} is a numeral system with four as its base. It uses the digits 0, 1, 2, and 3 to represent any real number. Conversion from binary is straightforward.

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 thirty-six), 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 systems== {| class="wikitable" |+ '''Numbers zero to sixty-four in standard quaternary (0 to 1000)''' |- align="center" !Decimal ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 |- align="center" !Binary | 0 || 1 || 10 || 11 || 100 || 101 || 110 || 111 || 1,000 || 1,001 || 1,010 || 1,011 || 1,100 || 1,101 || 1,110 || 1,111 |- align="center" style="background: SpringGreen;" | style="background: black; color: SpringGreen;" | Quaternary | 0 || 1 || 2 || 3 || 10 || 11 || 12 || 13 || 20 || 21 || 22 || 23 || 30 || 31 || 32 || 33 |- align="center" !Octal | 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 |- align="center" !Hexadecimal ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! A !! B !! C !! D !! E !! F |- align="center" ! Decimal ! 16 !! 17 !! 18 !! 19 !! 20 !! 21 !! 22 !! 23 !! 24 !! 25 !! 26 !! 27 !! 28 !! 29 !! 30 !! 31 |- align="center" ! Binary | 10,000 || 10,001 || 10,010 || 10,011 || 10,100 || 10,101 || 10,110 || 10,111 || 11,000 || 11,001 || 11,010 || 11,011 || 11,100 || 11,101 || 11,110 || 11,111 |- align="center" style="background: SpringGreen;" | style="background: black; color: SpringGreen;" | Quaternary | 100 || 101 || 102 || 103 || 110 || 111 || 112 || 113 || 120 || 121 || 122 || 123 || 130 || 131 || 132 || 133 |- align="center" ! Octal | 20 || 21 || 22 || 23 || 24 || 25 || 26 || 27 || 30 || 31 || 32 || 33 || 34 || 35 || 36 || 37 |- align="center" ! Hexadecimal ! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 1A !! 1B !! 1C !! 1D !! 1E !! 1F |- align="center" ! Decimal ! 32 !! 33 !! 34 !! 35 !! 36 !! 37 !! 38 !! 39 !! 40 !! 41 !! 42 !! 43 !! 44 !! 45 !! 46 !! 47 |- align="center" ! Binary | 100,000 || 100,001 || 100,010 || 100,011 || 100,100 || 100,101 || 100,110 || 100,111 || 101,000 || 101,001 || 101,010 || 101,011 || 101,100 || 101,101 || 101,110 || 101,111 |- align="center" style="background: SpringGreen;" | style="background: black; color: SpringGreen;" | Quaternary | 200 || 201 || 202 || 203 || 210 || 211 || 212 || 213 || 220 || 221 || 222 || 223 || 230 || 231 || 232 || 233 |- align="center" ! Octal | 40 || 41 || 42 || 43 || 44 || 45 || 46 || 47 || 50 || 51 || 52 || 53 || 54 || 55 || 56 || 57 |- align="center" ! Hexadecimal ! 20 !! 21 !! 22 !! 23 !! 24 !! 25 !! 26 !! 27 !! 28 !! 29 !! 2A !! 2B !! 2C !! 2D !! 2E !! 2F |- align="center" ! Decimal ! 48 !! 49 !! 50 !! 51 !! 52 !! 53 !! 54 !! 55 !! 56 !! 57 !! 58 !! 59 !! 60 !! 61 !! 62 !! 63 |- align="center" ! Binary | 110,000 || 110,001 || 110,010 || 110,011 || 110,100 || 110,101 || 110,110 || 110,111 || 111,000 || 111,001 || 111,010 || 111,011 || 111,100 || 111,101 || 111,110 || 111,111 |- align="center" style="background: SpringGreen;" | style="background: black; color: SpringGreen;" | Quaternary | 300 || 301 || 302 || 303 || 310 || 311 || 312 || 313 || 320 || 321 || 322 || 323 || 330 || 331 || 332 || 333 |- align="center" ! Octal | 60 || 61 || 62 || 63 || 64 || 65 || 66 || 67 || 70 || 71 || 72 || 73 || 74 || 75 || 76 || 77 |- align="center" ! Hexadecimal ! 30 !! 31 !! 32 !! 33 !! 34 !! 35 !! 36 !! 37 !! 38 !! 39 !! 3A !! 3B !! 3C !! 3D !! 3E !! 3F |- align="center" !Decimal ! colspan="16" | 64 |- align="center" !Binary | colspan="16" | 1,000,000 |- align="center" style="background: SpringGreen;" | style="background: black; color: SpringGreen;" | Quaternary | colspan="16" | 1,000 |- align="center" !Octal | colspan="16" | 100 |- align="center" !Hexadecimal ! colspan="16" | 40 |- |}

===Relation to binary and hexadecimal=== {| class="wikitable" style="float:right; text-align:right" |+ addition table |- | {{figure space}}+ || {{figure space}}'''0''' || '''1''' || '''2''' || '''3''' |- | '''0''' || 0 || 1 || 2 || 3 |- | '''1''' || 1 || 2 || 3 || 10 |- | '''2''' || 2 || 3 || 10 || 11 |- | '''3''' || 3 || 10 || 11 || 12 |} As with the octal and hexadecimal numeral systems, quaternary has a special relation to the binary numeral system. Each radix four, eight, and sixteen is a power of two, so the conversion to and from binary is implemented by matching each digit with two, three, or four binary digits, or bits. For example, in quaternary, :2302104 = 10 11 00 10 01 002. Since sixteen is a power of four, conversion between these bases can be implemented by matching each hexadecimal digit with two quaternary digits. In the above example, :23 02 104 = B2416

{| class="wikitable" style="float:right; text-align:right" |+ multiplication table |- | {{figure space}}× || {{figure space}}'''0''' || {{figure space}}'''1''' || '''2''' || '''3''' |- | '''0''' || 0 || 0 || 0 || 0 |- | '''1''' || 0 || 1 || 2 || 3 |- | '''2''' || 0 || 2 || 10 || 12 |- | '''3''' || 0 || 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. 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''.

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

{|class="wikitable" | colspan="3" align="center" | 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''' | colspan="3" align="center" | '''Quaternary base''' Prime factors of the base: '''2''' Prime factors of one below the base: '''3''' Prime factors of one above the base: '''5 (=114)''' Other prime factors: '''13 23 31 101 103 113 131 133''' |- | align="center" | Fraction | align="center" | Prime factors of the denominator | align="center" | Positional representation | align="center" | Positional representation | align="center" | Prime factors of the denominator | align="center" | Fraction |- | align="center" | {{sfrac|1|2}} | align="center" | '''2''' | '''0.5''' | '''0.2''' | align="center" | '''2''' | align="center" | {{sfrac|1|2}} |- | align="center" | {{sfrac|1|3}} | align="center" | '''3''' | bgcolor=#c0c0c0 | '''0.'''3333... = '''0.'''{{overline|3}} | bgcolor=#c0c0c0 | '''0.'''1111... = '''0.'''{{overline|1}} | align="center" | '''3''' | align="center" | {{sfrac|1|3}} |- | align="center" | {{sfrac|1|4}} | align="center" | '''2''' | '''0.25''' | '''0.1''' | align="center" | '''2''' | align="center" | {{sfrac|1|10}} |- | align="center" | {{sfrac|1|5}} | align="center" | '''5''' | '''0.2''' | bgcolor=#c0c0c0 | '''0.'''{{overline|03}} | align="center" | '''11''' | align="center" | {{sfrac|1|11}} |- | align="center" | {{sfrac|1|6}} | align="center" | '''2''', '''3''' | bgcolor=#c0c0c0 | '''0.1'''{{overline|6}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|2}} | align="center" | '''2''', '''3''' | align="center" | {{sfrac|1|12}} |- | align="center" | {{sfrac|1|7}} | align="center" | '''7''' | bgcolor=#c0c0c0 | '''0.'''{{overline|142857}} | bgcolor=#c0c0c0 | '''0.'''{{overline|021}} | align="center" | '''13''' | align="center" | {{sfrac|1|13}} |- | align="center" | {{sfrac|1|8}} | align="center" | '''2''' | '''0.125''' | '''0.02''' | align="center" | '''2''' | align="center" | {{sfrac|1|20}} |- | align="center" | {{sfrac|1|9}} | align="center" | '''3''' | bgcolor=#c0c0c0 | '''0.'''{{overline|1}} | bgcolor=#c0c0c0 | '''0.'''{{overline|013}} | align="center" | '''3''' | align="center" | {{sfrac|1|21}} |- | align="center" | {{sfrac|1|10}} | align="center" | '''2''', '''5''' | '''0.1''' | bgcolor=#c0c0c0 | '''0.0'''{{overline|12}} | align="center" | '''2''', '''11''' | align="center" | {{sfrac|1|22}} |- | align="center" | {{sfrac|1|11}} | align="center" | '''11''' | bgcolor=#c0c0c0 | '''0.'''{{overline|09}} | bgcolor=#c0c0c0 | '''0.'''{{overline|01131}} | align="center" | '''23''' | align="center" | {{sfrac|1|23}} |- | align="center" | {{sfrac|1|12}} | align="center" | '''2''', '''3''' | bgcolor=#c0c0c0 | '''0.08'''{{overline|3}} | bgcolor=#c0c0c0 | '''0.0'''{{overline|1}} | align="center" | '''2''', '''3''' | align="center" | {{sfrac|1|30}} |- | align="center" | {{sfrac|1|13}} | align="center" | '''13''' 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==Occurrence in human languages== {{See also|Quaternary counting system}} Many or all of the Chumashan languages (spoken by the Native American Chumash peoples) originally used a quaternary numeral system, in which the names for numbers were structured according to multiples of four and sixteen, instead of ten. There is a surviving list of Ventureño language number words up to thirty-two written down by a Spanish priest ca. 1819.<ref name="Beeler_1986"/>

The Kharosthi numerals (from the languages of the tribes of Pakistan and Afghanistan) have a partial quaternary numeral system from one to ten.

==Hilbert curves== 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 four sub-quadrants the number will be projected.

==Genetics== {{main|Bioinformatics}} 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.<ref name="CUHK_2010"/> 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). The human genome is 3.2 billion base pairs in length.<ref name="chial08">{{cite journal |last1=Chial |first1=Heidi |title=DNA Sequencing Technologies Key to the Human Genome Project |journal=Nature Education |date=2008 |volume=1 |issue=1 |page=219 |url=https://www.nature.com/scitable/topicpage/dna-sequencing-technologies-key-to-the-human-828/}}</ref>

==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.

The GDDR6X standard, developed by Nvidia and Micron, uses quaternary bits to transmit data.<ref>{{Cite web|url=https://www.nvidia.com/en-us/geforce/graphics-cards/30-series/|title = NVIDIA GeForce RTX 30 Series GPUs Powered by Ampere Architecture}}</ref>

==Computing== Some computers have used quaternary floating point arithmetic including the Illinois ILLIAC II (1962)<ref name="Beebe_2017"/> and the Digital Field System DFS IV and DFS V high-resolution site survey systems.<ref name="Parkinson_2000"/>

==See also== * Conversion between bases * Moser–de Bruijn sequence, the numbers that have only 0 or 1 as their base-4 digits

==References== {{Reflist|refs= <ref name="Beeler_1986">{{cite book |title=Native American Mathematics |chapter=Chumashan Numerals |author-first=Madison S. |author-last=Beeler |editor-first=Michael P. |editor-last=Closs |date=1986 |publisher=University of Texas Press |isbn=0-292-75531-7 |url-access=registration |url=https://archive.org/details/nativeamericanma0000unse }}</ref> <ref name="CUHK_2010">{{cite web |title=Bacterial based storage and encryption device |publisher=The Chinese University of Hong Kong |work=iGEM 2010 |date=2010 |url=http://2010.igem.org/files/presentation/Hong_Kong-CUHK.pdf |access-date=2010-11-27 |url-status=dead |archive-url=https://web.archive.org/web/20101214090439/http://2010.igem.org/files/presentation/Hong_Kong-CUHK.pdf |archive-date=2010-12-14}}</ref> <ref name="Beebe_2017">{{cite book |author-first=Nelson H. F. |author-last=Beebe |title=The Mathematical-Function Computation Handbook - Programming Using the MathCW Portable Software Library |chapter=Chapter H. Historical floating-point architectures |date=2017-08-22 |location=Salt Lake City, UT, USA |publisher=Springer International Publishing AG |edition=1 |lccn=2017947446 |isbn=978-3-319-64109-6 |doi=10.1007/978-3-319-64110-2 |page=948|s2cid=30244721 }}</ref> <ref name="Parkinson_2000">{{cite book |title=High Resolution Site Surveys |author-first=Roger |author-last=Parkinson |publisher=CRC Press |date=2000-12-07 |chapter=Chapter 2 - High resolution digital site survey systems - Chapter 2.1 - Digital field recording systems |edition=1 |isbn=978-0-20318604-6 |page=24 |chapter-url=https://books.google.com/books?id=Ocip5vpLD4wC&pg=PA24 |access-date=2019-08-18 |quote=[...] 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)</ref> }}

== External links == {{commons category}} * [https://web.archive.org/web/20110609101715/http://www.mathsisfun.com/numbers/convert-base.php?to=quaternary Quaternary Base Conversion], includes fractional part, from Math Is Fun * [https://web.archive.org/web/20110605062735/http://www.hauptmech.com/base42/wiki/index.php?title=Base4 Base42] Proposes unique symbols for Quaternary and Hexadecimal digits

Category:Power-of-two numeral systems Category:4 (number)