View source for Quaternary numeral system

{{Short description|Base-4 numeral system}}
{{Use dmy dates|date=August 2019|cs1-dates=l}}
{{Table Numeral Systems}}

'''Quaternary''' {{IPAc-en|k|w|ə|ˈ|t|ɜr|n|ər|i}} is a [[numeral system]] with [[4|four]] as its [[radix|base]]. It uses the [[numerical digit|digit]]s 0, 1, 2, and 3 to represent any [[real number]]. Conversion from [[Binary number|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 number]]s and [[irrational number]]s.  See [[decimal]] and [[binary numeral system|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 number|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]]
|-
| + || '''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]] 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 [[bit]]s.  For example, in quaternary,
:230210<sub>4</sub> = 10&nbsp;11&nbsp;00&nbsp;10&nbsp;01&nbsp;00<sub>2</sub>.
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&nbsp;02&nbsp;10<sub>4</sub> = B24<sub>16</sub>

{| class="wikitable" style="float:right; text-align:right"
|+ [[multiplication table]]
|-
| × || '''1''' || '''2''' || '''3'''
|-
| '''1''' || {{figure space}}1 || 2 || 3
|-
| '''2''' || {{figure space}}2 || 10 || 12
|-
| '''3''' || {{figure space}}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<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''3'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Magenta">'''11'''</span></SMALL><br><SMALL>Other prime factors: <span style="color:Red">'''7 13 17 19 23 29 31'''</span></SMALL>
| colspan="3" align="center" | '''Quaternary base'''<br><SMALL>Prime factors of the base: <span style="color:Green">'''2'''</span></SMALL><br><SMALL>Prime factors of one below the base: <span style="color:Blue">'''3'''</span></SMALL><br><SMALL>Prime factors of one above the base: <span style="color:Magenta">'''5 (=11<sub>4</sub>)'''</span></SMALL><br><SMALL>Other prime factors: <span style="color:Red">'''13 23 31 101 103 113 131 133'''</span></SMALL>
|-
| align="center" | Fraction
| align="center" | <SMALL>Prime factors of<br>the denominator</SMALL>
| align="center" | Positional<br>representation
| align="center" | Positional<br>representation
| align="center" | <SMALL>Prime factors of<br>the denominator</SMALL>
| align="center" | Fraction
|-
| align="center" | {{sfrac|1|2}}
| align="center" | <span style="color:Green">'''2'''</span>
| '''0.5'''
| '''0.2'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | {{sfrac|1|2}}
|-
| align="center" | {{sfrac|1|3}}
| align="center" | <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.'''3333... = '''0.'''{{overline|3}}
| bgcolor=#c0c0c0 | '''0.'''1111... = '''0.'''{{overline|1}}
| align="center" | <span style="color:Blue">'''3'''</span>
| align="center" | {{sfrac|1|3}}
|-
| align="center" | {{sfrac|1|4}}
| align="center" | <span style="color:Green">'''2'''</span>
| '''0.25'''
| '''0.1'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | {{sfrac|1|10}}
|-
| align="center" | {{sfrac|1|5}}
| align="center" | <span style="color:Green">'''5'''</span>
| '''0.2'''
| bgcolor=#c0c0c0 | '''0.'''{{overline|03}}
| align="center" | <span style="color:Magenta">'''11'''</span>
| align="center" | {{sfrac|1|11}}
|-
| align="center" | {{sfrac|1|6}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.1'''{{overline|6}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|2}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| align="center" | {{sfrac|1|12}}
|-
| align="center" | {{sfrac|1|7}}
| align="center" | <span style="color:Red">'''7'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|142857}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|021}}
| align="center" | <span style="color:Red">'''13'''</span>
| align="center" | {{sfrac|1|13}}
|-
| align="center" | {{sfrac|1|8}}
| align="center" | <span style="color:Green">'''2'''</span>
| '''0.125'''
| '''0.02'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | {{sfrac|1|20}}
|-
| align="center" | {{sfrac|1|9}}
| align="center" | <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|1}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|013}}
| align="center" | <span style="color:Blue">'''3'''</span>
| align="center" | {{sfrac|1|21}}
|-
| align="center" | {{sfrac|1|10}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
| '''0.1'''
| bgcolor=#c0c0c0 | '''0.0'''{{overline|12}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span>
| align="center" | {{sfrac|1|22}}
|-
| align="center" | {{sfrac|1|11}}
| align="center" | <span style="color:Magenta">'''11'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|09}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|01131}}
| align="center" | <span style="color:Red">'''23'''</span>
| align="center" | {{sfrac|1|23}}
|-
| align="center" | {{sfrac|1|12}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.08'''{{overline|3}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|1}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| align="center" | {{sfrac|1|30}}
|-
| align="center" | {{sfrac|1|13}}
| align="center" | <span style="color:Red">'''13'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|076923}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|010323}}
| align="center" | <span style="color:Red">'''31'''</span>
| align="center" | {{sfrac|1|31}}
|-
| align="center" | {{sfrac|1|14}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|714285}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|102}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span>
| align="center" | {{sfrac|1|32}}
|-
| align="center" | {{sfrac|1|15}}
| align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|6}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|01}}
| align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Magenta">'''11'''</span>
| align="center" | {{sfrac|1|33}}
|-
| align="center" | {{sfrac|1|16}}
| align="center" | <span style="color:Green">'''2'''</span>
| '''0.0625'''
| '''0.01'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | {{sfrac|1|100}}
|-
| align="center" | {{sfrac|1|17}}
| align="center" | <span style="color:Red">'''17'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0588235294117647}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|0033}}
| align="center" | <span style="color:Red">'''101'''</span>
| align="center" | {{sfrac|1|101}}
|-
| align="center" | {{sfrac|1|18}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|5}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|032}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| align="center" | {{sfrac|1|102}}
|-
| align="center" | {{sfrac|1|19}}
| align="center" | <span style="color:Red">'''19'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|052631578947368421}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|003113211}}
| align="center" | <span style="color:Red">'''103'''</span>
| align="center" | {{sfrac|1|103}}
|-
| align="center" | {{sfrac|1|20}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Green">'''5'''</span>
|'''0.05'''
| bgcolor=#c0c0c0 | '''0.0'''{{overline|03}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span>
| align="center" | {{sfrac|1|110}}
|-
| align="center" | {{sfrac|1|21}}
| align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''7'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|047619}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|003}}
| align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''13'''</span>
| align="center" | {{sfrac|1|111}}
|-
| align="center" | {{sfrac|1|22}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Magenta">'''11'''</span>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|45}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|02322}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''23'''</span>
| align="center" | {{sfrac|1|112}}
|-
| align="center" | {{sfrac|1|23}}
| align="center" | <span style="color:Red">'''23'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0434782608695652173913}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|00230201121}}
| align="center" | <span style="color:Red">'''113'''</span>
| align="center" | {{sfrac|1|113}}
|-
| align="center" | {{sfrac|1|24}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.041'''{{overline|6}}
| bgcolor=#c0c0c0 | '''0.00'''{{overline|2}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| align="center" | {{sfrac|1|120}}
|-
| align="center" | {{sfrac|1|25}}
| align="center" | <span style="color:Green">'''5'''</span>
|'''0.04'''
| bgcolor=#c0c0c0 | '''0.'''{{overline|0022033113}}
| align="center" | <span style="color:Magenta">'''11'''</span>
| align="center" | {{sfrac|1|121}}
|-
| align="center" | {{sfrac|1|26}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|384615}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|021312}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''31'''</span>
| align="center" | {{sfrac|1|122}}
|-
| align="center" | {{sfrac|1|27}}
| align="center" | <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|037}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|002113231}}
| align="center" | <span style="color:Blue">'''3'''</span>
| align="center" | {{sfrac|1|123}}
|-
| align="center" | {{sfrac|1|28}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''7'''</span>
| bgcolor=#c0c0c0 | '''0.03'''{{overline|571428}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|021}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''13'''</span>
| align="center" | {{sfrac|1|130}}
|-
| align="center" | {{sfrac|1|29}}
| align="center" | <span style="color:Red">'''29'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|0344827586206896551724137931}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|00203103313023}}
| align="center" | <span style="color:Red">'''131'''</span>
| align="center" | {{sfrac|1|131}}
|-
| align="center" | {{sfrac|1|30}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>, <span style="color:Green">'''5'''</span>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|3}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|02}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>, <span style="color:Magenta">'''11'''</span>
| align="center" | {{sfrac|1|132}}
|-
| align="center" | {{sfrac|1|31}}
| align="center" | <span style="color:Red">'''31'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|032258064516129}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|00201}}
| align="center" | <span style="color:Red">'''133'''</span>
| align="center" | {{sfrac|1|133}}
|-
| align="center" | {{sfrac|1|32}}
| align="center" | <span style="color:Green">'''2'''</span>
|'''0.03125'''
|'''0.002'''
| align="center" | <span style="color:Green">'''2'''</span>
| align="center" | {{sfrac|1|200}}
|-
| align="center" | {{sfrac|1|33}}
| align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Magenta">'''11'''</span>
| bgcolor=#c0c0c0 | '''0.'''{{overline|03}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|00133}}
| align="center" | <span style="color:Blue">'''3'''</span>, <span style="color:Red">'''23'''</span>
| align="center" | {{sfrac|1|201}}
|-
| align="center" | {{sfrac|1|34}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''17'''</span>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|2941176470588235}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|0132}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Red">'''101'''</span>
| align="center" | {{sfrac|1|202}}
|-
| align="center" | {{sfrac|1|35}}
| align="center" | <span style="color:Green">'''5'''</span>, <span style="color:Red">'''7'''</span>
| bgcolor=#c0c0c0 | '''0.0'''{{overline|285714}}
| bgcolor=#c0c0c0 | '''0.'''{{overline|001311}}
| align="center" | <span style="color:Magenta">'''11'''</span>, <span style="color:Red">'''13'''</span>
| align="center" | {{sfrac|1|203}}
|-
| align="center" | {{sfrac|1|36}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| bgcolor=#c0c0c0 | '''0.02'''{{overline|7}}
| bgcolor=#c0c0c0 | '''0.0'''{{overline|013}}
| align="center" | <span style="color:Green">'''2'''</span>, <span style="color:Blue">'''3'''</span>
| align="center" | {{sfrac|1|210}}
|}

==Occurrence in human languages==
{{See also|Quaternary counting system}}
Many or all of the [[Chumashan languages]] (spoken by the Native American [[Chumash people|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|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 curve]]s.  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 [[nucleotide]]s in [[alphabetical order]], abbreviated [[adenine|A]], [[cytosine|C]], [[guanine|G]], and [[thymine|T]], can be taken to represent the quaternary digits in [[Collation|numerical order]] 0, 1, 2, and 3. With this encoding, the [[:wikt:complementary|complementary]] digit pairs 0↔3, and 1↔2 (binary 00↔11 and 01↔10) match the complementation of the [[base pair]]s: 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 number|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 code]]s have been used for transmission, from the [[Electrical telegraph#Gauss-Weber telegraph and Carl Steinheil|invention of the telegraph]] to the [[2B1Q]] code used in modern [[ISDN]] circuits.

The GDDR6X standard, developed by [[Nvidia]] and [[Micron Technology|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&nbsp;II]] (1962)<ref name="Beebe_2017"/> and the Digital Field System DFS&nbsp;IV and DFS&nbsp;V high-resolution site survey systems.<ref name="Parkinson_2000"/>

==See also==
* [[Radix#Conversion among bases|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&nbsp;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]]