Quaternary numeral system

Quaternary/kwəˈtɜːrnəri/is anumeral systemwithfouras itsbase.It uses thedigits0, 1, 2, and 3 to represent anyreal number.Conversion frombinaryis straightforward.

Four is the largest number within thesubitizingrange and one of two numbers that is both a square and ahighly composite number(the other being thirty-six), making quaternary a convenient choice for a base at this scale. Despite being twice as large, itsradix economyis equal to that of binary. However, it fares no better in the localization of prime numbers (the smallest better base being theprimorialbase six,senary).

Quaternary shares with all fixed-radixnumeral systems many properties, such as the ability to represent any real number with a canonical representation (almost unique) and the characteristics of the representations ofrational numbersandirrational numbers.Seedecimalandbinaryfor a discussion of these properties.

Relation to other positional number systems

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

Relation to binary and hexadecimal

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

As with theoctalandhexadecimalnumeral systems, quaternary has a special relation to thebinary numeral system.Eachradixfour, eight, and sixteen is apower of two,so the conversion to and from binary is implemented by matching each digit with two, three, or four binary digits, orbits.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

multiplication table
×	1	2	3
1	1	2	3
2	2	10	12
3	3	12	21

Although octal and hexadecimal are widely used incomputingandcomputer programmingin 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 withbyteandnybble,a quaternary digit is sometimes called acrumb.

Fractions

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:5 (=11₄) 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 languages

Many or all of theChumashan languages(spoken by the Native AmericanChumash 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 ofVentureño languagenumber words up to thirty-two written down by a Spanish priest ca. 1819.^[1]

TheKharosthi 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 2DHilbert 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

Parallels can be drawn between quaternary numerals and the waygenetic codeis represented byDNA.The four DNAnucleotidesinAlpha betical order,abbreviatedA,C,G,andT,can be taken to represent the quaternary digits innumerical order0, 1, 2, and 3. With this encoding, thecomplementarydigit pairs 0↔3, and 1↔2 (binary 00↔11 and 01↔10) match the complementation of thebase 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 (=decimal9156 orbinary10 00 11 11 00 01 00). Thehuman genomeis 3.2 billion base pairs in length.^[3]

Data transmission

Quaternaryline codeshave been used for transmission, from theinvention of the telegraphto the2B1Qcode used in modernISDNcircuits.

The GDDR6X standard, developed byNvidiaandMicron,uses quaternary bits to transmit data.^[4]

Computing

Some computers have usedquaternary floating pointarithmetic including theIllinois ILLIAC II(1962)^[5]and the Digital Field System DFS IV and DFS V high-resolution site survey systems.^[6]

References

^Beeler, Madison S. (1986). "Chumashan Numerals". In Closs, Michael P. (ed.).Native American Mathematics.ISBN 0-292-75531-7.
^"Bacterial based storage and encryption device"(PDF).iGEM 2010.The Chinese University of Hong Kong.2010. Archived fromthe original(PDF)on 14 December 2010.Retrieved27 November2010.
^Chial, Heidi (2008)."DNA Sequencing Technologies Key to the Human Genome Project".Nature Education.1(1): 219.
^"NVIDIA GeForce RTX 30 Series GPUs Powered by Ampere Architecture".
^Beebe, Nelson H. F. (22 August 2017). "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.S2CID 30244721.
^Parkinson, Roger (7 December 2000)."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.Retrieved18 August2019.[...] 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)

External links

Quaternary Base Conversion,includes fractional part, fromMath Is Fun
Base42Proposes unique symbols for Quaternary and Hexadecimal digits

[Beeler_1986-1] Beeler, Madison S. (1986). "Chumashan Numerals". In Closs, Michael P. (ed.).Native American Mathematics.ISBN 0-292-75531-7.

[CUHK_2010-2] "Bacterial based storage and encryption device"(PDF).iGEM 2010.The Chinese University of Hong Kong.2010. Archived fromthe original(PDF)on 14 December 2010.Retrieved27 November2010.

[chial08-3] Chial, Heidi (2008)."DNA Sequencing Technologies Key to the Human Genome Project".Nature Education.1(1): 219.

[4] "NVIDIA GeForce RTX 30 Series GPUs Powered by Ampere Architecture".

[Beebe_2017-5] Beebe, Nelson H. F. (22 August 2017). "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.S2CID 30244721.

[Parkinson_2000-6] Parkinson, Roger (7 December 2000)."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.Retrieved18 August2019.[...] 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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