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 systems

**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 hexadecimal

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.

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: 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 languages

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 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 4 sub-quadrants the number will be projected.

Genetics

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 transmission

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

Computing

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]

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 from the original (PDF) on 2010-12-14. Retrieved 2010-11-27.CS1 maint: location (link)
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.
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)

External links

Quaternary Base Conversion, includes fractional part, from Math Is Fun
Base42 Proposes unique symbols for Quaternary and Hexadecimal digits

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[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 from the original (PDF) on 2010-12-14. Retrieved 2010-11-27.CS1 maint: location (link)

[Beebe_2017-3] 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.

[Parkinson_2000-4] 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)