نظام عد رباعي

نظام العد الرباعي (بالإنجليزية: Quaternary numeral system) هو نظام عد ذو رقم أساس 4، ويسمى هذا النظام عد رباعي وهي تستخدم غالباً تحت عنوان "أنظمة خاصة" أي أنها ليست شائعة في استخدامها كثيراً في تحويلها إلى نظام عد آخر.^[1]

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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			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
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

انظر أيضاً

نظام عد ثنائي
نظام عد ثماني
نظام عد خماسي
نظام عد سداسي
نظام عد ثماني
Conversion between bases
Moser–de Bruijn sequence, the numbers that have only 0 or 1 as their base-4 digits

الهامش

^ "Archived copy" (PDF). Archived from the original (PDF) on December 14, 2010. Retrieved November 27, 2010. {{cite web}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)CS1 maint: archived copy as title (link)

External links

Quaternary Base Conversion, includes fractional part, from Math Is Fun
Base42 Proposes unique symbols for Quaternary and Hexadecimal digits
Visualization of nucleotide sequence, Visualization of numeral systems

[1] "Archived copy" (PDF). Archived from the original (PDF) on December 14, 2010. Retrieved November 27, 2010. {{cite web}}: Unknown parameter |deadurl= ignored (|url-status= suggested) (help)CS1 maint: archived copy as title (link)

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