Mavzu: Shaxsiy kompyuterning axboriy-mantiqiy asoslari. Kompyuterda axborotlarni qayta ishlashning arifmetik asoslari Reja

Shu qonuniyatdan foydalanib, quyidagi birinchi 10 ta butun sonni hosil qilamiz:

10 lik sanoq sistemasida	0	1	2	3	4	5	6	7	8	9
2 lik sanoq sistemasida	0	1	10	11	100	101	110	111	1000	1001
5 lik sanoq sistemasida	0	1	2	3	4	10	11	12	13	14
8 lik sanoq sistemasida	0	1	2	3	4	5	6	7	10	11
16 lik sanoq sistemasida	0	1	2	3	4	5	6	7	8	9

Jadvaldan ko’rinadiki, turli sanoq sistemalarida o’xshash sonlar bor ekan. Shu sababli bu sonlarni farqlash uchun informatikada 10₂, 10₅, 10₈ kabi belgilash qabul qilingan.
Pozitsiyali bo’lmagan sanoq sistemalari:
Sonlarni boshqacha usul va qoidalar asosida shakllantiriladigan sanoq sistemalar ham mavjud. Bunday sistemalar jumlasiga rim raqamlari sistemasi kiradi. Mazkur sistemada raqamlar va sonlarni ifodalovchi maxsus belgilar to’plami bo’lib, ixtiyoriy son ana shu belgilar yordamida tuziladi. Bunda ishlatilayotgan belgilar raqam va sonlar guruhiga bo’linmaydi hamda shu belgi sonni ifodalovchi raqamlar ketma-ketligining qaysi o’rnida turishidan qat’i nazar o’z miqdorini bildiraveradi.
Masalan, rim raqamlari² sistemasida ma’lum belgilar mavjud bo’lib, ular quyidagilarni ifodalaydi:
1 raqami «I» belgi bilan;
5 raqami «V» belgi bilan;
10 soni «X» belgi bilan;
50 soni «L» belgi bilan;
100 soni «С» belgi bilan;
500 soni «D» belgi bilan;
1000 soni «М» belgi bilan ifodalanadi.
Ular orasidagi boshqa sonlar miqdor jihatdan katta bo’lganidan ma’lum sonni ayirish yoki miqdor jihatdan kichik bo’lganiga biror sonni qo’shish ko’rinishida belgilanadi.
Masalan:

Son o’qilishi	Belgisi	Rim raqamlari sistemasida	Mazmuni
To’rt	4	IV	5-1
Olti	6	VI	5+1
To’qqiz	9	IX	10-1
O’n to’rt	14	XIV	10+(5—1)
O’n to’qqiz	19	XIX	10+(10-1)
Qirq besh	45	XLV	10+10+10+10+5
Qirq to’qqiz	49	XLIX	50-1
Bir yuz oltmish sakkiz	168	CLXVIII	100+50+10+5+3
Bir ming to’qqiz yuz to’qson yeti	1997	MCMXCVII	1000+(1000-100)+(100-10)+5+2

Quyidagi jadvalda ba’zi birliklarni ko’rishlaring mumkin:

Arab raqamlari	Rim raqamlari	Arab raqamlari	Rim raqamlari	Arab raqamlari	Rim raqamlari	Arab raqamlari	Rim raqamlari
1	I	104	CIV	270	CCLXX	1966	MCMLXVI
2	II	105	CV	280	CCLXXX	1967	MCMLXVII
3	III	106	CVI	290	CCXC	1968	MCMLXVIII
4	IV	107	CVII	300	CCC	1969	MCMLXIX
5	V	108	CVIII	310	CCCX	1970	MCMLXX
6	VI	109	CIX	320	CCCXX	1971	MCMLXXI
7	VII	110	CX	330	CCCXXX	1972	MCMLXXII
8	VIII	111	CXI	340	CCCXL	1973	MCMLXXIII
9	IX	112	CXII	350	CCCL	1974	MCMLXXIV
10	X	113	CXIII	360	CCCLX	1975	MCMLXXV
11	XI	114	CXIV	370	CCCLXX	1976	MCMLXXVI
12	XII	115	CXV	380	CCCLXXX	1977	MCMLXXVII
13	XIII	116	CXVI	390	CCCXC	1978	MCMLXXVIII
14	XIV	117	CXVII	400	CD	1979	MCMLXXIX
15	XV	118	CXVIII	410	CDX	1980	MCMLXXX
16	XVI	119	CXIX	420	CDXX	1981	MCMLXXXI
17	XVII	120	CXX	430	CDXXX	1982	MCMLXXXII
18	XVIII	121	CXXI	440	CDXL	1983	MCMLXXXIII
19	XIX	122	CXXII	450	CDL	1984	MCMLXXXIV
20	XX	123	CXXIII	460	CDLX	1985	MCMLXXXV
21	XXI	124	CXXIV	470	CDLXX	1986	MCMLXXXVI
22	XXII	125	CXXV	480	CDLXXX	1987	MCMLXXXVII
23	XXIII	126	CXXVI	490	CDXC	1988	MCMLXXXVIII
24	XXIV	127	CXXVII	500	D	1989	MCMLXXXIX
25	XXV	128	CXXVIII	600	DC	1990	MCMXC
26	XXVI	129	CXXIX	700	DCC	1991	MCMXCI
27	XXVII	130	CXXX	800	DCCC	1992	MCMXCII
28	XXVIII	131	CXXXI	900	CM	1993	MCMXCIII
29	XXIX	132	CXXXII	1000	M	1994	MCMXCIV
30	XXX	133	CXXXIII	1100	MC	1995	MCMXCV
31	XXXI	134	CXXXIV	1200	MCC	1996	MCMXCVI
32	XXXII	135	CXXXV	1300	MCCC	1997	MCMXCVII
33	XXXIII	136	CXXXVI	1400	MCD	1998	MCMXCVIII
34	XXXIV	137	CXXXVII	1500	MD	1999	MCMXCIX
35	XXXV	138	CXXXVIII	1600	MDC	2000	MM
36	XXXVI	139	CXXXIX	1700	MDCC	2001	MMI
37	XXXVII	140	CXL	1800	MDCCC	2002	MMII
38	XXXVIII	141	CXLI	1900	MCM	2003	MMIII
39	XXXIX	142	CXLII	1901	MCMI	2004	MMIV
40	XL	143	CXLIII	1902	MCMII	2005	MMV
41	XLI	144	CXLIV	1903	MCMIII	2006	MMVI
42	XLII	145	CXLV	1904	MCMIV	2007	MMVII
43	XLIII	146	CXLVI	1905	MCMV	2008	MMVIII
44	XLIV	147	CXLVII	1906	MCMVI	2009	MMIX
45	XLV	148	CXLVIII	1907	MCMVII	2010	MMX
46	XLVI	149	CXLIX	1908	MCMVIII	2011	MMXI
47	XLVII	150	CL	1909	MCMIX	2012	MMXII
48	XLVIII	151	CLI	1910	MCMX	2013	MMXIII
49	XLIX	152	CLII	1911	MCMXI	2014	MMXIV
50	L	153	CLIII	1912	MCMXII	2015	MMXV
51	LI	154	CLIV	1913	MCMXIII	2016	MMXVI
52	LII	155	CLV	1914	MCMXIV	2017	MMXVII
53	LIII	156	CLVI	1915	MCMXV	2018	MMXVIII
54	LIV	157	CLVII	1916	MCMXVI	2019	MMXIX
55	LV	158	CLVIII	1917	MCMXVII	2020	MMXX
56	LVI	159	CLIX	1918	MCMXVIII	2021	MMXXI
57	LVII	160	CLX	1919	MCMXIX	2022	MMXXII
58	LVIII	161	CLXI	1920	MCMXX	2023	MMXXIII
59	LIX	162	CLXII	1921	MCMXXI	2024	MMXXIV
60	LX	163	CLXIII	1922	MCMXXII	2025	MMXXV
61	LXI	164	CLXIV	1923	MCMXXIII	2026	MMXXVI
62	LXII	165	CLXV	1924	MCMXXIV	2027	MMXXVII
63	LXIII	166	CLXVI	1925	MCMXXV	2028	MMXXVIII
64	LXIV	167	CLXVII	1926	MCMXXVI	2029	MMXXIX
65	LXV	168	CLXVIII	1927	MCMXXVII	2030	MMXXX
66	LXVI	169	CLXIX	1928	MCMXXVIII	2031	MMXXXI
67	LXVII	170	CLXX	1929	MCMXXIX	2032	MMXXXII
68	LXVIII	171	CLXXI	1930	MCMXXX	2033	MMXXXIII
69	LXIX	172	CLXXII	1931	MCMXXXI	2034	MMXXXIV
70	LXX	173	CLXXIII	1932	MCMXXXII	2035	MMXXXV
71	LXXI	174	CLXXIV	1933	MCMXXXIII	2036	MMXXXVI
72	LXXII	175	CLXXV	1934	MCMXXXIV	2037	MMXXXVII
73	LXXIII	176	CLXXVI	1935	MCMXXXV	2038	MMXXXVIII
74	LXXIV	177	CLXXVII	1936	MCMXXXVI	2039	MMXXXIX
75	LXXV	178	CLXXVIII	1937	MCMXXXVII	2040	MMXL
76	LXXVI	179	CLXXIX	1938	MCMXXXVIII	2041	MMXLI
77	LXXVII	180	CLXXX	1939	MCMXXXIX	2042	MMXLII
78	LXXVIII	181	CLXXXI	1940	MCMXL	2043	MMXLIII
79	LXXIX	182	CLXXXII	1941	MCMXLI	2044	MMXLIV
80	LXXX	183	CLXXXIII	1942	MCMXLII	2045	MMXLV
81	LXXXI	184	CLXXXIV	1943	MCMXLIII	2046	MMXLVI
82	LXXXII	185	CLXXXV	1944	MCMXLIV	2047	MMXLVII
83	LXXXIII	186	CLXXXVI	1945	MCMXLV	2048	MMXLVIII
84	LXXXIV	187	CLXXXVII	1946	MCMXLVI	2049	MMXLIX
85	LXXXV	188	CLXXXVIII	1947	MCMXLVII	2050	MML
86	LXXXVI	189	CLXXXIX	1948	MCMXLVIII	2100	MMC
87	LXXXVII	190	CXC	1949	MCMXLIX	2200	MMCC
88	LXXXVIII	191	CXCI	1950	MCML	2300	MMCCC
89	LXXXIX	192	CXCII	1951	MCMLI	2400	MMCD
90	XC	193	CXCIII	1952	MCMLII	2500	MMD
91	XCI	194	CXCIV	1953	MCMLIII	2600	MMDC
92	XCII	195	CXCV	1954	MCMLIV	2700	MMDCC
93	XCIII	196	CXCVI	1955	MCMLV	2800	MMDCCC
94	XCIV	197	CXCVII	1956	MCMLVI	2900	MMCM
95	XCV	198	CXCVIII	1957	MCMLVII	3000	MMM
96	XCVI	199	CXCIX	1958	MCMLVIII	5000	V
97	XCVII	200	CC	1959	MCMLIX	10,000	X
98	XCVIII	210	CCX	1960	MCMLX	50,000	L
99	XCIX	220	CCXX	1961	MCMLXI	100,000	C
100	C	230	CCXXX	1962	MCMLXII	500,000	D
101	CI	240	CCXL	1963	MCMLXIII	1,000,000	M
102	CII	250	CCL	1964	MCMLXIV
103	CIII	260	CCLX	1965	MCMLXV

Bir nechta o’nlik raqamlarni o’z ichiga olgan raqam quyidagi misollarda bo’lgani kabi, har biri uchun eng yuqori darajadan pastgacha rim raqamli ekvivalenti qo’shilishi bilan tuziladi:³

• 
  39 = XXX + IX = XXXIX.
  
• 
  246 = CC + XL + VI = CCXLVI.
  
• 
  789 = DCC + LXXX + IX = DCCLXXXIX.
  
• 
  2421 = MM + CD + XX + I = MMCDXXI.
  
• 
  160 = C + LX = CLX
  
• 
  207 = CC + VII = CCVII
  
• 
  1009 = M + IX = MIX
  
• 
  1066 = M + LX + VI = MLXVI
  
• 
  1776 = M + DCC + LXX + VI = MDCCLXXVI 
  
• 
  1918 = M + CM + X + VIII = MCMXVIII
  
• 
  1954 = M + CM + L + IV = MCMLIV
  
• 
  2014 = MM + X + IV = MMXIV
  
• 
  3999 = MMM + CM + XC + IX
  
• 
  LIV = L + IV = 50 + 5 – 1 = 54
  
• 
  MIX = M + IX = 1000 + 10 – 1 = 1009
  
• 
  MMXXIV = MM + XX + IV = 1000 + 1000 + 10 + 10 + 5 – 1 = 2024
  

Ushbu yozuvda namoyish etilishi mumkin bo’lgan eng katta raqam – 3,999,999 (MMM CMXCIXCMXCIX), g’arbda arab raqamlari paydo bo’lishidan oldin, tizimning qadimgi va o’rta asr foydalanuvchilari kattaroq raqamlarni yozish uchun turli xil vositalardan foydalanganlar.

Shuning uchun ham bunday sanoq sistema raqamlari sondagi o’rniga bog’liq bo ‘lmagan, ya’ni pozitsiyaga bog’liq bo’lmagan sanoq sistema deyiladi.