ﺍﻟﺴﻼﻡ ﻋﻠﻴﻜﻢ
ﺍﻻﻧﻈﻤﻪ ﻛﻠﻬﺎ ﻣﻀﺎﻋﻔﺎﺕ ﺍﻻﺛﻨﻴﻦ
......-32-16-8-4-2-1
ﻧﻈﺎﻡ ﻋﺪ ﺛﻨﺎﺋﻲ(binary)
ﻳﺘﺄﻟﻒ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ ﻣﻦ ﺭﻗﻤﻴﻦ ﻓﻘﻂ
ﻭﻫﻤﺎ ﺍﻟﺼﻔﺮ ﻭﺍﻟﻮﺍﺣﺪ .
ﺣﻴﻨﻤﺎ ﻧﻜﺘﺐ ﺃﻱ ﻋﺪﺩ ﻓﻲ ﺍﻟﻨﻈﺎﻡ ﺍﻟﺜﻨﺎﺋﻲ
ﻳﻤﺜﻞ ﻛﻞ ﺭﻗﻢ ﺑﻌﺎﻣﻞ ﻣﻀﺮﻭﺏ ﺑﺎﻟﻌﺪﺩ
(2) ﻣﺮﻓﻮﻉ ﺇﻟﻰ ﻗﻮﺓ ﻣﺎ .
ﻓﻤﺜﻼ: ﺍﻟﻌﺪﺩ(19) ﻳﻜﺘﺐ ﻓﻲ ﺍﻟﻨﻈﺎﻡ
ﺍﻟﺜﻨﺎﺋﻲ.(10011)
ﻧﻈﺎﻡ ﻋﺪ ﻋﺸﺮﻱ(Decima)
ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ ﻫﻮ ﺍﻟﻤﺴﺘﺨﺪﻡ ﻓﻲ ﺣﻴﺎﺗﻨﺎ
ﺍﻟﻴﻮﻣﻴﺔ ﻭﻳﺘﺄﻟﻒ ﻣﻦ ﻋﺸﺮ ﺃﺭﻗﺎﻡ ﻫﻲ
(9-8-7-6-5-4-3-2-1-0)
ﺍﺳﺎﺱ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ ﻫﻮ r =10
ﻗﺎﻋﺪﺓ: ﺍﻥ ﻋﺪﺩ ﺃﺭﻗﺎﻡ ﺃﻱ ﻧﻈﺎﻡ ﻫﻮ 0)
--------r-1(
ﻓﺎﻟﻨﻈﺎﻡ ﺍﻟﻌﺸﺮﻱ ﻳﺘﺄﻟﻒ (9-------0)
ﻧﻈﺎﻡ ﻋﺪ ﺛﻤﺎﻧﻲ (octal)
ﻳﺘﺄﻟﻒ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ ﻣﻦ ﺛﻤﺎﻧﻲ ﺃﺭﻗﺎﻡ
ﺍﻥ ﺍﻟﺮﺍﺑﻄﺔ ﺑﻴﻦ ﺍﻟﻨﻈﺎﻡ ﺍﻟﺜﻨﺎﺋﻲ ﻭﺍﻟﺜﻤﺎﻧﻲ
ﺃﻥ ﻛﻞ ﻋﺪﺩ ﻣﻦ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ
ﻳﻤﺜﻞ ﺑﺜﻼﺙ ﺃﺭﻗﺎﻡ ﺛﻨﺎﺋﻴﺔ ﻓﻤﺜﻼ 110)
(101 ﻓﻲ ﺍﻟﻨﻈﺎﻡ ﺍﻟﺜﻨﺎﺋﻲ ﻳﺴﺎﻭﻱ
ﺍﻟﻌﺪﺩ(56)ﺣﻴﺚ (101) ﺗﻤﺜﻞ(5) ﻭ(110)
ﺗﻤﺜﻞ. (6)
ﺍﺳﺎﺱ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ ) r =8 ﺃﺭﻗﺎﻣﻪ
.(7-----0
ﻧﻈﺎﻡ ﻋﺪ ﺳﺪﺍﺳﻲ ﻋﺸﺮ
(HEXADECIMAL)
ﻳﺘﺄﻟﻒ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ ﻣﻦ ﺳﺖ ﻋﺸﺮ ﺭﻗﻢ
ﺍﻷﻭﺍﺋﻞ ﻫﻲ(9-----0) ﻭﺍﻟﺒﺎﻗﻲ
ﻫﻲ ﺍﻟﺤﺮﻑ)A,B,C,D,E,F(
ﺍﻟﺮﺍﺑﻄﺔ ﺑﻴﻦ ﺍﻟﺜﻨﺎﺋﻲ ﻭﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ ﺃﻥ ﻛﻞ
ﺭﻗﻢ ﻣﻦ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ ﻳﻤﺜﻞ
ﺑﺄﺭﺑﻊ ﺃﺭﻗﺎﻡ ﺛﻨﺎﺋﻴﺔ
ﻓﻤﺜﻼ(1110 1010) ﻓﻲ ﺍﻟﺜﻨﺎﺋﻲ ﻳﺴﺎﻭﻱ
ﺍﻟﻌﺪﺩ(AE) ﺣﻴﺚ (1010)ﺗﻤﺜﻞ(A)
ﻭ(1110) ﺗﻤﺜﻞ.(E)
ﺍﺳﺎﺱ ﻫﺬﺍ ﺍﻟﻨﻈﺎﻡ ﻫﻮ r =16 .
ﻃﺮﻕ ﺍﻟﺘﺤﻮﻳﻞ ﺑﻴﻦ ﺍﻷﻧﻈﻤﺔ
ﺃﻭﻻ: ﻣﻦ ﺍﻟﻨﻈﺎﻡ ﺍﻟﻌﺸﺮﻱ ﺇﻟﻰ ﺍﻟﺜﻨﺎﺋﻲ
ﻧﻮﺿﺢ ﻃﺮﻳﻘﺔ ﺍﻟﺘﺤﻮﻳﻞ ﻣﻦ ﺧﻼﻝ ﺍﻟﻤﺜﺎﻝ
ﺍﻟﺘﺎﻟﻲ
ﻣﺜﺎﻝ: ﺣﻮﻝ ﺍﻟﻌﺪﺩ ﺍﻟﻌﺸﺮﻱ( 46)ﺍﻟﻰ
ﺍﻟﻨﻈﺎﻡ ﺍﻟﺜﻨﺎﺋﻲ
ﺍﻟﺤﻞ 46/2 0 23 =
23/2 1 11 =
11/2 1 5=
5/21 2=
2/20 1=
1/21 0=
ﻭﻣﻦ ﺛﻢ ﻧﺮﺗﺒﻬﺎ ﺣﺴﺐ ﺃﻫﻤﻴﺔ ﺍﻟﺨﺎﻧﺎﺕ
(101110)
ﺛﺎﻧﻴﺎ: ﻣﻦ ﺍﻟﺜﻨﺎﺋﻲ ﺇﻟﻰ ﺍﻟﻌﺸﺮﻱ
ﻣﺜﺎﻝ: ﺣﻮﻝ ﺍﻟﻌﺪ ﺍﻟﺜﻨﺎﺋﻲ (101110) ﺇﻟﻰ
ﻋﺪﺩ ﻓﻲ ﺍﻟﻨﻈﺎﻡ ﺍﻟﻌﺸﺮﻱ
ﺍﻟﺤﻞ 1*1^2 + 0*0^2) = (101110)
+ 0*4^2 + 1*3^2 + 1*2^2 +
(1*5^2
32+0+8+4+1+0 = (101110)
45 = (101110)
ﺛﺎﻟﺜﺎ:ﻣﻦ ﺍﻟﺜﻤﺎﻧﻲ ﺇﻟﻰ ﺍﻟﺜﻨﺎﺋﻲ
ﻟﻠﺘﺤﻮﻳﻞ ﻣﻦ ﺍﻟﺜﻤﺎﻧﻲ ﺇﻟﻰ ﺍﻟﺜﻨﺎﺋﻲ ﻧﻌﺘﺒﺮ
ﻛﻞ ﺭﻗﻢ ﺛﻤﺎﻧﻲ ﻳﻤﻜﻦ ﺗﻤﺜﻴﻠﻪ ﺑﻮﺍﺳﻄﺔ
ﺛﻼﺙ ﺃﺭﻗﺎﻡ ﺛﻨﺎﺋﻴﺔ
ﻣﺜﺎﻝ: ﺣﻮﻝ ﺍﻟﻌﺪﺩ ﺍﻟﺜﻤﺎﻧﻲ(56)ﺍﻟﻰ
ﺍﻟﻨﻈﺎﻡ ﺍﻟﺜﻨﺎﺋﻲ
ﺍﻟﺤﻞ
(56)=(101 100)
ﺣﻴﺚ5=(101) ﻭ6=(110)
ﺭﺍﺑﻌﺎ:ﻣﻦ ﺍﻟﻌﺸﺮﻱ ﺇﻟﻰ ﺍﻟﺜﻤﺎﻧﻲ
ﻣﺜﺎﻝ: ﺣﻮﻝ ﺍﻟﻌﺪﺩ ﺍﻟﻌﺸﺮﻱ(46)ﺍﻟﻰ
ﺍﻟﺜﻤﺎﻧﻲ
8/46 = 5,75 ------- 0,756 = 8*
5/8 = 0,625 ------- 0,6255 = 8*
ﺍﺫﻥ ﺍﻟﻌﺪﺩ ﺍﻟﻌﺸﺮﻱ (46) ﻳﺴﺎﻭﻱ ﻓﻲ
ﺍﻟﺜﻤﺎﻧﻲ(56)
ﺧﺎﻣﺴﺎ: ﻣﻦ ﺍﻟﺜﻤﺎﻧﻲ ﺇﻟﻰ ﺍﻟﻌﺸﺮﻱ
ﻟﻠﺘﺤﻮﻳﻞ ﻓﺎﻧﻨﺎ ﻧﺤﻮﻝ ﺍﻟﺜﻤﺎﻧﻲ ﺇﻟﻰ ﺍﻟﺜﻨﺎﺋﻲ
ﻭﺍﻟﺜﻨﺎﺋﻲ ﺇﻟﻰ ﺍﻟﻌﺸﺮﻱ
ﻣﺜﺎﻝ: ﺣﻮﻝ ﺍﻟﻌﺪﺩ ﺍﻟﺜﻤﺎﻧﻲ(56)ﺍﻟﻰ
ﺍﻟﻨﻈﺎﻡ ﺍﻟﻌﺸﺮﻱ
ﺍﻟﺤﻞ
= (101110) (56) = (101110)
0*5^2+1*4^2+1*3^2+1*2^2+1*1^2+0*0^2
32+0+8+4+2+0 = (101110)
46 = (101110) ﺍﺫﻥ ﺍﻟﺜﻤﺎﻧﻲ (56)
ﻳﺴﺎﻭﻱ ﻓﻲ ﺍﻟﻌﺸﺮﻱ (46)
ﻣﻨﻘﻮﻝ ﻟﻠﻔﺎﺋﺪﻩ