一、十进制数按位权展开
1、十进制数按位权展开的具体实例
( 456.75 ) 10 = 400 + 50 + 6 + 0.7 + 0.05 = 4 × 1 0 2 + 5 × 1 0 1 + 6 × 1 0 0 + 7 × 1 0 − 1 + 5 × 1 0 − 2 (456.75)_{10} = 400 + 50 + 6 + 0.7 + 0.05 = 4 × 10^2 + 5 × 10^1 + 6 × 10^0 + 7 × 10^{-1} + 5 × 10^{-2} (456.75)10=400+50+6+0.7+0.05=4×102+5×101+6×100+7×10−1+5×10−2
2、十进制数按位权展开的一般形式
a n − 1 … … a 3 a 2 a 1 a 0 . a − 1 a − 2 … … a − m = ∑ i = n − 1 − m a i ⋅ 1 0 i , a i ∈ { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } a_{n-1}……a_3a_2a_1a_0.a_{-1}a_{-2}……a_{-m}=\displaystyle \sum_{i=n-1}^{-m}a_i\cdot10^i,a_i \in \{0,1,2,3,4,5,6,7,8,9\} an−1……a3a2a1a0.a−1a−2……a−m=i=n−1∑−mai⋅10i,ai∈{ 0,1,2,3,4,5,6,7,8,9}
二、二进制数按位权展开
1、二进制数按位权展开的具体实例
( 1101.01 ) 2 = 1 × 2 3 + 1 × 2 2 + 0 × 2 1 + 1 × 2 0 + 0 × 2 − 1 + 1 × 2 − 2 = 8 + 4 + 1 + 0.25 = 13.25 (1101.01)_2 = 1\times2^3 + 1\times2^2 + 0\times2^1 + 1\times2^0 + 0\times2^{-1} + 1\times2^{-2}=8+4+1+0.25=13.25 (1101.01)2=1×23+1×22+0×21+1×20+0×2−1+1×2−2=8+4+1+0.25=13.25
2、二进制数按位权展开的一般形式
a n − 1 … … a 3 a 2 a 1 a 0 . a − 1 a − 2 … … a − m = ∑ i = n − 1 − m a i ⋅ 2 i , a i ∈ { 0 , 1 } a_{n-1}……a_3a_2a_1a_0.a_{-1}a_{-2}……a_{-m}=\displaystyle \sum_{i=n-1}^{-m}a_i\cdot2^i,a_i \in \{0,1\} an−1……a3a2a1a0.a−1a−2……a−m=i=n−1∑−mai⋅2i,ai∈{ 0,1}
三、八进制数和十六进制数按位权展开
1、八进制数按位权展开的一般形式
a n − 1 … … a 3 a 2 a 1 a 0 . a − 1 a − 2 … … a − m = ∑ i = n − 1 − m a i ⋅ 8 i , a i ∈ { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } a_{n-1}……a_3a_2a_1a_0.a_{-1}a_{-2}……a_{-m}=\displaystyle \sum_{i=n-1}^{-m}a_i\cdot8^i,a_i \in \{0,1,2,3,4,5,6,7\} an−1……a3a2a1a0.a−1a−2……a−m=i=n−1∑−mai⋅8i,ai∈{ 0,1,2,3,4,5,6,7}
例如: ( 267.5 ) 8 = 2 × 8 2 + 6 × 8 1 + 7 × 8 0 + 5 × 8 − 1 = 128 + 48 + 7 + 0.625 = 183.625 (267.5)_8=2\times8^2+6\times8^1+7\times8^0+5\times8^{-1}=128+48+7+0.625=183.625 (267.5)8=2×82+6×81+7×80+5×8−1=128+48+7+0.625=183.625
2、十六进制数按位权展开的一般形式
a n − 1 … … a 3 a 2 a 1 a 0 . a − 1 a − 2 … … a − m = ∑ i = n − 1 − m a i ⋅ 1 6 i , a i ∈ { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , A , B , C , D , E , F } a_{n-1}……a_3a_2a_1a_0.a_{-1}a_{-2}……a_{-m}=\displaystyle \sum_{i=n-1}^{-m}a_i\cdot16^i,a_i \in \{0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F\} an−1……a3a2a1a0.a−1a−2……a−m=i=n−1∑−mai⋅16i,ai∈{ 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F}
例如: ( 4 A 8 F . 5 ) 16 = 4 × 1 6 3 + 10 × 1 6 2 + 8 × 1 6 1 + 15 × 1 6 0 + 5 × 1 6 − 1 = 16384 + 2560 + 128 + 15 + 0.3125 = 19087.3125 (4A8F.5)_{16}=4\times16^3+10\times16^2+8\times16^1+15\times16^0+5\times16^{-1}=16384+2560+128+15+0.3125=19087.3125 (4A8F.5)16=4×163+10×162+8×161+15×160+5×16−1=16384+2560+128+15+0.3125=19087.3125