Convert the following positive binary numbers to their decimal number equivalents, showing all steps. a) 111111 b) 1110.11 c) 101111.101 d) 10111110.00001 e) 110011.011
The Correct Answer and Explanation is:
Conversion of Binary to Decimal:
a) 111111
Each binary digit (bit) is multiplied by 2 raised to the power of its position from right to left, starting at 0.1×25+1×24+1×23+1×22+1×21+1×20=32+16+8+4+2+1=631 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 \\ = 32 + 16 + 8 + 4 + 2 + 1 = \boxed{63}1×25+1×24+1×23+1×22+1×21+1×20=32+16+8+4+2+1=63
b) 1110.11
Integer part:1×23+1×22+1×21+0×20=8+4+2+0=141 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 8 + 4 + 2 + 0 = 141×23+1×22+1×21+0×20=8+4+2+0=14
Fractional part:1×2−1+1×2−2=0.5+0.25=0.751 \times 2^{-1} + 1 \times 2^{-2} = 0.5 + 0.25 = 0.751×2−1+1×2−2=0.5+0.25=0.75
Combined:14+0.75=14.7514 + 0.75 = \boxed{14.75}14+0.75=14.75
c) 101111.101
Integer part:1×25+0×24+1×23+1×22+1×21+1×20=32+0+8+4+2+1=471 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 0 + 8 + 4 + 2 + 1 = 471×25+0×24+1×23+1×22+1×21+1×20=32+0+8+4+2+1=47
Fractional part:1×2−1+0×2−2+1×2−3=0.5+0+0.125=0.6251 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} = 0.5 + 0 + 0.125 = 0.6251×2−1+0×2−2+1×2−3=0.5+0+0.125=0.625
Combined:47+0.625=47.62547 + 0.625 = \boxed{47.625}47+0.625=47.625
d) 10111110.00001
Integer part:1×27+0×26+1×25+1×24+1×23+1×22+1×21+0×20=128+0+32+16+8+4+2+0=1901 \times 2^7 + 0 \times 2^6 + 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 \\ = 128 + 0 + 32 + 16 + 8 + 4 + 2 + 0 = 1901×27+0×26+1×25+1×24+1×23+1×22+1×21+0×20=128+0+32+16+8+4+2+0=190
Fractional part:0×2−1+0×2−2+0×2−3+0×2−4+1×2−5=0.031250 \times 2^{-1} + 0 \times 2^{-2} + 0 \times 2^{-3} + 0 \times 2^{-4} + 1 \times 2^{-5} = 0.031250×2−1+0×2−2+0×2−3+0×2−4+1×2−5=0.03125
Combined:190+0.03125=190.03125190 + 0.03125 = \boxed{190.03125}190+0.03125=190.03125
e) 110011.011
Integer part:1×25+1×24+0×23+0×22+1×21+1×20=32+16+0+0+2+1=511 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 16 + 0 + 0 + 2 + 1 = 511×25+1×24+0×23+0×22+1×21+1×20=32+16+0+0+2+1=51
Fractional part:0×2−1+1×2−2+1×2−3=0+0.25+0.125=0.3750 \times 2^{-1} + 1 \times 2^{-2} + 1 \times 2^{-3} = 0 + 0.25 + 0.125 = 0.3750×2−1+1×2−2+1×2−3=0+0.25+0.125=0.375
Combined:51+0.375=51.37551 + 0.375 = \boxed{51.375}51+0.375=51.375
Explanation (300 Words)
Binary numbers operate on base 2, meaning each digit represents a power of 2. Converting from binary to decimal involves summing the value of each binary digit after multiplying it by 2 raised to its positional index. For integer parts, the index starts from 0 on the right and increases leftward. For fractional parts, the index begins at -1 moving rightward.
In example (a), all bits are 1, so each power of 2 from 2⁰ to 2⁵ contributes to the total. This results in a sum of 63.
In (b), both integer and fractional parts are evaluated separately. The integer part (1110) translates to 14, and the fractional part (0.11) adds 0.75, resulting in 14.75.
In (c), a similar process is followed. The integer portion (101111) equals 47, and the fractional part (0.101) adds 0.625, yielding a combined total of 47.625.
In (d), the longer binary number still follows the same structure. The integer part (10111110) gives 190, while the fractional component (0.00001) contributes a small value of 0.03125. Together, they total 190.03125.
Lastly, (e) contains a mixture of 1s and 0s on both sides of the decimal. The integer portion sums to 51, and the fractional part adds 0.375, giving a final result of 51.375.
This process ensures accurate conversion using place-value rules of binary positional notation.
