9 points) Converting from binary to decimal: a. (2 points) Convert 10010
to decimal. b. (3 points) Convert 110101
to decimal. c. (4 points) Convert 11001110
to decimal. 7) (9 points) Converting from decimal to binary: a. (2 points) Convert 11
to binary. b. (3 points) Convert 43
to binary. c. (4 points) Convert 298
to binary.
The Correct Answer and Explanation is:
Binary to Decimal Conversions:
a. Convert 10010210010_2100102 to Decimal:
To convert binary to decimal, we sum the values of the positions where there is a “1”, starting from the rightmost digit. The binary number 10010210010_2100102 can be expanded as: 1×24+0×23+0×22+1×21+0×201 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^01×24+0×23+0×22+1×21+0×20 =16+0+0+2+0=18= 16 + 0 + 0 + 2 + 0 = 18=16+0+0+2+0=18
So, 100102=181010010_2 = 18_{10}100102=1810.
b. Convert 1101012110101_21101012 to Decimal:
For 1101012110101_21101012, we expand similarly: 1×25+1×24+0×23+1×22+0×21+1×201 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^01×25+1×24+0×23+1×22+0×21+1×20 =32+16+0+4+0+1=53= 32 + 16 + 0 + 4 + 0 + 1 = 53=32+16+0+4+0+1=53
So, 1101012=5310110101_2 = 53_{10}1101012=5310.
c. Convert 11001110211001110_2110011102 to Decimal:
For 11001110211001110_2110011102, the expansion is: 1×27+1×26+0×25+0×24+1×23+1×22+1×21+0×201 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^01×27+1×26+0×25+0×24+1×23+1×22+1×21+0×20 =128+64+0+0+8+4+2+0=206= 128 + 64 + 0 + 0 + 8 + 4 + 2 + 0 = 206=128+64+0+0+8+4+2+0=206
So, 110011102=2061011001110_2 = 206_{10}110011102=20610.
Decimal to Binary Conversions:
a. Convert 111011_{10}1110 to Binary:
To convert decimal to binary, we repeatedly divide the number by 2, recording the remainders.
- 11÷2=511 \div 2 = 511÷2=5 remainder 1
- 5÷2=25 \div 2 = 25÷2=2 remainder 1
- 2÷2=12 \div 2 = 12÷2=1 remainder 0
- 1÷2=01 \div 2 = 01÷2=0 remainder 1
Now, write the remainders from bottom to top: 1110=1011211_{10} = 1011_21110=10112.
b. Convert 431043_{10}4310 to Binary:
Following the same method for 431043_{10}4310:
- 43÷2=2143 \div 2 = 2143÷2=21 remainder 1
- 21÷2=1021 \div 2 = 1021÷2=10 remainder 1
- 10÷2=510 \div 2 = 510÷2=5 remainder 0
- 5÷2=25 \div 2 = 25÷2=2 remainder 1
- 2÷2=12 \div 2 = 12÷2=1 remainder 0
- 1÷2=01 \div 2 = 01÷2=0 remainder 1
Reading the remainders: 4310=101011243_{10} = 101011_24310=1010112.
c. Convert 29810298_{10}29810 to Binary:
For 29810298_{10}29810:
- 298÷2=149298 \div 2 = 149298÷2=149 remainder 0
- 149÷2=74149 \div 2 = 74149÷2=74 remainder 1
- 74÷2=3774 \div 2 = 3774÷2=37 remainder 0
- 37÷2=1837 \div 2 = 1837÷2=18 remainder 1
- 18÷2=918 \div 2 = 918÷2=9 remainder 0
- 9÷2=49 \div 2 = 49÷2=4 remainder 1
- 4÷2=24 \div 2 = 24÷2=2 remainder 0
- 2÷2=12 \div 2 = 12÷2=1 remainder 0
- 1÷2=01 \div 2 = 01÷2=0 remainder 1
Reading the remainders from bottom to top: 29810=1001011102298_{10} = 100101110_229810=1001011102.
Explanation:
Binary to decimal involves multiplying each binary digit by 2 raised to the power of its position (starting from 0 for the rightmost bit) and summing the results. Decimal to binary is done by dividing the number by 2, keeping track of the remainders at each step, and reading them from bottom to top to form the binary number. These conversions are fundamental for understanding how data is represented in computers.
