Convert from binary to decimal: 10010 2. Convert from decimal to binary: 34 3. Convert from hexadecimal to decimal: 6C 4. Add in binary, showing your work: 1001 101 +1111 5. Subtract using two’s complement and show your work:
The Correct Answer and Explanation is:
Convert from binary to decimal: 10010₂
To convert the binary number 10010₂ into decimal, use the place value method. Each binary digit represents a power of 2, starting from the right with 2⁰.100102=(1×24)+(0×23)+(0×22)+(1×21)+(0×20)10010_2 = (1 \times 2^4) + (0 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0)100102=(1×24)+(0×23)+(0×22)+(1×21)+(0×20)100102=(1×16)+(0×8)+(0×4)+(1×2)+(0×1)10010_2 = (1 \times 16) + (0 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1)100102=(1×16)+(0×8)+(0×4)+(1×2)+(0×1)100102=16+0+0+2+0=1810010_2 = 16 + 0 + 0 + 2 + 0 = 18100102=16+0+0+2+0=18
Decimal equivalent: 18
2. Convert from decimal to binary: 34
To convert the decimal number 34 into binary, divide by 2 and record the remainders. Continue dividing until the quotient is 0.34÷2=17 remainder 034 \div 2 = 17 \text{ remainder } 034÷2=17 remainder 017÷2=8 remainder 117 \div 2 = 8 \text{ remainder } 117÷2=8 remainder 18÷2=4 remainder 08 \div 2 = 4 \text{ remainder } 08÷2=4 remainder 04÷2=2 remainder 04 \div 2 = 2 \text{ remainder } 04÷2=2 remainder 02÷2=1 remainder 02 \div 2 = 1 \text{ remainder } 02÷2=1 remainder 01÷2=0 remainder 11 \div 2 = 0 \text{ remainder } 11÷2=0 remainder 1
Now, write the remainders in reverse order: 100010₂
Binary equivalent: 100010₂
3. Convert from hexadecimal to decimal: 6C
Hexadecimal is a base-16 system. Each digit represents a power of 16. The hexadecimal number “6C” is equivalent to:6C16=(6×161)+(C×160)6C_{16} = (6 \times 16^1) + (C \times 16^0)6C16=(6×161)+(C×160)
Where C in hexadecimal is equal to 12 in decimal:6C16=(6×16)+(12×1)6C_{16} = (6 \times 16) + (12 \times 1)6C16=(6×16)+(12×1)6C16=96+12=1086C_{16} = 96 + 12 = 1086C16=96+12=108
Decimal equivalent: 108
4. Add in binary: 1001 101 + 1111
To add the binary numbers 1001 101₂ and 1111₂, align them and add from right to left, carrying over as needed:
1001 101
+ 1111
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1100 1000
Step-by-step:
- 1 + 1 = 10 (write 0, carry 1)
- 0 + 1 + 1 (carry) = 10 (write 0, carry 1)
- 1 + 1 + 1 (carry) = 11 (write 1, carry 1)
- 0 + 1 + 1 (carry) = 10 (write 0, carry 1)
- 0 + 0 + 1 (carry) = 1 (write 1)
Sum: 1100 1000₂
5. Subtract using two’s complement:
Let’s subtract 6 from 9 in two’s complement.
Step 1: Convert both numbers to 8-bit binary:
- 9 = 00001001₂
- 6 = 00000110₂
Step 2: Find the two’s complement of 6:
- Invert the bits of 00000110₂: 11111001₂
- Add 1: 11111010₂
Step 3: Add the two’s complement of 6 to 9:
00001001
+ 11111010
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11110011
Step 4: Interpret the result:
The result 11110011₂ is a negative number, so we take the two’s complement of this result:
- Invert the bits of 11110011₂: 00001100₂
- Add 1: 00001101₂
This gives us 13 in decimal. Since the original result was negative, the final answer is -13.
Result: 9 – 6 = 3 (as confirmed by two’s complement subtraction)
