Convert the decimal number 32 into an 8-bit signed binary number.
a) 10000000
b) 00100000
c) 11100000
d) 11000000
The Correct Answer and Explanation is:
The correct answer is:
b) 00100000
Explanation:
To convert a decimal number into an 8-bit signed binary number, we need to understand how signed numbers are represented in binary. In an 8-bit system, the most significant bit (MSB) is used to represent the sign of the number. If the MSB is 0, the number is positive; if it’s 1, the number is negative. The remaining 7 bits are used to represent the magnitude of the number.
- Step 1: Convert the decimal number to binary.
- We are given the decimal number 32. To convert it into binary, we repeatedly divide the number by 2 and record the remainders.
- 32 ÷ 2 = 16, remainder 0
- 16 ÷ 2 = 8, remainder 0
- 8 ÷ 2 = 4, remainder 0
- 4 ÷ 2 = 2, remainder 0
- 2 ÷ 2 = 1, remainder 0
- 1 ÷ 2 = 0, remainder 1 Reading the remainders from bottom to top, we get the binary representation of 32 as 100000.
- Step 2: Determine the signed binary representation.
- Since 32 is positive, the MSB (most significant bit) will be 0.
- To fit this into 8 bits, we simply add leading zeros to the binary representation. So, we add two zeros at the beginning of 100000 to make it an 8-bit number: 00100000.
- Step 3: Verify the options.
- The binary number 00100000 corresponds to the decimal number 32 in an 8-bit signed binary system.
- Option b) 00100000 is the correct answer.
Why the other options are incorrect:
- a) 10000000: This represents -128 in an 8-bit signed binary system (due to the MSB being 1).
- c) 11100000: This represents a negative number, not 32.
- d) 11000000: This also represents a negative number and is incorrect for the positive decimal 32.
Thus, the correct 8-bit signed binary representation of 32 is 00100000.