Convert the following to its Two’s Complement binary representation. If in decimal or hexadecimal, convert to binary first. Use as many bits as you deem necessary.
a. 0b 1000
b. 0b 1110
c. 0d 7
d. d 0
e. 0d 15
f. 0x C4
g. 0x FF
The correct answer and explanation is:
To convert the given numbers to their Two’s Complement binary representation, we need to follow a few basic steps. For this task, I’ll convert each value to binary (if it’s not already), and then determine its Two’s Complement representation.
a. 0b 1000 (Binary)
This is already in binary. For a positive number like this, the Two’s complement is simply the binary value itself, padded to the desired bit-width if necessary. Let’s assume a 8-bit representation for simplicity:
8-bit Two’s Complement: 00001000
b. 0b 1110 (Binary)
This is also in binary. To represent this number in Two’s Complement, it stays the same since it’s positive. For an 8-bit representation:
8-bit Two’s Complement: 00001110
c. 0d 7 (Decimal)
Convert 7 to binary: 0111 (in 4 bits for simplicity).
Since it’s positive, the Two’s complement representation is the same as the binary representation.
8-bit Two’s Complement: 00000111
d. 0d 0 (Decimal)
The number 0 in binary is 0. For the Two’s Complement, it remains zero.
8-bit Two’s Complement: 00000000
e. 0d 15 (Decimal)
Convert 15 to binary: 1111. To represent it in Two’s Complement:
8-bit Two’s Complement: 00001111
f. 0x C4 (Hexadecimal)
Convert C4 from hexadecimal to binary: 11000100.
Since it’s positive, the Two’s complement representation remains the same.
8-bit Two’s Complement: 11000100
g. 0x FF (Hexadecimal)
Convert FF from hexadecimal to binary: 11111111. This is a negative number in Two’s Complement. For an 8-bit representation, FF represents -1 in Two’s complement, as the most significant bit (MSB) is 1 (indicating negativity).
8-bit Two’s Complement: 11111111
Explanation
The Two’s Complement is used to represent both positive and negative numbers in binary. The key features are:
- For positive numbers, the Two’s complement representation is the same as the regular binary.
- For negative numbers, the Two’s complement is obtained by inverting all bits of the binary representation of the number and adding 1 to the result.
- The Two’s complement system allows both positive and negative numbers to be represented efficiently in binary, where the MSB indicates the sign of the number (0 for positive and 1 for negative).