Two's complement of -8 is 

Explanation:

Two's Complement Representation

Definition: Two's complement is a mathematical operation used in computer science to represent signed integers in binary. It is widely used because of its simplicity in handling arithmetic operations and representing negative numbers. In this system, positive numbers are represented in standard binary form, while negative numbers are obtained by inverting all the bits of the absolute value (in binary) and adding 1 to the least significant bit (LSB).

Steps to Find Two's Complement:

1. Start with the binary representation of the absolute value: Represent the number in binary. For example, the absolute value of -8 is 8, which in binary (for a 4-bit system) is 1000.
2. Invert all the bits: Flip all the bits of the binary representation. For 1000, the inverted bits become 0111.
3. Add 1 to the result: Adding 1 to the inverted binary number gives 0111 + 1 = 1000.

Thus, the two's complement representation of -8 (in a 4-bit system) is 1000.

Correct Option Analysis:

The correct option is:

Option 1: 1000

This is the correct two's complement representation of -8 in a 4-bit system. Following the steps outlined above, the binary value 1000 is derived by inverting the bits of the binary representation of 8 and adding 1.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: 1010

This option is incorrect because 1010 is not derived using the two's complement method for -8 in a 4-bit binary system. The inverted bits of 1000 are 0111, and adding 1 results in 1000, not 1010.

Option 3: 110

This option is incorrect because 110 does not represent -8 in two's complement form within a 4-bit binary system. It neither matches the expected binary format nor adheres to the two's complement operation rules.

Option 4: 1110

This option is incorrect as it does not represent the two's complement of -8 in a 4-bit binary system. The correct result, as derived earlier, is 1000. 1110 represents a different value altogether in two's complement.

Conclusion:

Understanding two's complement representation is essential for dealing with signed integers in binary arithmetic. The process involves inverting the bits of the absolute value and adding 1 to the least significant bit. This method simplifies arithmetic operations and provides a consistent way to handle negative numbers in binary systems.

In the context of the given problem, -8 is correctly represented as 1000 in two's complement form for a 4-bit binary system. This representation is derived systematically by following the steps outlined above, ensuring accuracy in binary arithmetic operations and signed integer representation.