Understanding 2's Complement for Negative Numbers

The question asks for the 2's complement representation of the number $(-100)_{10}$ in an 8-bit system. The 2's complement is a standard way computers represent negative numbers.

Step 1: Convert the Absolute Value to Binary

First, we find the binary representation of the absolute value of the number, which is $(100)_{10}$.

• Divide 100 by powers of 2:
  • $100 \div 64 = 1$ remainder $36$
  • $36 \div 32 = 1$ remainder $4$
  • $4 \div 16 = 0$ remainder $4$
  • $4 \div 8 = 0$ remainder $4$
  • $4 \div 4 = 1$ remainder $0$
  • $0 \div 2 = 0$ remainder $0$
  • $0 \div 1 = 0$ remainder $0$
• Reading the remainders from bottom up is incorrect. A better method is direct conversion:
  • $100 = 64 + 32 + 4$
  • $100 = 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0$
• So, the binary representation of $(100)_{10}$ is $1100100_{2}$.

Step 2: Represent in 8 Bits

Since we need an 8-bit representation, we pad the binary number $1100100_{2}$ with leading zeros to make it 8 bits long.

The 8-bit representation of $(+100)_{10}$ is $01100100_{2}$.

Step 3: Find the 1's Complement

To find the 2's complement, we first find the 1's complement by inverting all the bits (changing 0s to 1s and 1s to 0s).

Original 8-bit binary number for +100: 01100100

Invert the bits to get the 1's complement: 10011011

Step 4: Calculate the 2's Complement

Next, we add 1 to the 1's complement to get the 2's complement representation.

1's Complement: 10011011

Add 1:
10011011
+ 1
------------------
10011100

Therefore, the 2's complement representation of $(-100)_{10}$ in an 8-bit system is 10011100.

Step 5: Match with Options

Comparing our result with the given options:

1. 10011011 (This is the 1's complement)
2. 01100100 (This is the 8-bit representation of +100)
3. 11100100 (Incorrect)
4. 10011100 (This matches our calculated 2's complement)

The correct option is 10011100.