Understanding 2's Complement Representation

The question asks for the representation of the decimal number -147 using 2's complement notation on a 12-bit machine. The 2's complement is a standard method for representing signed integers in computing, allowing arithmetic operations like subtraction to be performed using addition circuitry.

Steps to Calculate 2's Complement

To find the 2's complement representation of a negative number like -147, we follow these steps:

1. Convert the absolute value of the number (147) to its binary equivalent.
2. Pad the binary number with leading zeros to match the specified bit length (12 bits).
3. Find the 1's complement by inverting all the bits (changing 0s to 1s and 1s to 0s).
4. Add 1 to the 1's complement to get the final 2's complement representation.

Step-by-Step Calculation

1. Convert 147 to Binary

First, we convert the absolute value, 147, into its binary form:

• $147 \div 2 = 73$ remainder 1
• $73 \div 2 = 36$ remainder 1
• $36 \div 2 = 18$ remainder 0
• $18 \div 2 = 9$ remainder 0
• $9 \div 2 = 4$ remainder 1
• $4 \div 2 = 2$ remainder 0
• $2 \div 2 = 1$ remainder 0
• $1 \div 2 = 0$ remainder 1

Reading the remainders from bottom to top, the binary representation of 147 is 10010011.

2. Pad to 12 Bits

The machine uses 12 bits. We pad the binary number 10010011 with leading zeros to make it 12 bits long:

000010010011

3. Find the 1's Complement

Next, we find the 1's complement by inverting each bit:

Original 12-bit binary: 000010010011

1's Complement: 111101101100

4. Find the 2's Complement

Finally, we add 1 to the 1's complement:

111101101100 (1's Complement)

+ 1

--------------------

111101101101 (2's Complement)

Conclusion

The 2's complement representation of the decimal number -147 on a 12-bit machine is 111101101101.

Comparing this result with the given options:

• Option 1: 111101101100 (Incorrect - This is the 1's complement)
• Option 2: 110001001101 (Incorrect)
• Option 3: 111101101101 (Correct)
• Option 4: 000001101101 (Incorrect - This represents a positive number)

Therefore, the correct representation is 111101101101.