GATE2025_CS1_Q25

The number −6 can be represented as 1010 in 4-bit 2's complement representation. Which of the following is/are CORRECT 2's complement representation(s) of −6?

Note: This is a Multiple Select Question (MSQ). One or more options may be correct.

✅ Final Answer:

The correct options are (B) and (D).

🔹 Short Answer:

To represent a negative number in 2's complement with more bits, we use sign extension. This means the Most Significant Bit (MSB), which is the sign bit, is replicated to fill the additional bits. For -6, the 4-bit representation is 1010. The sign bit is 1. Extending this to 8 bits gives 11111010, and extending to 16 bits gives 1111111111111010.

🔸 Long Answer:

This question tests the core concept of representing negative numbers in 2's complement and how to extend that representation to a larger number of bits.

1. Verifying the 4-bit Representation of -6

First, let's confirm the given 4-bit representation of -6 is correct.

1. Start with the positive value: +6.
2. Binary representation of 6 in 4 bits: 0110.
3. Find the 1's complement by inverting all bits: 1001.
4. Find the 2's complement by adding 1: 1001 + 1 = 1010.

The given 4-bit representation is indeed correct.

2. The Concept of Sign Extension

When we want to represent the same signed number using a larger number of bits (e.g., from 4-bit to 8-bit), we must perform sign extension to preserve its value. The rule is:

• Take the original number's Most Significant Bit (MSB), also known as the sign bit.
• Copy this sign bit to fill all the new bit positions to the left.

For a positive number, the sign bit is 0, so we pad with 0s. For a negative number, the sign bit is 1, so we pad with 1s.

3. Analyzing the 8-bit Options

• Original 4-bit number for -6: 1010.
• The sign bit is the MSB, which is 1.
• To extend to 8 bits, we add 4 new bits to the left and fill them with the sign bit (1).

New Sign Bits (4)	Original Bits (4)	Resulting 8-bit Number
1111	1010	11111010

Therefore, option (B) 1111 1010 is correct, and option (A) is incorrect as it pads with zeros.

4. Analyzing the 16-bit Options

• We can extend directly from the 4-bit or the 8-bit representation. Let's use the 8-bit one: 11111010.
• The sign bit is still 1.
• To extend to 16 bits, we add 8 new bits to the left and fill them with the sign bit (1).

New Sign Bits (8)	Original Bits (8)	Resulting 16-bit Number
11111111	11111010	1111111111111010

Therefore, option (D) 1111 1111 1111 1010 is correct, and option (C) is incorrect.

Bibliography:
- Stallings, W. (2016). Computer Organization and Architecture: Designing for Performance. Pearson. (Chapter on Computer Arithmetic).
- Mano, M. M., & Kime, C. R. (2015). Logic and Computer Design Fundamentals. Pearson. (Chapter on Number Systems and Conversion).

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