Understanding 4-Bit 2's Complement Overflow

This question asks us to identify which additions performed using 4-bit 2's complement arithmetic result in an overflow. Overflow occurs when the outcome of an arithmetic operation is outside the range that can be represented by the given number of bits.

In 4-bit 2's complement representation, the range of numbers is from $-(2^{4-1})$ to $(2^{4-1} - 1)$, which is -8 to +7.

A key rule for detecting overflow in signed number addition is:

• Overflow occurs if the addition of two operands with the same sign bit results in a value with the opposite sign bit.
• Specifically: Adding two positive numbers yields a negative result, OR adding two negative numbers yields a positive result.

Analyzing 4-Bit Additions

Addition (i): 1100 + 0100

First, let's determine the values represented by the binary numbers:

• $1100$: The sign bit is 1, indicating a negative number. To find its magnitude, we take the 2's complement: invert bits ($0011$) and add 1 ($0100$), which is 4. So, $1100$ represents -4.
• $0100$: The sign bit is 0, indicating a positive number. It directly represents +4.

Now, let's perform the addition:

  1  

  1100 

(-4)

+    0100 

(+4)

-------

10000

Discarding the carry-out bit (the 5th bit), the 4-bit result is $0000$, which represents 0.

We added a negative number (-4) and a positive number (+4). The result is 0. Since the sign bits of the operands were different (1 and 0), overflow does not occur.

Addition (ii): 1100 + 1111

Values:

• $1100$ represents -4.
• $1111$: The sign bit is 1. 2's complement: invert ($0000$) + 1 = $0001$, which is 1. So, $1111$ represents -1.

Addition:

  1  

  1100 

(-4)

+    1111 

(-1)

-------

11011

Discarding the carry-out bit, the 4-bit result is $1011$.

$1011$: Sign bit is 1. 2's complement: invert ($0100$) + 1 = $0101$, which is 5. So, $1011$ represents -5.

We added two negative numbers (-4 and -1). The result (-5) is negative. Since the sign bits were the same (1 and 1) and the result's sign bit is also 1, no overflow occurred.

Addition (iii): 0101 + 0100

Values:

• $0101$ represents +5.
• $0100$ represents +4.

Addition:

    0101 

(+5)

+    0100 

(+4)

-------

  1001

The 4-bit result is $1001$.

$1001$: Sign bit is 1. 2's complement: invert ($0110$) + 1 = $0111$, which is 7. So, $1001$ represents -7.

We added two positive numbers (+5 and +4). The expected result is +9. However, the 4-bit result $1001$ represents -7. Since two positive numbers resulted in a negative number, overflow occurred.

Addition (iv): 1010 + 1100

Values:

• $1010$: Sign bit is 1. 2's complement: invert ($0101$) + 1 = $0110$, which is 6. So, $1010$ represents -6.
• $1100$ represents -4.

Addition:

  1  

  1010 

(-6)

+    1100 

(-4)

-------

10110

Discarding the carry-out bit, the 4-bit result is $0110$.

$0110$: Sign bit is 0, representing a positive number. It directly represents +6.

We added two negative numbers (-6 and -4). The expected result is -10. However, the 4-bit result $0110$ represents +6. Since two negative numbers resulted in a positive number, overflow occurred.

Conclusion on Overflow

Based on the analysis:

• Addition (i) resulted in 0. No overflow.
• Addition (ii) resulted in -5. No overflow.
• Addition (iii) resulted in -7 (representing +9). Overflow occurred.
• Addition (iv) resulted in +6 (representing -10). Overflow occurred.

Therefore, additions (iii) and (iv) result in overflow.