Binary Addition and Overflow Analysis

Question 2 4

Assume each choice below shows the addition of two 4-bit 2's complement binary integers. Mark every choice that results in overflow. FYI: Be certain; Canvas deducts points for incorrect choices.

Choices:

  1. 1001 + 1101
  2. 0110 + 0111
  3. 1011 + 1001
  4. 0010 + 0011
  5. 1001 + 0111 (Overflow)

In 4-bit two's complement binary addition, overflow occurs when the sum of two positive numbers yields a negative result or when the sum of two negative numbers yields a positive result. From the choices, only the pair 1011 + 1001 results in an overflow.

Let's analyze each pair:

  1. 1001 + 1101: These are negative numbers (-7 & -3). Their sum is -10 (1110 in two's complement), which is a negative number — no overflow.
  2. 0110 + 0111: Here, we're dealing with positive numbers (6 & 7). Their sum is 13 (1101 in binary), another positive number — no overflow.
  3. 1011 + 1001: These are negative numbers (-5 & -7). Their sum is -12 (100 in two's complement), which is interpreted as a positive number — overflow occurs.
  4. 0010 + 0011: These are positive numbers (2 & 3). Their sum is 5 (0101 in binary) — no overflow.
  5. 1001 + 0111: This is a negative number (-7) and a positive number (7). Their sum is 0 (0000 in binary) — no overflow.
Question: How can we determine if the addition of two 4-bit 2's complement binary integers results in overflow? Answer: In 4-bit two's complement binary addition, overflow occurs if the sum of two positive numbers gives a negative result or if the sum of two negative numbers yields a positive result. To determine overflow, we calculate the sum of the binary integers and check whether it fits within the 4-bit range without changing sign. If the resulting sum exceeds the 4-bit representation, overflow occurs.
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