GATE CSE 1992 | Question: 4-a

Question

Consider addition in two's complement arithmetic. A carry from the most significant bit does not always correspond to an overflow. Explain what is the condition for overflow in two's complement arithmetic.

Comments

$Remark:$
if the sign bit of both the signed numbers is not same as the sign bit of the product, then there is an overflow.

---

Good read : #Number System - GATE Overflow  

Answer 1

XOR of $C_{\text{in}}$ with $C_{\text{out}}$ of the MSB position.

Comments

$C_{out} \bigoplus C_{in} = 1 (Overflow) $

$C_{out} \bigoplus C_{in} = 0 ( No Overflow)$

---

Or can also say

Let A and B be two numbers

                        a$_{(n)}$a$_{(n-1)}$a$_{(n-2)}$… a$_{1}$ a$_{0}$

                  +

                        b$_{(n)}$b$_{(n-1)}$b$_{(n-2)}$...b$_{1}$ b$_{0}$

result (R)=     r$_{(n)}$r$_{(n-1)}$r$_{(n-2)}$...r$_{1}$r$_{0}$

a$_{(n)}$  b$_{(n)}$ r’$_{(n)}$ + a’$_{(n)}$  b’$_{(n)}$r$_{(n)}$ =1 (overflow)

a$_{(n)}$  b$_{(n)}$ r’$_{(n)}$ + a’$_{(n)}$  b’$_{(n)}$r$_{(n)}$ =0 (no overflow)

Answer 2

(a) In 2's complement addition Overflow happens only when :

• Sign bit of two input numbers is 0, and the result has sign bit 1.
• Sign bit of two input numbers is 1, and the result has sign bit 0.

Comments

In 4 bits representation, 

$-3$ 1101 (2's complement of $-3$) $+$

$-2$ 1110 (2's complement of $-2$)

---

$-5$ 1011 (2's complement of $-5$)

This is an example where there is carry from MSB  but still there is no overflow. Is this right?

---

yes

more example here https://gateoverflow.in/231197/condition-for-overflow

Answer 3

FOR overflow to happen during addition of two number in 2's complement form they must have same sign and result is of opposite sign

(+A) + (+B)= -C

(-A) + (-B)= +C