GATE CSE 2004 | Question: 66

Question

Let $A = 1111 1010$ and $B = 0000 1010$ be two $8-bit$ $2’s$ complement numbers. Their product in $2’s$ complement is

• A. $1100 0100$
• B. $1001 1100$
• C. $1010 0101$
• D. $1101 0101$

Answer 1

$A = 1111\quad 1010 = -6$
$B = 0000\quad 1010 = 10$
$A\times B = -60 =1100\quad 0100$

Correct Answer: $A$

Comments

60 is represented as 111100. Make it 8 bits by adding 2 leading zeroes 00111100. As negative numbers are represented in 2’s complement, therefore Its 2's complement will be 1100 0100.

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$A*B=(-6)*(+10)=-60$

1. $11000100=-60$
2. $10011100=-100$
3. $10100101=-91$
4. $11010101=-43$

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If the MSB is 0,  number is +ve simply convert it to decimal.

If MSB is 1, number is -ve. Invert the bits and add 1. Now convert it to decimal (don’t forget the -ve sign).

Given, A*B = -60.

option A will satisfy it

Answer 2

Explanation: Here, we have

A = 1111 1010 =  – 6₁₀ (A is a 2’s complement number)

B = 0000 1010 =  10₁₀ (B is a 2’s complement number)

A x B = – 60₁₀ = 1 011 1100₂ = 1 100 0011 (1’s complement) = 1 100 0100 (2’s complement)

Thus, the product of A and B in 2’s complement is 1100 0100, which is option A.

So, A is the correct option

Comments

for this question wht I did first,simply multiply both numbers and the result which I got  000100111000100

after taking 2's complement of this -2500    came.plz someone verify this.

I long it is taking long time but I want to know my mistake .@shreshtha @debashish

Answer 3

In 2's complement form 

A = 11111010 = -128+64+32+16+8+2 = -6

B = 00001010 = 10

Now A* B = -60

-60 = -128+64+4 =11000100

Option : A

 

Comments

This seems somewhat understood able.

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Does that 1 at the MSB means a negative sign?

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arpit_18 yes, but only for signed number.

Answer 4

 A=11111010(in 2's compliment)

Convert into decimal then A= -6

There are sequence of 1's in left side delete all consecutive 1's except last one i.e;(delete four 1's from starting) This is shortcut of finding a decimal number from 2's compliment form

11111010=1010 and (1010= -8+2= -6)

B=00001010(in 2's compliment)

Convert into decimal then B=(10 ) in decimal

(00001010)= 8+2 = 10 in decimal

A×B= (-6)×10= (-60)

How to represent (-60) in 2's compliment form?

First write +60 in binary (using 8 bit because here all options are in 8 bits)

+60 in binary=00111100

Now take 2's compliment of (00111100)=11000100=(-60) in decimal

Hence option A is right.