Morris Mano Edition 3 Exercise 1 Question 18 (Page No. 34)

Question

Perform the arithmetic operations (+42)  +  (-13) and (-42)  –  (-13) using binary signed 2’s complement representation for the negative numbers.

Answer 1

For positive numbers; 2’s Complement representation is the binary representation of the given positive number. For negative numbers, 2’s complement representation is the 2’s complement of the binary representation of its absolute value

Hence 2’s complement representation of

• 42 = 0101010
• 13 = 01101
• -42 = 1010110
• -13 = 10111

Now we do 2’s complement addition and ignore any carry. So, 42 + (-13) = 0101010 + 1110011 we get 0011101 = 29.

For (-42) - (-13) = (-42) + 13 = 1010110 + 0001101 = 1100011 = –29.

(We append 0s before positive and 1s before negative numbers (sign extension) to make the number of bits the same while doing addition)

Answer 2

(+42)^10 + (-13)^10

= (101010)^2 + (-1101)^2

= (101010)^2 + (110011)^2

= (1011101)^2

= (11101)^2

= (29)^10...

 

(-42)^10 - (-13)^10

= -42 + (-(-13))

= -42 + 13

= (-101010)^2 + (1101)^2

= (-101010)^2 + (1101)^2

= (010110)^2 + (001101)^2

= (100011)^2 ....

 

There is no carry. Thus the answer is (-11101)^2 = (-29)^10.…