Question

Convert decimal +47 and +31 to binary, using the signed-2’s-complement representation and enough digits to accommodate the numbers. Then perform the binary equivalent of (+31)+(-47), (-31)+(+47), and (-31)+(-47). Convert the answers back to decimal and verify that they are correct.

Answer 1

(+31) + (-47) = -16

(-31) + (+47) = 16

(-31) + (-47) = -78

Explanation:

47 in binary = 101111₂

31 in binary = 11111₂

Let the signed number system be of 8 bit then

47 = 0010 1111

31 = 0001 1111

-47 = 1101 0001 (in 2's compliment)

-31 = 1110 0001 (in 2's compliment)

Conversion to 2's compliment explanation:

first convert decimal number to binary here 47 = 0010 1111

since the number is negative convert it to 1's compliment just flip the bits 1 to 0 and 0 to 1

1's compliment: 1101 0000

add 1 to 1's compliment we get 2's compliment

1101 0000 + 1

1101 0001

2's compliment conversion is same for -31 as well

Coming to questions:

(+31) + (-47) = -16

+31 = 0001 1111

-47 = 1101 0001

(+31) + (-47) =

= .1111 0000

since the MSB is 1 hence the number is negative. converting it to 2' compliment we get 0001 0000

in decimal 0001 0000 = 16 Since MSB is 1 hence it is -16

(-31) + (+47) = 16

-31 = 1110 0001

+47 = 0010 1111

(-31) + (+47) =

= 0001 0000

since MSB is 0 the number is positive no need for 2's compliment: 0001 0000 in decimal is 16

(-31) + (-47) = -78

-31 = 1110 0001

-47 = 1101 0001

(-31) + (-47) =

= 1011 0010

Since MSB is 1 hence the number is negative convert it to 2's compliment we get 0100 1110

In decimal it is -78.