Question

Determine for 5-bits system, if there is any overflow or carry produced from the addition of following decimal numbers: [4*5 = 20 Marks]
a. -9 and -10
b. -14 and 14
c. 12 and -13
d. 9 and 12

Answer 1

Ans 1. -9 and -10

In the above problem we have two signed numbers -9 and -10.

As we know to add two signed numbers in binary we have to take 2's complement of negative numbers.

Note: 2's Complement is done by first complimenting true binary digit and then adding +1 to it. eg. -9 is 11001 where MSB is a signed digit (0 for positive and 1 for negative numbers). First complement of 1001 is 0110 and then by adding +1, 2's complement is 0111.

Decimal Sign digit Binary form 2's Complement

    -9 1 1001 0111

    -10 1 1010 0110

On adding 2's complement of -9 and -10, we get

-9 = 1 0111

-10 = 1 0110

___________

         101101

Now, MSB is 1 denoting negative and 01101 is in 2's complement form. When we retake 2's complement form, we get 10011 that is 19 in decimal and hence the final output is -19 in decimal.

Note: The result is having an additional bit because in binary we need 5 bits required to represent 19 whereas 9 and 10 requires only 4 bits.

Ans 2. -14 and 14

In the above problem we have two equal and opposite numbers.

Decimal Sign digit Binary form 2's Complement

    -14 1 1110 0010

     14 0 1110 NA

On adding 2's complement of -14 and 14, we get

-14 = 1 0010

14 = 0 1110

___________

         1 0 0000

The extra 1 gets discarded/removed and the final result is 00000.

Ans 3. 12 and -13

In the above problem we have one positive and one larger negative number.

Decimal Sign digit Binary form 2's Complement

     12 0 1100 NA

    -13 1 1101 0011

On adding 2's complement of 12 and -13, we get

12 = 0 1100

-13 = 1 0011

___________

            1 1111

The final sum is represented in complemented form and on retaking 2's complement we get 0001 and MSB 1 shows negative numbers. Hence the final result is -1 in decimal.

Ans 4. 9 and 12

In the above problem we have both positive numbers or unsigned numbers.

Decimal Sign digit Binary form 2's Complement

        9 0 1001 NA

      12 0 1100 NA

On adding 2's complement of 12 and -13, we get

   9 = 0 1001

12 = 0 1100

___________

           010101

The final sum is 10101 that is 21 and MSB is 0 means positive number.

Note: The extra bit is because 21 requires 5 bits whereas 9 and 12 requires only 4 bits.