Question

a) Using Binary Signed Magnitude arithmetic, find the ‘sum’ of 58₁₀ + (-23₁₀). Show your work. (use 8 bits)

b) Using two’s complement binary arithmetic, find the sum of 45 and -16. Show your work. (use 8 bits)

Answer 1

a)Addition using 1's complement:-

Addition of the positive number with a negative number when the positive number has a greater magnitude.

Initially, calculate the 1's complement of the given negative number. Sum up with the given positive number. If we get the end-around carry 1, it gets added to the LSB.

(58)₁₀ --------->(00111010)₂

(-23)_{10----------->}(-00010111)₂

1. First, find the 1's complement of the negative number 00010111. So, for finding 1's complement, change all 0 to 1 and all 1 to 0. The 1's complement of the number 00010111 is 11101000.
2. Now, add both the numbers, i.e., 00111010 and 11101000;
   00111010+11101000 = 1 00100010
3. By adding both numbers, we get the end-around carry 1. We add this end around carry to the LSB of 00100010.
   00100010+1=00100011------->(35)₁₀

b)Addition using 2's complement:-

Case 1: Addition of the positive number with a negative number when the positive number has a greater magnitude.

Initially find the 2's complement of the given negative number. Sum up with the given positive number. If we get the end-around carry 1 then the number will be a positive number and the carry bit will be discarded and remaining bits are the final result.

(45)₁₀---------------->(00101101)₂

(-16)₁₀----------------->(-00010000)₂

1. First, find the 2's complement of the negative number 00010000. So, for finding 2's complement, change all 0 to 1 and all 1 to 0 or find the 1's complement of the number 00010000. The 1's complement of the number 00010000 is 11101111, and add 1 to the LSB of the result11101111. So the 2's complement of number 11101111 is 11101111+1=11110000
2. Add both the numbers, i.e., 00101101 and 11110000;
   00101101+11110000=1 00011101
3. By adding both numbers, we get the end-around carry 1. We discard the end-around carry. So, the addition of both numbers is 00011101------------>(29)₁₀