Question

1. Perform the following calculation in a 6-bit two’s complement system. Show your work. Indicate at the end if there will be overflow/underflow or not and why.
   18₁₀ – 10₁₀

Answer 1

Number: 18
Let's convert this to two's complement binary
18
Since this is a positive number. we can directly convert this into binary
Divide 18 successively by 2 until the quotient is 0
   > 18/2 = 9, remainder is 0
   > 9/2 = 4, remainder is 1
   > 4/2 = 2, remainder is 0
   > 2/2 = 1, remainder is 0
   > 1/2 = 0, remainder is 1
Read remainders from the bottom to top as 10010
So, 18 of decimal is 10010 in binary
so, 18 in 2's complement binary is 010010

Number: -10
Let's convert this to two's complement binary
-10
This is negative. so, follow these steps to convert this into a 2's complement binary
Step 1:
Divide 10 successively by 2 until the quotient is 0
   > 10/2 = 5, remainder is 0
   > 5/2 = 2, remainder is 1
   > 2/2 = 1, remainder is 0
   > 1/2 = 0, remainder is 1
Read remainders from the bottom to top as 1010
So, 10 of decimal is 1010 in binary
So, 10 in normal binary is 001010
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   001010 is flipped to 110101
Step 3:. Add 1 to above result
110101 + 1 = 110110
so, -10 in 2's complement binary is 110110

Adding 010010 and 110110 in binary
    010010
    110110
-----------
 (1)001000
-----------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 001000

Verification:
---------------
sum = 001000
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
Converting 1000 to decimal
1000
=> 1x2^3+0x2^2+0x2^1+0x2^0
=> 1x8+0x4+0x2+0x1
=> 8+0+0+0
=> 8
Answer: 8
This is correct since we can verify that 18+-10 = 8
So, there was no overflow.