Question

add the following numbers using 16-bit 2's complement.
show all the steps and calculations.
Please also show steps to verify that the answer is correct.

9288 and -7372

Answer 1

Number: 9288
Let's convert this to two's complement binary
Since this is a positive number. we can directly convert this into binary
Divide 9288 successively by 2 until the quotient is 0
   > 9288/2 = 4644, remainder is 0
   > 4644/2 = 2322, remainder is 0
   > 2322/2 = 1161, remainder is 0
   > 1161/2 = 580, remainder is 1
   > 580/2 = 290, remainder is 0
   > 290/2 = 145, remainder is 0
   > 145/2 = 72, remainder is 1
   > 72/2 = 36, remainder is 0
   > 36/2 = 18, remainder 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 10010001001000
So, 9288 of decimal is 10010001001000 in binary
Adding 2 zeros on left hand side of this number to make this of length 16
so, 9288 in 2's complement binary is 0010010001001000

Number: -7372
Let's convert this to two's complement binary
This is negative. so, follow these steps to convert this into a 2's complement binary
Step 1:
Divide 7372 successively by 2 until the quotient is 0
   > 7372/2 = 3686, remainder is 0
   > 3686/2 = 1843, remainder is 0
   > 1843/2 = 921, remainder is 1
   > 921/2 = 460, remainder is 1
   > 460/2 = 230, remainder is 0
   > 230/2 = 115, remainder is 0
   > 115/2 = 57, remainder is 1
   > 57/2 = 28, remainder is 1
   > 28/2 = 14, remainder is 0
   > 14/2 = 7, remainder is 0
   > 7/2 = 3, remainder is 1
   > 3/2 = 1, remainder is 1
   > 1/2 = 0, remainder is 1
Read remainders from the bottom to top as 1110011001100
So, 7372 of decimal is 1110011001100 in binary
Adding 3 zeros on left hand side of this number to make this of length 16
So, 7372 in normal binary is 0001110011001100
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   0001110011001100 is flipped to 1110001100110011
Step 3:. Add 1 to above result
1110001100110011 + 1 = 1110001100110100
so, -7372 in 2's complement binary is 1110001100110100

Adding 0010010001001000 and 1110001100110100 in binary
0010010001001000
1110001100110100
---------------------
(1)0000011101111100
---------------------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 0000011101111100

Verification:
---------------
sum = 0000011101111100
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 11101111100
=> 1x2^10+1x2^9+1x2^8+0x2^7+1x2^6+1x2^5+1x2^4+1x2^3+1x2^2+0x2^1+0x2^0
=> 1x1024+1x512+1x256+0x128+1x64+1x32+1x16+1x8+1x4+0x2+0x1
=> 1024+512+256+0+64+32+16+8+4+0+0
=> 1916
Answer: 1916
This is correct since we can verify that 9288+-7372 = 1916
So, there was no overflow.