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.

-7493 and -6372

Answer 1

Number: -7493
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 7493 successively by 2 until the quotient is 0
   > 7493/2 = 3746, remainder is 1
   > 3746/2 = 1873, remainder is 0
   > 1873/2 = 936, remainder is 1
   > 936/2 = 468, remainder is 0
   > 468/2 = 234, remainder is 0
   > 234/2 = 117, remainder is 0
   > 117/2 = 58, remainder is 1
   > 58/2 = 29, remainder is 0
   > 29/2 = 14, remainder is 1
   > 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 1110101000101
So, 7493 of decimal is 1110101000101 in binary
Adding 3 zeros on left hand side of this number to make this of length 16
So, 7493 in normal binary is 0001110101000101
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   0001110101000101 is flipped to 1110001010111010
Step 3:. Add 1 to above result
1110001010111010 + 1 = 1110001010111011
so, -7493 in 2's complement binary is 1110001010111011

Number: -6372
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 6372 successively by 2 until the quotient is 0
   > 6372/2 = 3186, remainder is 0
   > 3186/2 = 1593, remainder is 0
   > 1593/2 = 796, remainder is 1
   > 796/2 = 398, remainder is 0
   > 398/2 = 199, remainder is 0
   > 199/2 = 99, remainder is 1
   > 99/2 = 49, remainder is 1
   > 49/2 = 24, remainder is 1
   > 24/2 = 12, remainder is 0
   > 12/2 = 6, remainder is 0
   > 6/2 = 3, remainder is 0
   > 3/2 = 1, remainder is 1
   > 1/2 = 0, remainder is 1
Read remainders from the bottom to top as 1100011100100
So, 6372 of decimal is 1100011100100 in binary
Adding 3 zeros on left hand side of this number to make this of length 16
So, 6372 in normal binary is 0001100011100100
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   0001100011100100 is flipped to 1110011100011011
Step 3:. Add 1 to above result
1110011100011011 + 1 = 1110011100011100
so, -6372 in 2's complement binary is 1110011100011100

Adding 1110001010111011 and 1110011100011100 in binary
1110001010111011
1110011100011100
---------------------
(1)1100100111010111
---------------------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 1100100111010111

Verification:
---------------
sum = 1100100111010111
since left most bit is 1, this number is negative number.
so, follow these steps below to convert this into a decimal value.
I. first flip all the bits. Flip all 0's to 1 and all 1's to 0.
   1100100111010111 is flipped to 0011011000101000
II. Add 1 to above result
0011011000101000 + 1 = 0011011000101001
III. Now convert this result to decimal value
=> 11011000101001
=> 1x2^13+1x2^12+0x2^11+1x2^10+1x2^9+0x2^8+0x2^7+0x2^6+1x2^5+0x2^4+1x2^3+0x2^2+0x2^1+1x2^0
=> 1x8192+1x4096+0x2048+1x1024+1x512+0x256+0x128+0x64+1x32+0x16+1x8+0x4+0x2+1x1
=> 8192+4096+0+1024+512+0+0+0+32+0+8+0+0+1
=> 13865
Answer: -13865
This is correct since we can verify that -7493+-6372 = -13865
So, there was no overflow.