Question

(5 points) Convert the following decimal numbers to 8-bit two's complement binary numbers and carry out the additions in binary. Indicate whether the sum overflows the 8-bit result. If not show the result as a decimal number. a) 39 + (-78) b) -43 + (-92)

Answer 1

Answer

a)
Number: 39
Let's convert this to two's complement binary
Since this is a positive number. we can directly convert this into binary
Divide 39 successively by 2 until the quotient is 0
   > 39/2 = 19, remainder is 1
   > 19/2 = 9, remainder is 1
   > 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 100111
So, 39 of decimal is 100111 in binary
Adding 2 zeros on left hand side of this number to make this of length 8
so, 39 in 2's complement binary is 00100111

Number: -78
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 78 successively by 2 until the quotient is 0
   > 78/2 = 39, remainder is 0
   > 39/2 = 19, remainder is 1
   > 19/2 = 9, remainder is 1
   > 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 1001110
So, 78 of decimal is 1001110 in binary
Adding 1 zeros on left hand side of this number to make this of length 8
So, 78 in normal binary is 01001110
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   01001110 is flipped to 10110001
Step 3:. Add 1 to above result
10110001 + 1 = 10110010
so, -78 in 2's complement binary is 10110010

Adding 00100111 and 10110010 in binary
00100111
10110010
-------------
(0)11011001
-------------
Sum does not produces a carry
So, sum of these numbers in binary is 11011001

Verification:
---------------
sum = 11011001
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.
   11011001 is flipped to 00100110
II. Add 1 to above result
00100110 + 1 = 00100111
III. Now convert this result to decimal value
=> 100111
=> 1x2^5+0x2^4+0x2^3+1x2^2+1x2^1+1x2^0
=> 1x32+0x16+0x8+1x4+1x2+1x1
=> 32+0+0+4+2+1
=> 39
Answer: -39
This is correct since we can verify that 39+-78 = -39
So, there was no overflow.

b)
Number: -43
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 43 successively by 2 until the quotient is 0
   > 43/2 = 21, remainder is 1
   > 21/2 = 10, remainder is 1
   > 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 101011
So, 43 of decimal is 101011 in binary
Adding 2 zeros on left hand side of this number to make this of length 8
So, 43 in normal binary is 00101011
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00101011 is flipped to 11010100
Step 3:. Add 1 to above result
11010100 + 1 = 11010101
so, -43 in 2's complement binary is 11010101

Number: -92
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 92 successively by 2 until the quotient is 0
   > 92/2 = 46, remainder is 0
   > 46/2 = 23, remainder is 0
   > 23/2 = 11, remainder is 1
   > 11/2 = 5, remainder is 1
   > 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 1011100
So, 92 of decimal is 1011100 in binary
Adding 1 zeros on left hand side of this number to make this of length 8
So, 92 in normal binary is 01011100
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   01011100 is flipped to 10100011
Step 3:. Add 1 to above result
10100011 + 1 = 10100100
so, -92 in 2's complement binary is 10100100

Adding 11010101 and 10100100 in binary
11010101
10100100
-------------
(1)01111001
-------------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 01111001

Verification:
---------------
sum = 01111001
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 1111001
=> 1x2^6+1x2^5+1x2^4+1x2^3+0x2^2+0x2^1+1x2^0
=> 1x64+1x32+1x16+1x8+0x4+0x2+1x1
=> 64+32+16+8+0+0+1
=> 121
Answer: 121
-43+-92 must be -135
This is not correct since we can verify that -43+-92 not equals 121
So, there was an overflow.

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