Question

1. (16%) Convert decimal +47 and +38 to binary, using the signed-2’s-complement representation and enough digits to accommodate the numbers, Then, perform the binary equivalent of (+47)+(-38) and (-47)+(-38) using addition. Convert the answers back to decimal and verify that they are correct.

Answer 1

1)
Number: 47
Let's convert this to two's complement binary
Since this is a positive number. we can directly convert this into binary
Divide 47 successively by 2 until the quotient is 0
   > 47/2 = 23, remainder is 1
   > 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 101111
So, 47 of decimal is 101111 in binary
Adding 2 zeros on left hand side of this number to make this of length 8
so, 47 in 2's complement binary is 00101111

Number: -38
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 38 successively by 2 until the quotient is 0
   > 38/2 = 19, remainder is 0
   > 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 100110
So, 38 of decimal is 100110 in binary
Adding 2 zeros on left hand side of this number to make this of length 8
So, 38 in normal binary is 00100110
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00100110 is flipped to 11011001
Step 3:. Add 1 to above result
11011001 + 1 = 11011010
so, -38 in 2's complement binary is 11011010

Adding 00101111 and 11011010 in binary
    00101111
    11011010
-------------
 (1)00001001
-------------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 00001001

Verification:
---------------
sum = 00001001
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 1001
=> 1x2^3+0x2^2+0x2^1+1x2^0
=> 1x8+0x4+0x2+1x1
=> 8+0+0+1
=> 9
Answer: 9
This is correct since we can verify that 47+-38 = 9
So, there was no overflow.

2)
Number: -47
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 47 successively by 2 until the quotient is 0
   > 47/2 = 23, remainder is 1
   > 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 101111
So, 47 of decimal is 101111 in binary
Adding 2 zeros on left hand side of this number to make this of length 8
So, 47 in normal binary is 00101111
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00101111 is flipped to 11010000
Step 3:. Add 1 to above result
11010000 + 1 = 11010001
so, -47 in 2's complement binary is 11010001

Number: -38
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 38 successively by 2 until the quotient is 0
   > 38/2 = 19, remainder is 0
   > 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 100110
So, 38 of decimal is 100110 in binary
Adding 2 zeros on left hand side of this number to make this of length 8
So, 38 in normal binary is 00100110
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00100110 is flipped to 11011001
Step 3:. Add 1 to above result
11011001 + 1 = 11011010
so, -38 in 2's complement binary is 11011010

Adding 11010001 and 11011010 in binary
    11010001
    11011010
-------------
 (1)10101011
-------------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 10101011

Verification:
---------------
sum = 10101011
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.
   10101011 is flipped to 01010100
II. Add 1 to above result
01010100 + 1 = 01010101
III. Now convert this result to decimal value
=> 1010101
=> 1x2^6+0x2^5+1x2^4+0x2^3+1x2^2+0x2^1+1x2^0
=> 1x64+0x32+1x16+0x8+1x4+0x2+1x1
=> 64+0+16+0+4+0+1
=> 85
Answer: -85
This is correct since we can verify that -47+-38 = -85
So, there was no overflow.