Question

2) Perform the following Mathematical operations Using 2's complement, Indicate where overflow occurs, (You must convert to s-bit Binary and then do the Math, A=+ll B = +6 c= -4 е) В - А c) a) A-B b) CIA C- A- d) C

Answer 1

a)
Number: 11
Let's convert this to two's complement binary
Since this is a positive number. we can directly convert this into binary
Divide 11 successively by 2 until the quotient is 0
   > 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 1011
So, 11 of decimal is 1011 in binary
Adding 4 zeros on left hand side of this number to make this of length 8
so, 11 in 2's complement binary is 00001011

Number: -6
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 6 successively by 2 until the quotient 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 110
So, 6 of decimal is 110 in binary
Adding 5 zeros on left hand side of this number to make this of length 8
So, 6 in normal binary is 00000110
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00000110 is flipped to 11111001
Step 3:. Add 1 to above result
11111001 + 1 = 11111010
so, -6 in 2's complement binary is 11111010

Adding 00001011 and 11111010 in binary
    00001011
    11111010
-------------
 (1)00000101
-------------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 00000101

Verification:
---------------
sum = 00000101
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 101
=> 1x2^2+0x2^1+1x2^0
=> 1x4+0x2+1x1
=> 4+0+1
=> 5
Answer: 5
This is correct since we can verify that 11+-6 = 5
So, there was no overflow.

b)
Number: -4
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 4 successively by 2 until the quotient is 0
   > 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 100
So, 4 of decimal is 100 in binary
Adding 5 zeros on left hand side of this number to make this of length 8
So, 4 in normal binary is 00000100
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00000100 is flipped to 11111011
Step 3:. Add 1 to above result
11111011 + 1 = 11111100
so, -4 in 2's complement binary is 11111100

Number: 11
Let's convert this to two's complement binary
Since this is a positive number. we can directly convert this into binary
Divide 11 successively by 2 until the quotient is 0
   > 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 1011
So, 11 of decimal is 1011 in binary
Adding 4 zeros on left hand side of this number to make this of length 8
so, 11 in 2's complement binary is 00001011

Adding 11111100 and 00001011 in binary
    11111100
    00001011
-------------
 (1)00000111
-------------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 00000111

Verification:
---------------
sum = 00000111
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 111
=> 1x2^2+1x2^1+1x2^0
=> 1x4+1x2+1x1
=> 4+2+1
=> 7
Answer: 7
This is correct since we can verify that -4+11 = 7
So, there was no overflow.

c)
Number: -4
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 4 successively by 2 until the quotient is 0
   > 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 100
So, 4 of decimal is 100 in binary
Adding 5 zeros on left hand side of this number to make this of length 8
So, 4 in normal binary is 00000100
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00000100 is flipped to 11111011
Step 3:. Add 1 to above result
11111011 + 1 = 11111100
so, -4 in 2's complement binary is 11111100

Number: -11
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 11 successively by 2 until the quotient is 0
   > 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 1011
So, 11 of decimal is 1011 in binary
Adding 4 zeros on left hand side of this number to make this of length 8
So, 11 in normal binary is 00001011
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00001011 is flipped to 11110100
Step 3:. Add 1 to above result
11110100 + 1 = 11110101
so, -11 in 2's complement binary is 11110101

Adding 11111100 and 11110101 in binary
    11111100
    11110101
-------------
 (1)11110001
-------------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 11110001

Verification:
---------------
sum = 11110001
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.
   11110001 is flipped to 00001110
II. Add 1 to above result
00001110 + 1 = 00001111
III. Now convert this result to decimal value
=> 1111
=> 1x2^3+1x2^2+1x2^1+1x2^0
=> 1x8+1x4+1x2+1x1
=> 8+4+2+1
=> 15
Answer: -15
This is correct since we can verify that -4+-11 = -15
So, there was no overflow.

d)
Number: 11
Let's convert this to two's complement binary
Since this is a positive number. we can directly convert this into binary
Divide 11 successively by 2 until the quotient is 0
   > 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 1011
So, 11 of decimal is 1011 in binary
Adding 4 zeros on left hand side of this number to make this of length 8
so, 11 in 2's complement binary is 00001011

Number: 4
Let's convert this to two's complement binary
Since this is a positive number. we can directly convert this into binary
Divide 4 successively by 2 until the quotient is 0
   > 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 100
So, 4 of decimal is 100 in binary
Adding 5 zeros on left hand side of this number to make this of length 8
so, 4 in 2's complement binary is 00000100

Adding 00001011 and 00000100 in binary
    00001011
    00000100
-------------
 (0)00001111
-------------
Sum does not produces a carry
So, sum of these numbers in binary is 00001111

Verification:
---------------
sum = 00001111
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 1111
=> 1x2^3+1x2^2+1x2^1+1x2^0
=> 1x8+1x4+1x2+1x1
=> 8+4+2+1
=> 15
Answer: 15
This is correct since we can verify that 11+4 = 15
So, there was no overflow.

e)
Number: 6
Let's convert this to two's complement binary
Since this is a positive number. we can directly convert this into binary
Divide 6 successively by 2 until the quotient 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 110
So, 6 of decimal is 110 in binary
Adding 5 zeros on left hand side of this number to make this of length 8
so, 6 in 2's complement binary is 00000110

Number: -11
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 11 successively by 2 until the quotient is 0
   > 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 1011
So, 11 of decimal is 1011 in binary
Adding 4 zeros on left hand side of this number to make this of length 8
So, 11 in normal binary is 00001011
Step 2: flip all the bits. Flip all 0's to 1 and all 1's to 0.
   00001011 is flipped to 11110100
Step 3:. Add 1 to above result
11110100 + 1 = 11110101
so, -11 in 2's complement binary is 11110101

Adding 00000110 and 11110101 in binary
    00000110
    11110101
-------------
 (0)11111011
-------------
Sum does not produces a carry
So, sum of these numbers in binary is 11111011

Verification:
---------------
sum = 11111011
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.
   11111011 is flipped to 00000100
II. Add 1 to above result
00000100 + 1 = 00000101
III. Now convert this result to decimal value
=> 101
=> 1x2^2+0x2^1+1x2^0
=> 1x4+0x2+1x1
=> 4+0+1
=> 5
Answer: -5
This is correct since we can verify that 6+-11 = -5
So, there was no overflow.