Question

Assume a 5 bit machine using two's complement. (Show your work)

14.) If 01010₂ is added to 01001₂ an overflow will occur (True or false)

15.) If 00111₂ is added to 01110₂ an over flow will occur. (True or false)

If the following addition produces an overflow write "error." If it does not, then write the decimal value of the sum of the addition

17.)

11011₂ + 01111₂

18.)

11001₂ + 11010₂

19.)

11110₂ + 11101₂

20.)

11011₂ + 01011₂

21.)

00101₂ + 01010₂

22.)

10001₂ +10010₂

23.)

10000₂ + 10000₂

24.)

11111₂ + 11111₂

Answer 1

14)
Adding 01010 and 01001 in binary
01010
01001
----------
(0)10011
----------
Sum does not produces a carry
So, sum of these numbers in binary is 10011

Verification
---------------
first = 01010
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 1010
=> 1x2^3+0x2^2+1x2^1+0x2^0
=> 1x8+0x4+1x2+0x1
=> 8+0+2+0
=> 10
Answer: 10

second = 01001
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

sum = 10011
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.
   11110011 is flipped to 00001100
II. Add 1 to above result
00001100 + 1 = 00001101
III. Now convert this result to decimal value
=> 1101
=> 1x2^3+1x2^2+0x2^1+1x2^0
=> 1x8+1x4+0x2+1x1
=> 8+4+0+1
=> 13
Answer: -13
10+9 must be 19
This is not correct since we can verify that 10+9 not equals -13
So, there was an overflow.

15)
Adding 00111 and 01110 in binary
00111
01110
----------
(0)10101
----------
Sum does not produces a carry
So, sum of these numbers in binary is 10101

Verification
---------------
first = 00111
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

second = 01110
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 1110
=> 1x2^3+1x2^2+1x2^1+0x2^0
=> 1x8+1x4+1x2+0x1
=> 8+4+2+0
=> 14
Answer: 14

sum = 10101
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.
   11110101 is flipped to 00001010
II. Add 1 to above result
00001010 + 1 = 00001011
III. Now convert this result to decimal value
=> 1011
=> 1x2^3+0x2^2+1x2^1+1x2^0
=> 1x8+0x4+1x2+1x1
=> 8+0+2+1
=> 11
Answer: -11
7+14 must be 21
This is not correct since we can verify that 7+14 not equals -11
So, there was an overflow.

17)
Adding 11011 and 01111 in binary
11011
01111
----------
(1)01010
----------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 01010

Verification
---------------
first = 11011
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.
   11011 is flipped to 00100
II. Add 1 to above result
00100 + 1 = 00101
III. Now convert this result to decimal value
=> 101
=> 1x2^2+0x2^1+1x2^0
=> 1x4+0x2+1x1
=> 4+0+1
=> 5
Answer: -5

second = 01111
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

sum = 01010
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 1010
=> 1x2^3+0x2^2+1x2^1+0x2^0
=> 1x8+0x4+1x2+0x1
=> 8+0+2+0
=> 10
Answer: 10
This is correct since we can verify that -5+15 = 10
So, there was no overflow.

18)
Adding 11001 and 11010 in binary
11001
11010
----------
(1)10011
----------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 10011

Verification
---------------
first = 11001
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.
   11001 is flipped to 00110
II. Add 1 to above result
00110 + 1 = 00111
III. Now convert this result to decimal value
=> 111
=> 1x2^2+1x2^1+1x2^0
=> 1x4+1x2+1x1
=> 4+2+1
=> 7
Answer: -7

second = 11010
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.
   11010 is flipped to 00101
II. Add 1 to above result
00101 + 1 = 00110
III. Now convert this result to decimal value
=> 110
=> 1x2^2+1x2^1+0x2^0
=> 1x4+1x2+0x1
=> 4+2+0
=> 6
Answer: -6

sum = 10011
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.
   11110011 is flipped to 00001100
II. Add 1 to above result
00001100 + 1 = 00001101
III. Now convert this result to decimal value
=> 1101
=> 1x2^3+1x2^2+0x2^1+1x2^0
=> 1x8+1x4+0x2+1x1
=> 8+4+0+1
=> 13
Answer: -13
This is correct since we can verify that -7+-6 = -13
So, there was no overflow.

19)
Adding 11110 and 11101 in binary
11110
11101
----------
(1)11011
----------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 11011

Verification
---------------
first = 11110
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.
   11110 is flipped to 00001
II. Add 1 to above result
00001 + 1 = 00010
III. Now convert this result to decimal value
=> 10
=> 1x2^1+0x2^0
=> 1x2+0x1
=> 2+0
=> 2
Answer: -2

second = 11101
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.
   11101 is flipped to 00010
II. Add 1 to above result
00010 + 1 = 00011
III. Now convert this result to decimal value
=> 11
=> 1x2^1+1x2^0
=> 1x2+1x1
=> 2+1
=> 3
Answer: -3

sum = 11011
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 -2+-3 = -5
So, there was no overflow.

20)
Adding 11011 and 01011 in binary
11011
01011
----------
(1)00110
----------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 00110

Verification
---------------
first = 11011
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.
   11011 is flipped to 00100
II. Add 1 to above result
00100 + 1 = 00101
III. Now convert this result to decimal value
=> 101
=> 1x2^2+0x2^1+1x2^0
=> 1x4+0x2+1x1
=> 4+0+1
=> 5
Answer: -5

second = 01011
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 1011
=> 1x2^3+0x2^2+1x2^1+1x2^0
=> 1x8+0x4+1x2+1x1
=> 8+0+2+1
=> 11
Answer: 11

sum = 00110
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 110
=> 1x2^2+1x2^1+0x2^0
=> 1x4+1x2+0x1
=> 4+2+0
=> 6
Answer: 6
This is correct since we can verify that -5+11 = 6
So, there was no overflow.

21)
Adding 00101 and 01010 in binary
00101
01010
----------
(0)01111
----------
Sum does not produces a carry
So, sum of these numbers in binary is 01111

Verification
---------------
first = 00101
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

second = 01010
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 1010
=> 1x2^3+0x2^2+1x2^1+0x2^0
=> 1x8+0x4+1x2+0x1
=> 8+0+2+0
=> 10
Answer: 10

sum = 01111
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 5+10 = 15
So, there was no overflow.

22)
Adding 10001 and 10010 in binary
10001
10010
----------
(1)00011
----------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 00011

Verification
---------------
first = 10001
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.
   10001 is flipped to 01110
II. Add 1 to above result
01110 + 1 = 01111
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

second = 10010
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.
   10010 is flipped to 01101
II. Add 1 to above result
01101 + 1 = 01110
III. Now convert this result to decimal value
=> 1110
=> 1x2^3+1x2^2+1x2^1+0x2^0
=> 1x8+1x4+1x2+0x1
=> 8+4+2+0
=> 14
Answer: -14

sum = 00011
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=> 11
=> 1x2^1+1x2^0
=> 1x2+1x1
=> 2+1
=> 3
Answer: 3
-15+-14 must be -29
This is not correct since we can verify that -15+-14 not equals 3
So, there was an overflow.

23)
Adding 10000 and 10000 in binary
10000
10000
----------
(1)00000
----------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 00000

Verification
---------------
first = 10000
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.
   10000 is flipped to 01111
II. Add 1 to above result
01111 + 1 = 10000
III. Now convert this result to decimal value
=> 10000
=> 1x2^4+0x2^3+0x2^2+0x2^1+0x2^0
=> 1x16+0x8+0x4+0x2+0x1
=> 16+0+0+0+0
=> 16
Answer: -16

second = 10000
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.
   10000 is flipped to 01111
II. Add 1 to above result
01111 + 1 = 10000
III. Now convert this result to decimal value
=> 10000
=> 1x2^4+0x2^3+0x2^2+0x2^1+0x2^0
=> 1x16+0x8+0x4+0x2+0x1
=> 16+0+0+0+0
=> 16
Answer: -16

sum = 00000
since left most bit is 0, this number is positive
so, we can directly convert this into a decimal value
=>
=>
=>
=>
=> 0
Answer: 0
-16+-16 must be -32
This is not correct since we can verify that -16+-16 not equals 0
So, there was an overflow.

24)
Adding 11111 and 11111 in binary
11111
11111
----------
(1)11110
----------
Sum produces a carry of 1. We can ignore that carry.
So, sum of these numbers in binary is 11110

Verification
---------------
first = 11111
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.
   11111 is flipped to 00000
II. Add 1 to above result
00000 + 1 = 00001
III. Now convert this result to decimal value
=> 1
=> 1x2^0
=> 1x1
=> 1
=> 1
Answer: -1

second = 11111
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.
   11111 is flipped to 00000
II. Add 1 to above result
00000 + 1 = 00001
III. Now convert this result to decimal value
=> 1
=> 1x2^0
=> 1x1
=> 1
=> 1
Answer: -1

sum = 11110
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.
   11111110 is flipped to 00000001
II. Add 1 to above result
00000001 + 1 = 00000010
III. Now convert this result to decimal value
=> 10
=> 1x2^1+0x2^0
=> 1x2+0x1
=> 2+0
=> 2
Answer: -2
This is correct since we can verify that -1+-1 = -2
So, there was no overflow.