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Compute the following binaryarithmetic operations using 2’s complement method. Show all the steps clearly. (LO 01)(2*5 = 10 Marks)a) Sum of -30 and 17b) Sum of -42 and -15c) Sum of -13 and -16d) Multiplication of -8 by 6e) Multiplication of -11 by -9

Question

Compute the following binaryarithmetic operations using 2’s complement method. Show all the steps clearly. (LO 01)(2*5 = 10 Marks)a) Sum of -30 and 17b) Sum of -42 and -15c) Sum of -13 and -16d) Multiplication of -8 by 6e) Multiplication of -11 by -9

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Solution 1

a) Sum of -30 and 17

  1. Convert -30 and 17 to 8-bit binary numbers:

    • 30 in binary: 00011110
    • To find -30 in 2's complement, invert the bits and add 1: 11100001 + 1 = 11100010
    • 17 in binary: 00010001

  2. Add the two binary numbers:

      11100010
+ 00010001
-----------
  11110011

  3. The result is 11110011, which is in 2's complement form. Convert it back to decimal:

    • Invert the bits: 00001100
    • Add 1: 00001100 + 1 = 00001101
    • The result is -13.

So, the sum of -30 and 17 is -13.

b) Sum of -42 and -15

  1. Convert -42 and -15 to 8-bit binary numbers:

    • 42 in binary: 00101010
    • To find -42 in 2's complement, invert the bits and add 1: 11010101 + 1 = 11010110
    • 15 in binary: 00001111
    • To find -15 in 2's complement, invert the bits and add 1: 11110000 + 1 = 11110001

  2. Add the two binary numbers:

      11010110
+ 11110001
-----------
 111100111

  3. The result is 111100111, which is a 9-bit number. Discard the overflow bit:

      11100111

  4. The result is 11100111, which is in 2's complement form. Convert it back to decimal:

    • Invert the bits: 00011000
    • Add 1: 00011000 + 1 = 00011001
    • The result is -57.

So, the sum of -42 and -15 is -57.

c) Sum of -13 and -16

  1. Convert -13 and -16 to 8-bit binary numbers:

    • 13 in binary: 00001101
    • To find -13 in 2's complement, invert the bits and add 1: 11110010 + 1 = 11110011
    • 16 in binary: 00010000
    • To find -16 in 2's complement, invert the bits and add 1: 11101111 + 1 = 11110000

  2. Add the two binary numbers:

      11110011
+ 11110000
-----------
 111100011

  3. The result is 111100011, which is a 9-bit number. Discard the overflow bit:

      11100011

  4. The result is 11100011, which is in 2's complement form. Convert it back to decimal:

    • Invert the bits: 00011100
    • Add 1: 00011100 + 1 = 00011101
    • The result is -29.

So, the sum of -13 and -16 is -29.

d) Multiplication of -8 by 6

  1. Convert -8 and 6 to 8-bit binary numbers:

    • 8 in binary: 00001000
    • To find -8 in 2's complement, invert the bits and add 1: 11110111 + 1 = 11111000
    • 6 in binary: 00000110

  2. Multiply the two binary numbers:

      11111000
x 00000110
-----------
  00000000
 11111000
11111000
-----------
111000000

  3. The result is 111000000, which is a 9-bit number. Discard the overflow bit:

      11000000

  4. The result is 11000000, which is in 2's complement form. Convert it back to decimal:

    • Invert the bits: 00111111
    • Add 1: 00111111 + 1 = 01000000
    • The result is -48.

So, the multiplication of -8 by 6 is -48.

e) Multiplication of -11 by -9

  1. Convert -11 and -9 to 8-bit binary numbers:

    • 11 in binary: 00001011
    • To find -11 in 2's complement, invert the bits and add 1: 11110100 + 1 = 11110101
    • 9 in binary: 00001001
    • To find -9 in 2's complement, invert the bits and add 1: 11110110 + 1 = 11110111

  2. Multiply the two binary numbers:

      11110101
x 11110111
-----------
  11110101
 11110101
11110101

11110101 11110101 11110101 11110101 11110101

111000000111


3. The result is 111000000111, which is a 12-bit number. Discard the overflow bits:

 00000111


4. The result is 00000111, which is in 2's complement form. Convert it back to decimal:
- The result is 99.

So, the multiplication of -11 by -9 is 99.

This problem has been solved

Solution 2

a) Sum of -30 and 17

Step 1: Convert the numbers to binary -30 = 11101 (in 2's complement form) 17 = 10001

Step 2: Add the numbers 11101 +10001

101110

Step 3: Ignore the carry 01110

Step 4: Convert back to decimal -14

b) Sum of -42 and -15

Step 1: Convert the numbers to binary -42 = 101010 (in 2's complement form) -15 = 1111

Step 2: Add the numbers 101010

  • 1111

110101

Step 3: Ignore the carry 10101

Step 4: Convert back to decimal -21

c) Sum of -13 and -16

Step 1: Convert the numbers to binary -13 = 1101 (in 2's complement form) -16 = 10000

Step 2: Add the numbers 1101 +10000

11101

Step 3: Ignore the carry 1101

Step 4: Convert back to decimal -29

d) Multiplication of -8 by 6

Step 1: Convert the numbers to binary -8 = 1000 (in 2's complement form) 6 = 110

Step 2: Multiply the numbers 1000 x 110

00000 1000 1000

11000

Step 3: Convert back to decimal -48

e) Multiplication of -11 by -9

Step 1: Convert the numbers to binary -11 = 1011 (in 2's complement form) -9 = 1001

Step 2: Multiply the numbers 1011 x1001

1011 0000 1011

11111

Step 3: Convert back to decimal 99

This problem has been solved

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