Binary Addition and Overflow

Binary - Basic Principles

There are 4 rules for carrying out addition of two binary digits these are:

1. 0 + 0 = 0

2. 0 + 1 = 1 

3. 1 + 0 = 1

4. 1 + 1 = 0 carry 1 but 1 + 1 + 1 = 1 carry 1

There are 8 rules for carrying out addition of three binary digits these are:

1. 0 + 0 + 0 = 0

2. 0 + 0 + 1 = 1

3. 0 + 1 + 0 = 1

4. 0 + 1 + 1 = 0 carry 1

5. 1 + 0 + 0  = 1

6. 1 + 0 + 1 = 0 carry 1

7. 1 + 1 + 0 = 0 carry 1

8. 1 + 1 + 1 = 1 carry 1

Add 01011011 + 00111010  (91 + 58)

Answer: 10010101 = 149

Task 1

Add up the following pairs of binary numbers. Use the table below to help.

1. 0110  + 0101 = 1011 = 11

2. 0101  + 0101 = 1010 = 10

3. 1011  + 1110  =  11001 = 25

4. 1001 + 0110  = 1111 = 15

5. 1010 + 1011  =  10101 =21

6. 1111 + 1001 = 11000 = 24

7. 0110 + 0101=  1011 = 11

5. 0 + 0 = 0

6. 0 + 1 = 1 

7. 1 + 0 = 1

8. 1 + 1 = 0 carry 1 but 1 + 1 + 1 = 1 carry 1

8 Bit numbers 

8 bit numbers follow exactly the same rule set as 4 bit numbers, the difference being that you can work with values up to 255. 

Task 3

Add up the following pairs of 8 bit binary numbers. Use the table below to help.

1. 01001110 + 00111100 = 1001010 

2. 10011101 + 10001111 = 10010110

3. 11010011 + 01010110 = 100101011

4. 10010011 + 10111001 =  101001100

5. 11111110 + 10110101 = 110110011 = 507

6. 10111101 + 11100111 = 110100100 = 420

0 + 0 + 0 = 0

7. 0 + 0 + 1 = 1

8. 0 + 1 + 0 = 1

9. 0 + 1 + 1 = 0 carry 1

10. 1 + 0 + 0  = 1

11. 1 + 0 + 1 = 0 carry 1

12. 1 + 1 + 0 = 0 carry 1

13. 1 + 1 + 1 = 1 carry 1

Adding three binary numbers

To add three binary numbers, add the first two, and then add the third to the result.  E.g to add the three binary numbers 00101100, 00010001 and 10000101:

00101100

00010001 +

00111101  

10000101 +

11000010

Overflow

Overflow occurs when the result of adding two binary numbers is greater than the number of bits allowed. The maximum value that can be held in an 8-bit register is 255.  

The computer would need 9 bits to represent 332 so this 9th bit doesn’t fit in the byte allocated.  Which is an overflow error.

Questions

1. Add the following binary numbers, leaving the answer as binary numbers.

   1. 00111011 + 10000110                                                                    [1]

011000001

2. 10011100 + 01111110                                                                     [1]

0100011010

3. 00110011 + 01000110       [1]

01111001

4. 00011100 + 01110011 + 01001101       [1]

011011100

2. Explain the problem that would occur in part 1(b) if the result was to be stored as an 8-bit number.                     It would be classed as an overflow                                                      [2]