Arithmetics for Computers (Number Systems and Operations)

Subtopic in this chapter:

# Operation on integer

                 * Addition and subtraction

                    * Multiplication and division

# Floating-point real number

                        *  Representation and operations

OPERATION ON INTEGER

Example for Binary Addition

Example for Binary Substraction

Example for Binary Multiplication

Example for Binary Division

by Nor Izzati Binti Setopa

                                                 

 Example for Hexadecimal Addition

Example for Hexadecimal Substraction

Example for Hexadecimal Multiplication

  Step Hexadecimal Multiplication:

                                                - convert hexadecimal to decimal

                                                - convert decimal to hexadecimal back
              

Example for Hexadecimal Division

Step Hexadecimal Division:

- convert hexadecimal to decimal first

- convert decimal to hexadecimal back

by Siti Nazirah Binti Abu Hasan

FLOATING-POINT REAL NUMBERS

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Floating Point Addition

      

    

              Add the following two decimal numbers in scientific notation:

8.70 × 10^-1 with 9.95 × 10¹

          1. Rewrite the smaller number such that its exponent matches with the                        exponent of the larger number.

8.70 × 10^-1 = 0.087 × 10¹

2.  Add the mantissas

9.95 + 0.087 = 10.037 and write the sum 10.037 × 10¹

3. Put the result in Normalised Form

10.037 × 10¹ = 1.0037 × 10² (shift mantissa, adjust exponent)

check for overflow/underflow of the exponent after normalisation

4. Round the result

    If the mantissa does not fit in the space reserved for it, it has to be                 rounded off.

    For Example: If only 4 digits are allowed for mantissa

1.0037 × 10² ===> 1.004 × 10²

   (only have a hidden bit with binary floating point numbers)

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Example addition in binary

                
           Perform 0.5 + (-0.4375)

0.5 = 0.1 × 2⁰ = 1.000 × 2^-1 (normalised)

-0.4375 = -0.0111 × 2⁰ = -1.110 × 2^-2 (normalised)

1. Rewrite the smaller number such that its exponent matches with the exponent of the larger number.
   
   -1.110 × 2^-2 = -0.1110 × 2^-1



2. Add the mantissas:
   
   1.000 × 2^-1 + -0.1110 × 2^-1 = 0.001 × 2^-1



3. Normalise the sum, checking for overflow/underflow:
   
   0.001 × 2^-1 = 1.000 × 2^-4
   -126 <= -4 <= 127 ===> No overflow or underflow



4. Round the sum:
   
   The sum fits in 4 bits so rounding is not required
   Check: 1.000 × 2^-4 = 0.0625 which is equal to 0.5 - 0.4375

Correct!

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 Floating Point Multiplication

      

     Multiply the following two numbers in scientific notation by hand:

1.110 × 10¹⁰ × 9.200 × 10^-5

1. Add the exponents to find
   
   New Exponent = 10 + (-5) = 5

   If we add biased exponents, bias will be added twice. Therefore we need to subtract it once to compensate: (10 + 127) + (-5 + 127) = 259 259 - 127 = 132 which is (5 + 127) = biased new exponent




2. Multiply the mantissas
   
   1.110 × 9.200 = 10.212000
   Can only keep three digits to the right of the decimal point, so the result is
   
   10.212 × 10⁵



3. Normalise the result
   
   1.0212 × 10⁶



4. Round it
   
   1.021 × 10⁶

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Example multiplication in binary:

1.000 × 2^-1 × -1.110 × 2^-2

1. Add the biased exponents
   
   (-1 + 127) + (-2 + 127) - 127 = 124 ===> (-3 + 127)



2. Multiply the mantissas
3. 
   

   1.000 × 1.110 ----------- 0000 1000 1000 + 1000 ----------- 1110000 ===> 1.110000

   The product is 1.110000 × 2-3

   Need to keep it to 4 bits 1.110 × 2-3




4. Normalise (already normalised)
   At this step check for overflow/underflow by making sure that
   
   -126 <= Exponent <= 127
   1 <= Biased Exponent <= 254



5. Round the result (no change)



6. Adjust the sign.
   
   Since the original signs are different, the result will be negative
   -1.110 × 2^-3

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by Robael Adawiyah dan Wan Nur Ulaiya




Floating Point Standart IEEE 754

by Siti Hajar

source:

• Lecture note COA 2013
• Lab companion COA Edition 2013
• http://www.doc.ic.ac.uk/~eedwards/compsys/float/
• http://www.youtube.com/watch?v=MIrQtuoT5Ak