Fundamentals of data representation Year 2

• We have learned the following methods of representing numbers:
  • unsinged integer
  • signed integer
  • two's compliment for signed integers
  • fix point representation of fraction numbers.
• We have also learned that with fixed number of bits, there are always numbers that cannot be represented accurately. For example, the number 0.1 cannot be represented in binary exactly.

• FPF (Floating Point Form) is a way of representing numbers that increases the precision and the range of numbers for given number of bits.
• Using the scientific notation (normalised form) to represent large numbers so they are easy to read. For example:

  Decimal Value		Noramlised As	Mantissa	Exponent
  2100	--->	2.1 x 103	2.1	3
  5500000	--->	5.5 x 106	5.5	6

  
  
• To represent numbers in binary using the scientific/normalised form, we allocate certain number of bits for the Mantissa and certain number of bits for the exponent. Throughout this notes, we will use 8 bits for Mantissa and 5 bits for exponent.

  

• Some examples of normalisation of binary numbers:

  Binary Value	Normalized As	Exponent
  1101.101	1.101101	3
  .00101	1.01	-3
  1.0001	1.0001	0
  10000011.0	1.0000011	7

• To workout the FPF of a decimal number using 2's complement:
  1. Using 67 as an example, convert the decimal number to binary using 2's complement.

     67 in binary (2's complement):   
                   0 100 0011.  ----- the first bit is the sign bit.
                                    The point  is for later normalisation convenience
     normalise by moving the point to the place indicated below:
                   0.100 0011  ----- the point is imaginary only.  
     The number of places the point moved is the exponent: 7
                   7 in binary using 5-bit pattern: 00111
     The final answer is:
                  01000011 00111
     

  2. Using 6.75 as an example, convert the decimal number to binary using 2's complement.

     6.75 in binary (2's complement):   
                 0 110.1100  ----- the first bit is the sign bit.Two more
                                   0s padded to the end o make the 8 bits
                                   Mantissa.
                                  
     Normalise by moving the point to the place indicated below:
                 0.110 1100  
     The number of places the point moved is the exponent: 3
                 3 in binary using 5-bit pattern: 00011
     The final answer is:
                 01101100 00011
     

  3. Now, lets try negative number, -6.75 as an example, convert the decimal number to binary using 2's complement.

     -6.75 in binary (2's complement):
              1. 6.75 in binary using the 7 bits (leave the sign bit for now):
                 110.1100  ----- Two more 0s padded to the end o make the 8 bits
                                   Mantissa. 
              2. add 1 and then flip:
                 001.0100
              3. add the sign bit back:
                 1001.0100
                                  
     Normalise by moving the point to the place indicated below:
                 1.001 0100  
     The number of places the point moved is the exponent: 3
                 3 in binary using 5-bit pattern: 00011
     The final answer is:
                 10010100 00011
     

• All numbers are using 2's complement, 8-bit Mantissa and 5-bit exponent
• DO NOT forget the invisible decimal point after the first sign bit
  1. 01100001 00110
  2. A negative FPF: 10110001 00101
  3. A negative FPF with negative exponent: 10110000 11101

• Using limited/fixed number of bits, there are always numbers that cannot be represented accurately. For example, representing 0.5 can be accurate, but representing 1/3 cannot be.
• The following image illustrates the limitations using 8 bits signed,fixed point representation.
• To represent the number 0.51, the closest we can get is: 0.5 + 0.0078125 = 0.5078125
• Rounding errors:

Use the above example, to calculate the errors as a result of the not-so-accurately represented number using limited number of bits:

• absolute error = 0.51 - 0.5078125 = 0.0021875
• relative error = absolute error/0.51 = 0.0021875/0.51 = 0.00428922 or 0.428922%

• We have learned representing numbers using fixed point and floating point scheme. Here are the pros and cons of those two schemes in terms of range and precision:
  • Using the number of bits, FPF allows greater range of numbers which means from smaller number to larger number. Larger Mantissa, better precision, larger exponent, greater range.
  • Fixed point is simpler, therefore faster to process.
  • Fixed point can have varies range and precisions depending on how many points before and after the point. More bits before the point, the greater the range, more bits after the point, the greater precision.

See above on FPF and normalisation

• Underflow is when a number is too small to be represented in the given number of bits. This can happen when a small number is divided by a number that is larger than 1, this may be represented by 0.
• Overflow is when a number is too large to be represented in the given number of bits. For example, the product of two larger than 1 numbers.