Floating point representation Denary base 10 version

1
Floating point representation Denary (base 10
version)

• A way of representing decimal numbers of any size
  could be SEEMMMMM where S is the sign which is
  0 for positive and 5 for negative. E is the
  exponent value added to 50 and M is the mantissa.
•  
• Consider the number 29156 this could be
  represented as 0.29156 ? 105 In this example
  the mantissa is 29156 and the exponent is 50 5
  55, also the number is positive so the
  representation would be 05529156.
•  
• Similarly the number -0.000000052 -0.52 ? 10-7
  which would be represented as 54352000

2
Floating point representation Denary Exercise
(base 10 version)
Convert to SEEMMMMM form

• 45
• 0.00567

• - 679000
• -0.078

Convert to an ordinary Number from SEEMMMMM form

• 05645000
• 54956700

• 55023679
• 04790000

3
Floating point representation Denary Exercise
(base 10 version)

• Answers
• 05245000
• 04856700
• 55667900
• 54978000
• 560000
• - 0.0567
• - 0.23679
• 0.0009

4
ADDING FLOATING POINT NUMBERS IN DENARY

• To add the floating point numbers 05199520 ?
  04967850
• First it is necessary to make the exponents the
  same by moving the decimal point in the mantissa
  of the second number right 2 places. (This is
  like aligning units, tens and hundreds etc.). We
  now get 05199520
• 0510067850
• Adding mantissa only we get (1)0019850
• The (1) in brackets is a carry bit. To
  accommodate this bit, it is necessary to move the
  mantissa to the right and increase the exponent
  by one to get the answer.
• i.e. 05210019(850) as we need only a 5 digit
  mantissa, by rounding we get 05210020
• Check the answer

5
MULTIPLYING IN DENARY FLOATING POINT

• To multiply the floating point numbers
• 05220000 ? 04712500
• First add the exponents and subtract 50
• 52 47 50 49
• Now multiply the mantissa
• 0.20000 ? 0.12500 0.025
• Normalizing we get a mantissa of 0.25 and
  therefore need to decrease the exponent by 1
• (remember we increased the mantissa by
  multiplying by 10 so we compensate by reducing
  the exponent by 10)
• The answer is 04825000
• Check the answer

6
Floating point representation Binary (base 2
version IEEE standard 754)

• A practical method in a computer (using binary)
  could be using 32 bits as follows
• bit 31 is a sign bit using 0 for positive and 1
  for negative
• bits 23 to 30 (i.e. 8 bits) for the exponent in
  excess of 12710
• bits 0 to 22 representing the mantissa less one
  (it assumes all mantissas start with 1, so MSB is
  ignored)
•  
• Thus 10101.0110 1.0101011 ? 24 The sign bit is
  0 (it is a positive number), the exponent is
  13110 (this comes from 127 4 131 and 13110
  which is 10000011 in binary). This leaves the
  mantissa as 0101011 remember we ignore the 1 as
  all binary numbers can have a 1 to the left of
  the decimal point when normalised. The Precision
  floating point number is 0?10000011?0101011000000
  0000000000 the symbol ? separates the sign,
  exponent and mantissa for your ease.