Real Number Representation

Real Number Representation

• 
  Fixed point notation
  
  • 
    Top bit specifies the sign (as in signed magnitude): 0 = positive, 1 = negative.
    
  • 
    Some bits = integer part (normal format), some bits = fractional part
    
    • 
      The fractional columns are 2^-1, 2^-2, ...
      
    • 
      To convert the fractional part:
      

• 
  eg: -37.90625
  
  • 
    Sign bit is: 1
    
  • 
    Integer part:
    

Base	Num	Rem
2	37
	18	1
	9	0
	4	1
	2	0
	1	0
	0	1

• 
  Fractional part :
  

Base	Num	Int
2	0.90625
	1.8125	1
	1.625	1
	1.25	1
	0.5	0
	1.0	1
	0

• 
   -37.90625₁₀ = 1100101.11101₂
  
• 
  Problem is how many bits to assign to integer & fractional parts
  
  • 
    More bits in integer part allows larger magnitude numbers
    
  • 
    More bits in fractional part is more accurate
    
  • 
    note: Some decimal fractions have infinite binary representations
    
    • 
      eg: 0.4
      

Base	Num	Int
2	0.4
	0.8	0
	1.6	1
	1.2	1
	0.4	0
	0.8	0
etc………

• 
  Scientific Notation
  
  • 
    eg: -37.90625₁₀ = -0.3790625E+2₁₀
    
  • 
    The format is Sign Significand E Exponent
    
  • 
    The signed significand is multiplied by Base^Exponent
    
  • 
    Any number can be normalized so it starts 0.
    
  • 
    This can be done base 2
    
  • 
    eg: 37.90625₁₀ = 100101.11101₂ = 0.10010111101 * 2⁶
    

• 
  Floating Point Representation
  
  • 
    The top bit specifies the sign: 0 = positive, 1 = negative.
    
  • 
    Some bits = exponent (in biased notation), some bits = significand
    
    • 
      note: Every normalized significand starts 0.1
      
    • 
      The 0.1 is not stored, ie. one free bit
      
  • 
    eg: -37.90625, 7 bit exponent, 8 bit significand
    
    • 
      Sign bit is 1
      
    • 
      Exponent is 1000110
      
    • 
      Significand is 00101111
      
  • 
    More bits in the exponent part allows larger and smaller magnitude numbers
    
    • 
      Very large numbers cannot be represented  overflow
      
    • 
      Numbers close to 0 cannot be represented  underflow
      
    • 
      0 cannot be stored, due to the implicit 0.1. How is this handled?
      
  • 
    More bits in the significand is more accurate
    
  • 
    Typically: 8 bits exponent, 23 bits significand (plus 1 bit for the sign)  gives you a 32 bit floating point number.
    
• 
  An Example:
  

• 
  1 bit sign, 3 bits exponent, 4 bits significand
  

+3.5 in binary is +11.1

+11.1 is normalized to become  0.111*2¹⁰
(normalized numbers always start with 0.1, so the first bit can be assumed)

The sign is +, represented by 0

The exponent is 2 (10), and in biased-4 that is 6 (110)

The significand bits are 111 from 0.111