Binary representation of real numbers

• irrational numbers
• rational numbers = p/q = vulgar fraction
  • proper fractions
    • vulgar fraction  with even   denominator   
      • p / 2 ⁿ   =   dyadic numbers
      • q*(2^k) with q odd
    • vulgar fraction with odd  denominator    =2k+1, where k is an integer
      • = p / ( 2ⁿ - 1)
      • = p / ( 2ⁿ + 1)
      • other
  • improper fractions

• Finite binary decimals = finite binary representation ( extension) of decimal numbers
  All dyadic rational numbers = p/2ⁿ  have a binary representation with a finite number  n of binary digits after the radix point.
  for example :
  1/2₁₀=1/2¹₁₀=0.1₂
  1/4₁₀=1/2²₁₀=0.01₂
  1/8₁₀=1/2³₁₀=0.001₂
  1/16₁₀=1/2⁴₁₀=0.0001₂
  1/32₁₀=1/2⁵₁₀=0.00001₂
  ...
  1/2ⁿ₁₀=  0.00..1₂
  For every positive integer n vulgar fraction 1/denominator with denominator with only powers of 2 as factors
  will be represented by 0.00..1₂ with binary digit 1 on n-th place after the radix point.
  
  1/8₁₀=0.001₂
  3/8₁₀=0.011₂
  7/8₁₀=0.111₂
  
  
  ???? Equivalent conditions:
  • denominator = p / 2 ⁿ
  • every prime factor of denominator = 2


• Mixed recurring binary decimals
  The binary expansion for a fraction with denominator = (2^k)*q
  with q odd consists of k digits followed by a repeating decimal with period n .
  Period n is computed in the same way as in pure recurring binary decimals ( Vulgar fraction with odd denominator ).
  1/6₁₀ = 1/ (2*3) ₁₀= 0.0(01)...₂
  1/10₁₀ = 1/(2*5)₁₀ =0.0(0011)... _{2 1/12 10= 1/(2²*3)10=0.00(01)...2 1/1410 = 1/(2*7)10 =0.0(001)... 2 1/2010 = 1/( 2²*5)10 =0.00(0011)...2}

• Pure recurring binary decimals
  have infinite binary representation ( not terminating ) with a finite sequence of binary digits repeating indefinitely.
  1/3₁₀ = 1/(2¹+1)₁₀ = 0.3₁₀ = 0.01₂= 1₂/11₂    period = 2
  1/5₁₀ = 1/(2²+1)₁₀ = 0.2₁₀ = 0.0011₂= 1₂/11₂    period = 4
  
  The binary expansion for a fraction with denominator 2^n - 1 is a repeating decimal with period n.
  
  ???? Equivalent conditions:
  • denominator has only factors other then power of 2
  • denominator mod 2 = 1
  • gcd(denominator,2)=1
  • denominator and 2 are relatively prime = coprime
  • Odd(denominator) = true
  • denominator = 2k+1, where k is an integer
  
  length of digit sequence = period of external angle under doubling map = period of hyperbolic component =
  
  

• Binary fractions by Michael Borgwardt
• Base Number - Decimal Number Conversion
• BINARY DECIMAL NUMBERS ... by Thomas Kim-wai YEUNG and Eric Kin-keung POON
• Floating Point Decimal to Integer Fraction Conversion byErik Oosterwal
• binary numbers by Nathan Louis Gibson
• External angles in the Mandelbrot set by Professor Douglas C. Ravenel
• "Fractions in Different Bases" from The Wolfram Demonstrations Project. Download free player and demonstration
• IEEE 754 Converter by Harald Schmidt

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