Fractals/Mathematics/Numbers

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Number type[edit]

Number ( for example angle in turns ) can be used as a :

  • decimal number (base = 10 ) [1]
    • ratio = rational number ( if number can not be represented as a ratio then it is irrational number )
      • in lowest terms ( irreducible form ) : \tfrac{1}{21}
      • reducible form
        • in explicit normalized form ( only when denominator is odd )[2] : \tfrac{3}{63} = \tfrac{3}{2^{6}-1}
    • decimal floating point number 0.\overline{047619}
      • finite expansion
      • endless expansion
        • continue infinitely without repeating (in which case the number is called irrational)
        • Recurring or repeating
          • (strictly) periodic ( preperiod = 0 , preiod > 0 )
          • mixed = eventually periodic ( preperiod > 0 , period > 0 )
  • binary number ( base = 2 )
    • binary rational number ( ratio) \tfrac{1}{10101}
    • binary real number
      • binary floating point number ( scientific notation )
      • binary fixed point number ( notation)
        • with repeating sequences : 0.\overline{000011}
        • with endless expansion 0.000011000011000011000011...


Binary expansion of \tfrac{1}{21} has a period= 6 under doubling map

Examples[edit]

For 3923/6173 decimal expansion has period= 3086 and preperiod 0 [3]

preperiodic[edit]

1/12= 1/(3*2^2) 0.08(3) preperiod=2 period=1

1/6=1/(2*3) = 0.1()6) preperiod=1 preriod=2 ;

77/600 = 77/(2^3*3*5^2) = 0.128(3) preperiod=3 period = 1

periodic[edit]

1/3 = 0.(3) preperiod=0 preriod=2 ;

1/7 = 0.142857  ; 6 repeating digits

1/17 = 0.05882352 94117647  ; 16 repeating digits

1/19 = 0.052631578 947368421  ; 18 repeating digits

1/23 = 0.04347826086 95652173913  ; 22 repeating digits

1/29 = 0.0344827 5862068 9655172 4137931  ; 28 repeating digits

1/97 = 0.01030927 83505154 63917525 77319587 62886597 93814432 98969072 16494845 36082474 22680412 37113402 06185567  ; 96 repeating digits

finite[edit]

1/2 = 0.2 finite preperiod =0 period = 0 ; terminating decimal


How to find number type[edit]

/*

Maxima CAS batch file




*/

remvalue(all);
kill(all);


/*
input = ratio, which automaticaly changed to lowest terms by Maxima CAS
output = string describing a type of decimal expansion

---------------------------------------------------------------------------------
" The rules that determine whether a fraction has recurring decimals or 
not are really quite simple.

1. First represent the fraction in its simplest form, by dividing both 
numerator and denominator by common factors.

2. Now, look at the denominator.

3.
3.1 If the prime factorization of the denominator contains only the 
factors 2 and 5, then the decimal fraction of that fraction will not 
have recurring digits. In other words : Terminating decimals represent 
rational numbers of the form k/(2^n*5^m)


3.2
  A fraction in lowest terms with a prime denominator other than 2 or 5 
(i.e. coprime to 10) always produces a repeating decimal.

3.2.1
  If the prime factorization yields factors like 3, 7, 11 or other 
primes (other than 2 and 5), then that fraction will have a decimal 
representation that includes recurring digits.

3.2.2
   Moreover, if the denominator's prime factors include 2 and/or 5 in 
addition to other prime factors like 3, 7, etc., the decimal 
representation of the fraction will start with a few non-recurring 
decimals before the recurring part."

http://blogannath.blogspot.com/2010/04/vedic-mathematics-lesson-49-recurring.html


check :
http://www.knowledgedoor.com/2/calculators/convert_a_ratio_of_integers.html




---------------------------------------------------------------------------------------

*/


GiveRatioType(ratio):=
block
(
   [numerator:denom(ratio),
    FactorsList ,
    Factor,
    Has25:false,
    HasAlsoOtherPrimes:false,
    type ], /* type of decimal expansion of the ratio of integers */

   /* compute list of prime factors ofd denominator */
   FactorsList:ifactors(numerator),
   FactorsList:map(first,FactorsList),
   print(numerator, FactorsList),
   /* check factors type :
          only 2 or 5
          also other primes then 2 or 5
  */
   if (member(2,FactorsList) or member(5,FactorsList)) then Has25:true,


   for Factor in FactorsList do
    if (not member(Factor,[2,5])) then
          HasAlsoOtherPrimes:true,
   print(Has25, HasAlsoOtherPrimes),

   /* find type of decimal expansion */
   if (not Has25 and HasAlsoOtherPrimes)     then type:"periodic",
   if (Has25 and HasAlsoOtherPrimes)     then type:"preperiodic",
   if (Has25 and not HasAlsoOtherPrimes) then type:"finite",


   return(type)
)$

compile(all)$

/* input numbers*/
a:1 $
b:3 $

r:a/b$

type :  GiveRatioType(r);

Conversions[edit]

Conversion between :

  • bases
    • using bc / dc
  • ratio and expansion
  • expansion and rational form [6]
    • repeating decimals and ratio [7]
    • Recognizing Rational Numbers From Their Decimal Expansion[8]: "to compute the simple continued fraction of the approximation, and truncate it before a large partial quotient a_n, then compute the value of the truncated continued fraction."

References[edit]

  1. wikipedia : Number base
  2. HOW TO WORK WITH ONE-DI MENSIONAL QUADRATIC MAPS G. Pastor , M. Romera, G. Álvarez, and F. Montoya
  3. wolframalpha : 3923/6173
  4. knowledgedoor calculators: convert_a_ratio_of_integers
  5. R.Knott : Fractions – Decimals Calculator
  6. Recognizing Rational Numbers From Their Decimal Expansion by William Stein
  7. basic-mathematics : converting-repeating-decimals-to-fractions
  8. Rational Numbers From Their Decimal Expansion by William Stein