Fractals/Mathematics/Numbers

"Many questions concerning (discrete) dynamical systems are of a number theoretic or combinatorial nature." Christian Krattenthaler thumb|right|The set of real numbers Intro Basic Number Theory Every Programmer Should Know...by CodeChef Quora : What important topics of number theory should every programmer know? science4all : numbers-and-constructibility Number Theory How Computers Use Numbers by Mabi19 Number types Number can be used as : a numerical values used in numerical computations a symbols used in symbolic computations Number ( for example angle in turns ) can be:exploring binary: nine-ways-to-display-a-floating-point-number decimal number (base = 10 ) wikipedia : Number base integer real number ratio = fraction ( Finite continued fraction ) = rational number ( the irrationality measure of any rational number is 1)math.stackexchange question: are-there-real-numbers-that-are-neither-rational-nor-irrational in lowest terms ( irreducible form ) : reducible form in explicit normalized form ( only when denominator is odd ):HOW TO WORK WITH ONE-DI MENSIONAL QUADRATIC MAPS G. Pastor , M. Romera, G. Álvarez, and F. Montoya irrational number = infinite continued fraction ( if number can not be represented as a ratio then it is irrational number ) algebraic ( irrationality measure = 2) transcendental ( irrationality measure > 2) decimal floating point number What Every Programmer Should Know About Floating-Point Arithmetic Stackoverflow : Why Are Floating Point Numbers Inaccurate? finite expansion infinite (endless) expansion continue infinitely without repeating (in which case the number is called irrational = non-repeating non-terminating decimal numbershome school math : The fascinating irrational numbers) Recurring or repeating (strictly) periodic ( preperiod = 0 , period > 0 ) mixed = eventually periodic ( preperiod > 0 , period > 0 ) binary number ( base = 2 )Tutorial: Floating-Point Binary by Kip Irvine binary rational number ( ratio) binary real number binary floating point number ( scientific notation ) Raw binary ( raw IEEE format ) binary fixed point number ( notation) with repeating sequences : with endless expansion dimension 1D: real 2D dualDual Numbers & Automatic Differentiation complexvideos: Imaginary Numbers are Real from Welch Labs 4Dmath stackexchange question: is-there-a-third-dimension-of-numbers quaterions way to represent real numbers using a finite number of bits IEEE floating point fixed point unums (universal numbers) by John Gustafson Posits are a hardware-friendly version of unumsBeating Floating Point at its Own Game: Posit Arithmetic by John L. Gustafson , Isaac Yonemotofractalforums.org : posits expansion/representation finite = terminating infinite = non-terminating periodic = infite repeating preperiodic = eventually periodic non-periodic: binary numerals which neither terminate nor recur represent irrational numbers base radix or base of a positional numeral systemwikipedia: Radix 2 ( binary number) 8 ( octal number) 10 ( decimal umber) 16 ( hexadecimal) form/notation NotationSurvey of Floating-Point Formats Robert Munafo's home pages on AWS   © 1996-2022 Robert P. Munafo.   sequence of digits and radix infinite sequence in general form is denoted by ellipsis ( = 3 dots): Infinitely repeating part of expansion denoted by round brackets : . overline: finite sequence 1.23 with trailing zeros: 1.2300 for indicating the number of significant figures, for example in a measurement. with absolute measurement error: mean ± range Physics 132 Lab Manual: how-to-write-numbers-significant-figures by Brokk Toggerson and Aidan Philbin , for example: 72.20 ± 0.02 ratio of integers in lowest terms ( irreducible form ) : reducible form in explicit normalized form ( only when denominator is odd ):HOW TO WORK WITH ONE-DI MENSIONAL QUADRATIC MAPS G. Pastor , M. Romera, G. Álvarez, and F. Montoya , for example : The explicit normalized form of formula for denominator of angle : continued fraction : scientific (exponential) form or notation: calculatorsoup: scientific-notation-converter A computer number formats ( storage forms) floating point form ( expansion) : the number's radix point can "float" anywhere to the left, right, or between the significant digits of the number. fixed point format Round brackets brackets with exponent ( superscript) denotes how many times the series repeats A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set M. Romera,1 G. Pastor, A. B. Orue,1 A. Martin, M.-F. Danca,and F. Montoya Trailing zeros Trailing zeros to the right of a decimal point, as in 12.3400, do not affect the value of a number and may be omitted if all that is of interest is its numerical value. This is true even if the zeros recur infinitely. For example, in pharmacy, trailing zeros are omitted from dose values to prevent misreading. However, trailing zeros may be useful for indicating the number of significant figures, for example in a measurement. In such a context, "simplifying" a number by removing trailing zeros would be incorrect. The number of trailing zeros in a non-zero base-b integer n equals the exponent of the highest power of b that divides n. For example, 14000 has three trailing zeros and is therefore divisible by 1000 = 103, but not by 104. This property is useful when looking for small factors in integer factorization. Some computer architectures have a count trailing zeros operation in their instruction set for efficiently determining the number of trailing zero bits in a machine word. Examples of binary expansions First check if the ratio is in the lowest terms ( reducible) Binary expansion can be : finite infinite periodic : preperiod = 0, period > 0 preperiodic ( = eventually periodic) : preperiod > 0, period > 0 aperiodic : preperiod = 0, period = 0 Conversions Conversion between : bases ( from binary to decimal, ...) forms ( rational to expansion, ...) Converting fractions to strings of digits by Claude Heiland-Allen Recognizing Rational Numbers From Their Decimal Expansion:Rational Numbers From Their Decimal Expansion by William Stein "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." converting-repeating-decimals-to-fractions basic-mathematics : converting-repeating-decimals-to-fractions fraction to recurring decimalRecurring decimal of a rational number by Yurii Lahodiuk use of Floyd's Cycle Detection Algorithm for finding of the first repetitive remainder recursive division and collection of remainders (associated with pieces of decimal fraction) convert-repeating-fractions-to-different-basesquora : How-do-you-convert-repeating-fractions-to-different-bases Using : online calculators knowledgedoor calculators: convert_a_ratio_of_integersR.Knott : Fractions – Decimals Calculator Base Number - Decimal Number ConversionDecimal to Floating-Point Converter By Rick Reganwolframalpha binary to decimal conversion code Algorithms Find strictly repeating patterns (that you do not know in advance) in a binary string/sequencestackoverflow question : best-algorithm-to-find-a-repeating-pattern "If there is a pattern => its length must divide the string length "AnotherGeekstackoverflow questions : method-to-find-repeated-pattern-in-string-apart-from-regex?noredirect=1&lq=1 using regexstackoverflow question: finding-a-repeated-pattern-in-a-string javascriptstackoverflow question: finding-a-pattern-in-a-binary-string jsfiddle by Jan Turoń non-repeating and strictly repeating patterns (that you do not know in advance) in a binary string/sequence convert a number with a repeating fractional part Reducing Fractions to Lowest Terms A fraction that is reducible can be reduced by dividing both the numerator and denominator by a common factor. It can be fully reduced to lowest terms if both are divided by their greatest common divisor Algorithms for finding the greatest common divisor: the Euclidean algorithm prime factorization The Euclidean algorithm is commonly preferred because it allows one to reduce fractions with numerators and denominators too large to be easily factored Examples: desmos: graphical fraction reduction convert decimal fraction to binary "... we repeatedly multiply the decimal fraction by 2. If the result is greater than or equal to 1, we add a 1 to our answer. If the result is less than 1, we add a 0 to our answer." (from Virginia Tech Online CS module Virginia Tech Online CS module ) Algorithm:Stackoverflow : How do you convert a fraction to binary ? Multiply the input decimal fraction by two from above result take integer part as the binary digit take the fractional part as the starting point for the next step repeat until you either get to 0 or a periodic number read the number starting from the top - the first binary digit is the first digit after the comma Example of conversion 0.1 decimal fraction to binary fraction : 0.1 * 2 = 0.2 -> 0 0.2 * 2 = 0.4 -> 0 0.4 * 2 = 0.8 -> 0 0.8 * 2 = 1.6 -> 1 0.6 * 2 = 1.2 -> 1 0.2 * 2 = 0.4 -> 0 0.4 * 2 = 0.8 -> 0 0.8 * 2 = 1.6 -> 1 0.6 * 2 = 1.2 -> 1 0.2 * 2 = 0.4 -> 0 0.4 * 2 = 0.8 -> 0 0.8 * 2 = 1.6 -> 1 0.6 * 2 = 1.2 -> 1 0.2 * 2 = 0.4 -> 0 Result: Repeating fractions :Converting a repeating binary number to decimal (express as a series?) 0.(567) = 567/999 = 189/333 = 63/111 0.(0011) = 0011 / 1111 =(in decimal) 3/15 = 1/5 thumbnail|right|Graphical conversion Code geeksforgeeks: convert-decimal-fraction-binary-number primary-bulb by Claude Heiland-Allen Convert binary fraction to decimal ratio Geometric series (Pre)periodic binary fraction can be split into 2 fractions: finite infinite: periodic with empty or filled with zeros preperiodic part Formula for the geometric series when |r|<1 :wikipedia :Geometric_series For the infinite periodic binary fraction with empty or filled with zeros preperiodic part this formula isstackoverflow question: converting-a-repeating-binary-number-to-decimal-express-as-a-series where : b is a binary digit : 0 or 1 t is a length of preperiodic block p is a length of the periodic block the value of a is simply the value of the first occurrence of the repeating block the value of so Full formula is now: Examples : code examples bc Conversion from decimal ratio to binarymath.stackexchange question: find-a-fraction-given-the-repeating-binary-expansions using bc – arbitrary–precision arithmetic language bc 1.06 Copyright 1991-1994, 1997, 1998, 2000 Free Software Foundation, Inc. This is free software with ABSOLUTELY NO WARRANTY. For details type `warranty'. obase=2 3/14 .0011011011011011011011011011011011011011011011011011011011011011010 1/5 .0011001100110011001100110011001100110011001100110011001100110011001 C itoa snprintf Where is the itoa function in Linux? Binary integer constant Macro BOOST_BINARY gmp mandelbrot-symbolics lbrary by Claude Heiland-Allen Fractals/mandelbrot-symbolics#m-binangle-from-rational binangle-from-rational Conversion numbers to binary representation by Wojciech Muła itoa itoa function itoa with GCC by Stuart /* itoa example http://www.cplusplus.com/reference/cstdlib/itoa/ */ #include <stdio.h> #include <stdlib.h> int main () { int i; char buffer [33]; printf ("Enter a number: "); scanf ("%d",&i); itoa (i,buffer,10); printf ("decimal: %s\n",buffer); itoa (i,buffer,16); printf ("hexadecimal: %s\n",buffer); itoa (i,buffer,2); printf ("binary: %s\n",buffer); return 0; } Binary integer constant Binary integer constantgcc - Binary-constants "Integer constants can be written as binary constants, consisting of a sequence of ‘0’ and ‘1’ digits, prefixed by ‘0b’ or ‘0B’. This is particularly useful in environments that operate a lot on the bit level (like microcontrollers). The following statements are identical: i = 42; i = 0x2a; i = 052; i = 0b101010; The type of these constants follows the same rules as for octal or hexadecimal integer constants, so suffixes like ‘L’ or ‘UL’ can be applied." gmp GMP libraryProgramowanie_w_systemie_UNIX: GMP in polish wikibooks /* C programme using gmp gcc r.c -lgmp -Wall http://gmplib.org/manual/Rational-Number-Functions.html#Rational-Number-Functions */ #include <stdio.h> #include <gmp.h> int main () { // input = binary fraction as a string char *sbr = "01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010/11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111"; mpq_t q; // rational number; int b =2 ; // base of numeral system mpz_t n ; mpz_t d ; mpf_t f; // init and set variables mpq_init (q); // Initialize r and set it to 0/1. mpq_set_str (q, sbr , b); mpq_canonicalize (q); // It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable. mpq_canonicalize (q); // It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable. // n , d mpz_inits(n,d,NULL); mpq_get_num(n,q); mpq_get_den(d, q); // mpf_init2(f, 100); // http://stackoverflow.com/questions/12804362/gmp-division-precision-or-printing-issue mpf_set_q(f,q); // There is no rounding, this conversion is exact. // print gmp_printf ("decimal fraction = %Zd / %Zd \ndecimal canonical form = %Qd\n",n,d, q); // gmp_printf ("binary fraction = %s \n", sbr); // gmp_printf ("decimal floating point number : %.30Ff \n", f); // // clear memory mpq_clear (q); mpz_clear (n); mpz_clear (d); mpf_clear (f); return 0; } Output : decimal fraction = 179622968672387565806504266 / 618970019642690137449562111 decimal canonical form = 179622968672387565806504266/618970019642690137449562111 binary fraction = 01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010/11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 decimal floating point number : 0.290196557138708685358212602171 Haskell Code by Claude Heiland-Allen:Converting fractions to strings of digits by Claude Heiland-Allen -- http://mathr.co.uk/blog/2014-10-13_converting_fractions_to_strings_of_digits.html import Data.Fixed (mod') import Data.List (nub) import Data.Ratio ((%), denominator) import Data.Numbers.Primes (primeFactors) import System.Environment (getArgs) data Digits = Digits { dNegative :: Bool , dInteger , dPreperiodic , dPeriodic :: [Int] } deriving Show preperiod :: Digits -> Int preperiod = length . dPreperiodic period :: Digits -> Int period = length . dPeriodic digitsAtBase :: Int -> Rational -> Digits digitsAtBase base rational = Digits { dNegative = rational < 0 , dInteger = int , dPreperiodic = pre , dPeriodic = per } where integer :: Integer fraction :: Rational (integer, fraction) = properFraction (abs rational) int | integer == 0 = [0] | otherwise = goInt integer [] goInt i ds | i == 0 = ds | otherwise = goInt i' (fromInteger d : ds) where (i', d) = i `divMod` baseZ factors :: [Integer] factors = map fromIntegral . nub . primeFactors $ base isPreperiodic :: Rational -> Bool isPreperiodic x = any (`divides` denominator x) factors baseZ :: Integer baseZ = fromIntegral base baseQ :: Rational baseQ = fromIntegral base (pre, per) = goPre fraction where goPre :: Rational -> ([Int], [Int]) goPre x | isPreperiodic x = first (d:) (goPre x') | otherwise = ([], d : goPer x x') where (d, x') = properFraction (baseQ * x) goPer :: Rational -> Rational -> [Int] goPer x0 x | x0 == x = [] | otherwise = d : goPer x0 x' where (d, x') = properFraction (baseQ * x) first :: (a -> c) -> (a, b) -> (c, b) first f (a, b) = (f a, b) divides :: Integer -> Integer -> Bool factor `divides` number = number `mod` factor == 0 digitsToString :: [String] -> Digits -> String digitsToString digits Digits { dNegative = sign , dInteger = int , dPreperiodic = pre , dPeriodic = per } = (if sign then "-" else "") ++ d int ++ "." ++ d pre ++ "(" ++ d per ++ ")" where d = concatMap (digits !!) atBase :: Int -> Rational -> String atBase base rational = digitsToString ds (digitsAtBase base rational) where ds | base <= 62 = map (:[]) $ ['0'..'9'] ++ ['A'..'Z'] ++ ['a'..'z'] | otherwise = [ "<" ++ show d ++ ">" | d <- [0 .. base - 1] ] main :: IO () main = do [sbase, sfraction] <- getArgs let (snum, _:sden) = break ('/' ==) sfraction base = read sbase num = read snum den = read sden rational = num % den putStrLn (atBase base rational) Python # https://wiki.python.org/moin/BitManipulation # binary string to integer >>> int('00100001', 2) 33 # conversion from binary string to hex string >>> print "0x%x" % int('11111111', 2) 0xff >>> print "0x%x" % int('0110110110', 2) 0x1b6 >>> print "0x%x" % int('0010101110101100111010101101010111110101010101', 2) 0xaeb3ab57d55 Other methods stackoverflow : python int to binary How to use numbers in computer programs ? First read: article "What Every Computer Scientist Should Know About Floating-Point Arithmetic" by DAVID GOLDBERG : What Every Computer Scientist Should Know About Floating-Point Arithmetic by DAVID GOLDBERG Josh HabermanFloating Point Demystified, Part 1 by floating-point-demystified-part2 integer types limits and overflow Limit /* gcc l.c -lm -Wall ./a.out http://stackoverflow.com/questions/29592898/do-long-long-and-long-have-same-range-in-c-in-64-bit-machine */ #include <stdio.h> #include <math.h> // M_PI; needs -lm also #include <limits.h> // INT_MAX, http://pubs.opengroup.org/onlinepubs/009695399/basedefs/limits.h.html int main(){ double lMax; lMax = log2(INT_MAX); printf("INT_MAX \t= %25d ; lMax = log2(INT_MAX) \t= %.0f \n",INT_MAX, lMax); lMax = log2(UINT_MAX); printf("UINT_MAX \t= %25u ; lMax = log2(UINT_MAX) \t= %.0f \n", UINT_MAX, lMax); lMax = log2(LONG_MAX); printf("LONG_MAX \t= %25ld ; lMax = log2(LONG_MAX) \t= %.0f \n",LONG_MAX, lMax); lMax = log2(ULONG_MAX); printf("ULONG_MAX \t= %25lu ; lMax = log2(ULONG_MAX) \t= %.0f \n",ULONG_MAX, lMax); lMax = log2(LLONG_MAX); printf("LLONG_MAX \t= %25lld ; lMax = log2(LLONG_MAX) \t= %.0f \n",LLONG_MAX, lMax); lMax = log2(ULLONG_MAX); printf("ULLONG_MAX \t= %25llu ; lMax = log2(ULLONG_MAX) \t= %.0f \n",ULLONG_MAX, lMax); return 0; } Results : INT_MAX = 2147483647 ; lMax = log2(INT_MAX) = 31 UINT_MAX = 4294967295 ; lMax = log2(UINT_MAX) = 32 LONG_MAX = 9223372036854775807 ; lMax = log2(LONG_MAX) = 63 ULONG_MAX = 18446744073709551615 ; lMax = log2(ULONG_MAX) = 64 LLONG_MAX = 9223372036854775807 ; lMax = log2(LLONG_MAX) = 63 ULLONG_MAX = 18446744073709551615 ; lMax = log2(ULLONG_MAX) = 64 For example Wolf Jung in program Mandel makes a silent bounds check:wikipedia : Bounds checking // mndynamo.h by Wolf Jung (C) 2007-2014 typedef unsigned long long int qulonglong; // mndcombi.cpp by Wolf Jung (C) 2007-2014 qulonglong mndAngle::wake(int k, int r, qulonglong &n) { if (k <= 0 || k >= r || r > 64) return 0LL; If r is to big for unsigned long long int type it returns 0 to prevent integer overflow. GMP library has arbitrary precision rationals. floating point precision Precision GMP : The mantissa of each float has a user-selectable precision ( variable prec type mp_bitcnt_t ). Counts of bits of a multi-precision number are represented in the C type mp_bitcnt_t. Currently this is always an unsigned long MPFR : The precision is the number of bits used to represent the significand ( mantissa) of a floating-point number; the corresponding C data type is mpfr_prec_t. Rational thumb|right|The rational points on a circle correspond, under stereographic projection, to the rational points of the line. "Any number with a finite decimal expansion is a rational number. " In other words : "any floating point number can be converted to a rational number." stackoverflow questions : check-if-a-number-is-rational-in-python So in numerical computations one can use only integer of floating points numbers ( rational ). Decimal Binary Exploring Binary Numbers With PARI/GP Calculator By Rick Regan Wolfram Alpha Numbers In C one can use : bitwise operators Joe McCullough : bitwise operators In Maxima CAS one can use : (%i1) ibase; (%o1) 10 (%i2) obase; (%o2) 10 (%i3) ibase:2; (%o3) 2 (%i4) x=1001110; (%o4) x=78 String Calculation of binary numbers with as a string with replicating parts in Haskell (ghci): -- by Claude Heiland-Allen -- http://mathr.co.uk/blog/haskell.html Prelude> let rep n s = concat (replicate n s) Prelude> putStrLn $ ".(" ++ rep 88 "001" ++ "010)" .(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010) putStrLn $ ".(" ++ rep 87 "001" ++ "010001)" .(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010001) Prelude> putStrLn $ ".(" ++ rep 88 "001" ++ "0001)" .(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010001) Prelude> putStrLn $ ".(" ++ rep 88 "001" ++ "0010)" .(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010) In Python : >>> bin(173) '0b10101101' >>> int('01010101111',2) 687 Literal In python one can use binary literals :Stackoverflow : How do you express binary literals in Python? python Python 2.7.5+ (default, Feb 27 2014, 19:37:08) [GCC 4.8.1] on linux2 Type "help", "copyright", "credits" or "license" for more information. >>> 0b101111 47 Irrational = not rational thumb|right|Irrational numbers The problem is that we are exploring environments based upon irrational numbers through computer machinery which works with finite rationals ! ( Alessandro Rosa ) Expansion is non terminating and non repeating Types: Algebraic Numbers = roots of Algebraic Equations. Example : sqrt(2), transcendental numbers = non algebraic If one wants use irrational number then should check : symbolic computations : exact number can be used as a symbol, but "you cannot print the whole irrational number" infinite continued fraction numerical computations : close rational approximations to irrational numbers John D Cook : best-rational-approximation (the Diophantine Approximation DISCOVERING EXACTLY WHEN A RATIONAL IS A BEST APPROXIMATE OF AN IRRATIONAL By KARI LOCK) ratio of integers floating point number finite continued fractions Inverse golden mean The most irrational numberams : The Most Irrational Number In a continued fraction all numbers are 1 = the slowest convergence of all the irrational numbers Using Maxima CAS : (%i10) print(float(%phi-1)); (%o10).6180339887498949 (%i11) rationalize(float(%phi-1)); (%o11) 347922205179541/562949953421312 and : (%i14) print(float(1/%phi)); (%o14) .6180339887498948 (%i15) rationalize(float(1/%phi)); (%o15) 5566755282872655/9007199254740992 where denominator : complex the multi-valued nature of complex powers can cause big troubles ( artifacts of branch cuts, arbitrary principal value of arg) domain coloring complex beauties : math-calendarDavid Bau : complex function viewer Examples VBA Complex Numbers in VBA by Pfadintegral Maxima CAS How to find number type Note that in numerical computations with finite precision ( on computer) : if number is represented as a ratio ( of integers) then it is a rational number if number has a floating point representation the it is also a rational number because of limited precision = finite expansion /* Maxima CAS batch file */ remvalue(all); kill(all); /* input = ratio, which automatically 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 wikipedia: Repeating_decimal " A fraction in lowest terms with a prime denominator other than 2 or 5 (i.e. coprime to 10) always produces a repeating decimal. The length of the repetend (period of the repeating decimal) of 1/p is equal to the order of 10 modulo p. If 10 is a primitive root modulo p, the repetend length is equal to p − 1; if not, the repetend length is a factor of p − 1. This result can be deduced from Fermat's little theorem, which states that 10p−1 = 1 (mod p)." --------------------------------------------------------------------------------------- */ GiveRatioType(ratio):= block ( [denominator: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(denominator), FactorsList:map(first,FactorsList), print(denominator, 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); tools dumpfp: A Tool to Inspect Floating-Point Numbers by Joshua Habermandumpfp: A Tool to Inspect Floating-Point Numbers by Joshua Haberman float.exposed - and blog by Bartosz Ciechanowski More sequences orbits period series cardinality partition random numbers Cardinality In mathematic ( theory) : "... the rational numbers are a countable set whereas the irrational numbers are an uncountable set. In other words, there are more irrational numbers than there are rational. " home school math : The fascinating irrational numbers "... in the set of real numbers there is continuum of irrational numbers and only aleph-zero rational numbers. Thus probability that any random number is irrational is 1;" ( Bartek Ogryczak) stackoverflow questions : irrational-number-check-function "To be pedantically correct you should have said almost certainly is 1. " – David Hammen height of a rational number in lowest term thumb|right|Thomae function = 1/q "a “height function” is some real-valued function that defines the “arithmetic complexity” of a point ... " Brian LawrenceIntroduction to Heights by Brian Lawrence Types of the height functions defined on the set of rational numbers : for the (multiplicative) height of a rational numbersagemath : Rational.global_height also called naive height the logarithmic height or additiveHeight Functions by Michael Tepper where: p/q is a rational number in lowest term "How complicated is a rational number? Its size is not a very good indicator for this. For instance, 1987985792837/1987985792836 is approximately 1, but so much more complicated than 1. We'll explain how to measure the complexity of a rational number using various notions of height. We'll then see how heights are used to prove some basic finiteness theorems in number theory. One example will be the Mordell-Weil theorem: that on any rational elliptic curve, the group of rational points is finitely generated. " Alina Bucur (UCSD): Size Doesn't Matter: Heights in Number Theory Key words: number field Height Functions in Number Theory Paritition paritition function : "partition numbers behave like fractals, possessing an infinitely-repeating structure" Fractal Structure to Partition Function. Random number The probability that any random number : is irrational is almost 1 ( in theory because of cardinality ) is rational is 1 ( in numerical computations because of limited precision ) Fields generalisation : scalar / vector / tensor fields : scalar , vector, tensor References An Illustrated Theory of Numbers by Martin H. Weissman See also Divisor Plot : Explore composite number patterns Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions by Daniel Bahrdt, Martin P. Seybold Discrete Mathematics : Number representations A-level_Computing : Binary_fractions sequence_of_fraction_in_the_elephant_valley Fractals : Iterations_in_the_complex_plane , wake