"Many questions concerning (discrete) dynamical systems are of a number theoretic or combinatorial nature." Christian Krattenthaler

• 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 can be used as :

• a numerical values used in numerical computations
• a symbols used in symbolic computations

Number ( for exampleangle in turns ) can be:^[1]

• decimal number (base = 10 )^[2]
  • integer
  • real number
    • ratio = fraction ( Finitecontinued fraction ) = rational number ( the irrationality measure of any rational number is 1)^[3]
      • in lowest terms ( irreducible form ) :${\displaystyle \frac{1}{21}}$
      • reducible form
        • in explicit normalized form ( only when denominator is odd ):^[4]${\displaystyle \frac{3}{63}=\frac{3}{2^6-1}}$
    • 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${\displaystyle 0.\overline{047619}}$^[5][6]
    • finite expansion
    • infinite (endless) expansion
      • continue infinitely without repeating (in which case the number is called irrational = non-repeating non-terminating decimal numbers^[7])
      • Recurring or repeating
        • (strictly) periodic ( preperiod = 0 , period > 0 )
        • mixed = eventually periodic ( preperiod > 0 , period > 0 )
• binary number ( base = 2 )^[8]
  • binary rational number ( ratio)${\displaystyle \frac{1}{10101}}$
  • binary real number
    • binary floating point number ( scientific notation )
    • Raw binary ( raw IEEE format )
    • binary fixed point number ( notation)
      • with repeating sequences :${\displaystyle 0.\overline{000011}}$
      • with endless expansion${\displaystyle \mathrm{0.000011000011000011000011...}}$

• 1D: real
• 2D
  • dual^[9]
  • complex^[10]
• 4D^[11]
  • quaterions

• IEEE floating point
• fixed point
• unums (universal numbers) by John Gustafson
  • Posits are a hardware-friendly version of unums^[12][13]

• finite = terminating
• infinite = non-terminating
  • periodic = infite repeating
  • preperiodic = eventually periodic
  • non-periodic: binary numerals which neither terminate nor recur represent irrational numbers

radix or base of a positional numeral system^[14]

• 2 ( binary number)
• 8 ( octal number)
• 10 ( decimal umber)
• 16 ( hexadecimal)

Notation^[15]

• sequence of digits and radix
  • infinite sequence
    • in general form is denoted byellipsis ( = 3 dots):${\displaystyle r=d_0.d_1{d}_2{d}_3\dots }$
    • Infinitely repeating part of expansion denoted by
      • round brackets :${\displaystyle 0.10(00101010)=0.10001010100010101000101010001010100010101000101010\dots }$.
      • overline:${\displaystyle 0.10\overline{00101010}=0.10001010100010101000101010001010100010101000101010\dots }$
  • finite sequence
    • 1.23
      • with trailing zeros: 1.2300 for indicating the number of significant figures, for example in a measurement.
    • with absolute measurment error: mean ± range^[16], for example: 72.20 ± 0.02
• ratio of integers${\displaystyle r=\frac{m}{n}}$
  • in lowest terms ( irreducible form ) :${\displaystyle \frac{1}{21}}$
  • reducible form
    • in explicit normalized form ( only when denominator is odd ):${\displaystyle \frac{n}{2^p-1}}$^[17], for example :${\displaystyle \frac{3}{63}=\frac{3}{2^6-1}}$
    • The explicit normalized form of formula for denominator of angle :${\displaystyle \frac{n}{(2^p-1)\ast 2^k}}$
• continued fraction :${\displaystyle [a_0;a_1,a_2,a_3]}$
• scientific (exponential) form or notation:^[18]

${\displaystyle 6.022e23=6.022\ast {10}^{23}}$

${\displaystyle 1.2345e-3=1.2345+{10}^{-3}}$

${\displaystyle \frac{3}{63}=\frac{3}{2^6-1}=\frac{1}{21}=0.04761904761904762}$

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

brackets with exponent ( superscript) denotes how many times the series repeats^[19]

${\displaystyle 0.11(11101010)=0.11(11(10{)}^3)}$

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.

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

Conversion between :

• bases ( from binary to decimal, ...)
• forms ( rational to expansion, ...)^[20]
  • Recognizing Rational Numbers From Their Decimal Expansion:^[21] "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^[22]
  • fraction to recurring decimal^[23]
    • 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-bases^[24]

Using :

• online calculators^{[25][26][27][28][29]}
• code

• Find
  • strictly repeating patterns (that you do not know in advance) in a binary string/sequence^[30] "If there is a pattern => its length must divide the string length "AnotherGeek^[31]
    • using regex^[32]
    • javascript^[33][34]
  • 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

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

"... 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^[35])

Algorithm:^[36]

• 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:

${\displaystyle {0.1}_{10}=0.0(0011{)}_2}$

Repeating fractions :^[37]

0.(567) = 567/999 = 189/333 = 63/1110.(0011) = 0011 / 1111 =(in decimal) 3/15 = 1/5

Code

• geeksforgeeks: convert-decimal-fraction-binary-number
• primary-bulb by Claude Heiland-Allen

(Pre)periodicbinary fraction can be split into 2 fractions:

• finite
• infinite: periodic with empty or filled with zeros preperiodic part

${\displaystyle 0.\stackrel{t}{\overbrace{b...b}}(\stackrel{p}{\overbrace{b...b}})=0.\stackrel{t}{\overbrace{b...b}}+0.\stackrel{t}{\overbrace{\mathrm{0...0}}}(\stackrel{p}{\overbrace{b...b}})}$

Formula for the geometric series when |r|<1 :^[38]

${\displaystyle a+ar+a{r}^2+a{r}^3+a{r}^4+\cdots =\sum _{k=0}^{\mathrm{\infty }}a{r}^k=\frac{a}{1-r}}$

For the infinite periodic binary fraction with empty or filled with zeros preperiodic part this formula is^[39]

${\displaystyle 0.\stackrel{t}{\overbrace{\mathrm{0...0}}}(\stackrel{p}{\overbrace{b...b}})=\frac{a}{1-r}}$

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${\displaystyle a=0.\stackrel{t}{\overbrace{\mathrm{0...0}}}\stackrel{p}{\overbrace{b...b}}}$
• the value of${\displaystyle r=\frac{1}{2^p}}$ so${\displaystyle 1-r=\frac{2^p-1}{2^p}}$

${\displaystyle 0.\stackrel{t}{\overbrace{b...b}}(\stackrel{p}{\overbrace{b...b}})=0.\stackrel{t}{\overbrace{b...b}}+\frac{a}{1-r}=0.\stackrel{t}{\overbrace{b...b}}+\frac{0.\stackrel{t}{\overbrace{\mathrm{0...0}}}\stackrel{p}{\overbrace{b...b}}}{\frac{2^p-1}{2^p}}}$

Full formula is now:

${\displaystyle 0.\stackrel{t}{\overbrace{b...b}}(\stackrel{p}{\overbrace{b...b}})=0.\stackrel{t}{\overbrace{b...b}}+\frac{0.\stackrel{t}{\overbrace{\mathrm{0...0}}}\stackrel{p}{\overbrace{b...b}}}{\frac{2^p-1}{2^p}}}$

Examples :

• ${\displaystyle 0.(0011)=\frac{0.0011}{\frac{2^4-1}{2^4}}=\frac{3/16}{15/16}=\frac{1}{5}}$
• ${\displaystyle 0.00(0011)=\frac{0.000011}{\frac{2^4-1}{2^4}}=\frac{3/64}{15/16}=\frac{1}{20}}$
• ${\displaystyle 0.01(1100)=0.01+\frac{0.0011}{\frac{2^4-1}{2^4}}=\frac{1}{4}+\frac{3/16}{15/16}=\frac{1}{4}+\frac{1}{5}=\frac{9}{20}}$

${\displaystyle 0.0101(11001)=0.0101+\frac{0.000011001}{\frac{2^5-1}{2^5}}=\frac{5}{16}+\frac{25/512}{31/32}=\frac{5}{16}+\frac{25}{496}=\frac{45}{124}}$

Conversion from decimal ratio to binary^[40] usingbc – arbitrary–precision arithmetic language

bc 1.06Copyright 1991-1994, 1997, 1998, 2000 Free Software Foundation, Inc.This is free software with ABSOLUTELY NO WARRANTY.For details type `warranty'. obase=23/14.00110110110110110110110110110110110110110110110110110110110110110101/5.0011001100110011001100110011001100110011001100110011001100110011001

• itoa
• snprintf^[41]
• 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 function^[42]

/* itoa example http://www.cplusplus.com/reference/cstdlib/itoa/*/#include<stdio.h>#include<stdlib.h>intmain(){inti;charbuffer[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);return0;}

Binary integer constant^[43]

"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 library^[44]

/*C programme using gmpgcc r.c -lgmp -Wallhttp://gmplib.org/manual/Rational-Number-Functions.html#Rational-Number-Functions*/#include<stdio.h>#include<gmp.h>intmain(){// input = binary fraction as a stringchar*sbr="01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010/11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111";mpq_tq;// rational number;intb=2;// base of numeral systemmpz_tn;mpz_td;mpf_tf;// init and set variablesmpq_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 , dmpz_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-issuempf_set_q(f,q);// There is no rounding, this conversion is exact.// printgmp_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 memorympq_clear(q);mpz_clear(n);mpz_clear(d);mpf_clear(f);return0;}

Output :

decimal fraction =  179622968672387565806504266 / 618970019642690137449562111 decimal canonical form =  179622968672387565806504266/618970019642690137449562111binary fraction  = 01001010010010100101001001010010010100101001001010010100100101001001010010100100101001010/11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 decimal floating point number : 0.290196557138708685358212602171

Code by Claude Heiland-Allen:^[45]

--  http://mathr.co.uk/blog/2014-10-13_converting_fractions_to_strings_of_digits.htmlimportData.Fixed(mod')importData.List(nub)importData.Ratio((%),denominator)importData.Numbers.Primes(primeFactors)importSystem.Environment(getArgs)dataDigits=Digits{dNegative::Bool,dInteger,dPreperiodic,dPeriodic::[Int]}derivingShowpreperiod::Digits->Intpreperiod=length.dPreperiodicperiod::Digits->Intperiod=length.dPeriodicdigitsAtBase::Int->Rational->DigitsdigitsAtBasebaserational=Digits{dNegative=rational<0,dInteger=int,dPreperiodic=pre,dPeriodic=per}whereinteger::Integerfraction::Rational(integer,fraction)=properFraction(absrational)int|integer==0=[0]|otherwise=goIntinteger[]goIntids|i==0=ds|otherwise=goInti'(fromIntegerd:ds)where(i',d)=i`divMod`baseZfactors::[Integer]factors=mapfromIntegral.nub.primeFactors$baseisPreperiodic::Rational->BoolisPreperiodicx=any(`divides`denominatorx)factorsbaseZ::IntegerbaseZ=fromIntegralbasebaseQ::RationalbaseQ=fromIntegralbase(pre,per)=goPrefractionwheregoPre::Rational->([Int],[Int])goPrex|isPreperiodicx=first(d:)(goPrex')|otherwise=([],d:goPerxx')where(d,x')=properFraction(baseQ*x)goPer::Rational->Rational->[Int]goPerx0x|x0==x=[]|otherwise=d:goPerx0x'where(d,x')=properFraction(baseQ*x)first::(a->c)->(a,b)->(c,b)firstf(a,b)=(fa,b)divides::Integer->Integer->Boolfactor`divides`number=number`mod`factor==0digitsToString::[String]->Digits->StringdigitsToStringdigitsDigits{dNegative=sign,dInteger=int,dPreperiodic=pre,dPeriodic=per}=(ifsignthen"-"else"")++dint++"."++dpre++"("++dper++")"whered=concatMap(digits!!)atBase::Int->Rational->StringatBasebaserational=digitsToStringds(digitsAtBasebaserational)whereds|base<=62=map(:[])$['0'..'9']++['A'..'Z']++['a'..'z']|otherwise=["<"++showd++">"|d<-[0..base-1]]main::IO()main=do[sbase,sfraction]<-getArgslet(snum,_:sden)=break('/'==)sfractionbase=readsbasenum=readsnumden=readsdenrational=num%denputStrLn(atBasebaserational)

# 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^[46]

First read:

• article "What Every Computer Scientist Should Know About Floating-Point Arithmetic" by DAVID GOLDBERG :^[47]
• Josh HabermanFloating Point Demystified, Part 1 by
• floating-point-demystified-part2

• types
• limits and overflow

/*gcc l.c -lm -Wall./a.outhttp://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.htmlintmain(){doublelMax;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);return0;}

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 inprogram Mandel makes a silent bounds check:^[48]

// mndynamo.h  by Wolf Jung (C) 2007-2014typedefunsignedlonglongintqulonglong;// mndcombi.cpp  by Wolf Jung (C) 2007-2014qulonglongmndAngle::wake(intk,intr,qulonglong&n){if(k<=0||k>=r||r>64)return0LL;

If r is to big for unsigned long long int type it returns 0 to prevent integer overflow.

GMP library has arbitrary precision rationals.

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.

"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."^[49]

So in numerical computations one can use only integer of floating points numbers ( rational ).

• Exploring Binary Numbers With PARI/GP Calculator By Rick Regan
• Wolfram Alpha

InC one can use :

• bitwise operators^[50]

InMaxima CAS one can use :

(%i1) ibase;(%o1) 10(%i2) obase;(%o2) 10(%i3) ibase:2;(%o3) 2(%i4) x=1001110;(%o4) x=78

Calculation of binary numbers with as a string with replicating parts inHaskell (ghci):

-- by Claude Heiland-Allen-- http://mathr.co.uk/blog/haskell.htmlPrelude>letrepns=concat(replicatens)Prelude>putStrLn$".("++rep88"001"++"010)".(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010)putStrLn$".("++rep87"001"++"010001)".(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010001)Prelude>putStrLn$".("++rep88"001"++"0001)".(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010001)Prelude>putStrLn$".("++rep88"001"++"0010)".(0010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010010)

InPython :

>>>bin(173)'0b10101101'>>>int('01010101111',2)687

In python one can use binary literals :^[51]

pythonPython2.7.5+(default,Feb272014,19:37:08)[GCC4.8.1]onlinux2Type"help","copyright","credits"or"license"formoreinformation.>>>0b10111147

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^[52] (the Diophantine Approximation^[53])
  • ratio of integers
  • floating point number
  • finite continued fractions

The most irrational number^[54] 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 :

${\displaystyle 562949953421312=1+2^{49}}$

• the multi-valued nature of complex powers can cause big troubles ( artifacts of branch cuts, arbitrary principal value of arg)
• domain coloring^[55][56]

Examples

• VBA^[57]
• Maxima CAS

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 CASoutput = 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.htmlcheck :http://www.knowledgedoor.com/2/calculators/convert_a_ratio_of_integers.htmlwikipedia: 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);

• dumpfp: A Tool to Inspect Floating-Point Numbers by Joshua Haberman^[58]
• float.exposed - andblog by Bartosz Ciechanowski

• sequences
  • orbits
    • period
• series
• cardinality
• partition
• random numbers

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. "^[59]
• "... 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)^[60] "To be pedantically correct you should have said almost certainly is 1. " – David Hammen

  "a “height function” is some real-valued function that defines the “arithmetic complexity” of a point ... " Brian Lawrence[61]

Types of the height functions defined on the set of rational numbers${\displaystyle \mathbb{Q}}$:

• ${\displaystyle H(p/q)=max\{|p|,|q|\}}$ for the (multiplicative) height of a rational number^[62] also callednaive height
• ${\displaystyle h(p/q)=\log H(p/q)}$ the logarithmic height or additive^[63]

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 function : "partition numbers behave like fractals, possessing an infinitely-repeating structure"^[64]

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 )

• generalisation : scalar / vector / tensor
• fields : scalar ,vector, tensor

1. ↑exploring binary: nine-ways-to-display-a-floating-point-number
2. ↑wikipedia : Number base
3. ↑math.stackexchange question: are-there-real-numbers-that-are-neither-rational-nor-irrational
4. ↑HOW TO WORK WITH ONE-DI MENSIONAL QUADRATIC MAPS G. Pastor , M. Romera, G. Álvarez, and F. Montoya
5. ↑What Every Programmer Should Know About Floating-Point Arithmetic
6. ↑Stackoverflow : Why Are Floating Point Numbers Inaccurate?
7. ↑home school math  : The fascinating irrational numbers
8. ↑Tutorial: Floating-Point Binary by Kip Irvine
9. ↑Dual Numbers & Automatic Differentiation
10. ↑videos: Imaginary Numbers are Real from Welch Labs
11. ↑math stackexchange question: is-there-a-third-dimension-of-numbers
12. ↑Beating Floating Point at its Own Game: Posit Arithmetic by John L. Gustafson , Isaac Yonemoto
13. ↑fractalforums.org : posits
14. ↑wikipedia: Radix
15. ↑Survey of Floating-Point Formats Robert Munafo's home pages on AWS   © 1996-2022 Robert P. Munafo.  
16. ↑Physics 132 Lab Manual: how-to-write-numbers-significant-figures by Brokk Toggerson and Aidan Philbin
17. ↑HOW TO WORK WITH ONE-DI MENSIONAL QUADRATIC MAPS G. Pastor , M. Romera, G. Álvarez, and F. Montoya
18. ↑calculatorsoup: scientific-notation-converter
19. ↑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
20. ↑Converting fractions to strings of digits by Claude Heiland-Allen
21. ↑Rational Numbers From Their Decimal Expansion by William Stein
22. ↑basic-mathematics : converting-repeating-decimals-to-fractions
23. ↑Recurring decimal of a rational number by Yurii Lahodiuk
24. ↑quora : How-do-you-convert-repeating-fractions-to-different-bases
25. ↑knowledgedoor calculators: convert_a_ratio_of_integers
26. ↑R.Knott : Fractions – Decimals Calculator
27. ↑Base Number - Decimal Number Conversion
28. ↑Decimal to Floating-Point Converter By Rick Regan
29. ↑wolframalpha binary to decimal conversion
30. ↑stackoverflow question : best-algorithm-to-find-a-repeating-pattern
31. ↑stackoverflow questions : method-to-find-repeated-pattern-in-string-apart-from-regex?noredirect=1&lq=1
32. ↑stackoverflow question: finding-a-repeated-pattern-in-a-string
33. ↑stackoverflow question: finding-a-pattern-in-a-binary-string
34. ↑jsfiddle by Jan Turoń
35. ↑Virginia Tech Online CS module
36. ↑Stackoverflow : How do you convert a fraction to binary ?
37. ↑Converting a repeating binary number to decimal (express as a series?)
38. ↑wikipedia :Geometric_series
39. ↑stackoverflow question: converting-a-repeating-binary-number-to-decimal-express-as-a-series
40. ↑math.stackexchange question: find-a-fraction-given-the-repeating-binary-expansions
41. ↑Where is the itoa function in Linux?
42. ↑itoa with GCC by Stuart
43. ↑gcc - Binary-constants
44. ↑Programowanie_w_systemie_UNIX: GMP in polish wikibooks
45. ↑Converting fractions to strings of digits by Claude Heiland-Allen
46. ↑stackoverflow : python int to binary
47. ↑What Every Computer Scientist Should Know About Floating-Point Arithmetic by DAVID GOLDBERG
48. ↑wikipedia : Bounds checking
49. ↑stackoverflow questions : check-if-a-number-is-rational-in-python
50. ↑Joe McCullough : bitwise operators
51. ↑Stackoverflow : How do you express binary literals in Python?
52. ↑John D Cook : best-rational-approximation
53. ↑DISCOVERING EXACTLY WHEN A RATIONAL IS A BEST APPROXIMATE OF AN IRRATIONAL By KARI LOCK
54. ↑ams : The Most Irrational Number
55. ↑complex beauties : math-calendar
56. ↑David Bau : complex function viewer
57. ↑Complex Numbers in VBA by Pfadintegral
58. ↑dumpfp: A Tool to Inspect Floating-Point Numbers by Joshua Haberman
59. ↑home school math : The fascinating irrational numbers
60. ↑stackoverflow questions : irrational-number-check-function
61. ↑Introduction to Heights by Brian Lawrence
62. ↑sagemath : Rational.global_height
63. ↑Height Functions by Michael Tepper
64. ↑Fractal Structure to Partition Function.

• An Illustrated Theory of Numbers by Martin H. Weissman

• 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

• Book:Fractals