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

Intro

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:^[1]

decimal number (base = 10 ) ^[2]
- integer
- real number
  - ratio = fraction ( Finite continued fraction ) = rational number ( the irrationality measure of any rational number is 1)^[3]
    - in lowest terms ( irreducible form ) : ${\tfrac {1}{21}}$
    - reducible form
      - in explicit normalized form ( only when denominator is odd ):^[4] ${\tfrac {3}{63}}={\tfrac {3}{2^{6}-1}}$
  - 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 $0.{\overline {047619}}$ ^[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) ${\tfrac {1}{10101}}$
- binary real number
  - binary floating point number ( scientific notation )
  - Raw binary ( raw IEEE format )
  - binary fixed point number ( notation)
    - with repeating sequences : $0.{\overline {000011}}$
    - with endless expansion $0.000011000011000011000011...$

dimension

1D: real
2D
- dual^[9]
- complex^[10]
4D^[11]
- 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 unums^[12]^[13]

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 system^[14]

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

form/notation

Notation^[15]

sequence of digits and radix
- infinite sequence
  - in general form is denoted by ellipsis ( = 3 dots): $r=d_{0}.d_{1}d_{2}d_{3}\dots$
  - Infinitely repeating part of expansion denoted by
    - round brackets : $0.10(00101010)=0.10001010100010101000101010001010100010101000101010\dots$ .
    - overline: $0.10{\overline {00101010}}=0.10001010100010101000101010001010100010101000101010\dots$
- 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 $r={\frac {m}{n}}$
- in lowest terms ( irreducible form ) : ${\tfrac {1}{21}}$
- reducible form
  - in explicit normalized form ( only when denominator is odd ): ${\frac {n}{2^{p}-1}}$ ^[17], for example : ${\tfrac {3}{63}}={\tfrac {3}{2^{6}-1}}$
  - The explicit normalized form of formula for denominator of angle : ${\frac {n}{(2^{p}-1)*2^{k}}}$
continued fraction : $[a_{0};a_{1},a_{2},a_{3}]$
scientific (exponential) form or notation: ^[18]

6.022e23=6.022*10^{23}

1.2345e-3=1.2345+10^{-3}

{\tfrac {3}{63}}={\tfrac {3}{2^{6}-1}}={\tfrac {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

Round brackets

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

0.11(11101010)=0.11(11(10)^{3})

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, ...) ^[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

Algorithms

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

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 ^[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:

 $0.1_{10}=0.0(0011)_{2}$

Repeating fractions :^[37]

0.(567) = 567/999 = 189/333 = 63/111
0.(0011) = 0011 / 1111 =(in decimal) 3/15 = 1/5

Graphical conversion

Code

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

0.\overbrace {b...b} ^{t}(\overbrace {b...b} ^{p})=0.\overbrace {b...b} ^{t}+0.\overbrace {0...0} ^{t}(\overbrace {b...b} ^{p})

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

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

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

0.\overbrace {0...0} ^{t}(\overbrace {b...b} ^{p})={\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 $a=0.\overbrace {0...0} ^{t}\overbrace {b...b} ^{p}$
the value of $r={\frac {1}{2^{p}}}$ so $1-r={\frac {2^{p}-1}{2^{p}}}$

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

Full formula is now:

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

Examples :

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

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

code examples

bc

Conversion from decimal ratio to binary^[40] 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^[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

itoa function ^[42]

/* 
 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 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

GMP library^[44]

/*

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:^[45]

--  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 ^[46]

How to use numbers in computer programs ?

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

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:^[48]

// 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 ineger 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

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." ^[49]

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

Decimal

Binary

Numbers

In C one can use :

bitwise operators ^[50]

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 :^[51]

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

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. Examplle : 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

Inverse golden mean

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 :

 $562949953421312=1+2^{49}$

complex

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

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 Haberman^[58]
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. " ^[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

height of a rational number in lowest term

Thomae function = 1/q

  "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 $\mathbb {Q}$ :

$H(p/q)=\max\{|p|,|q|\}$ for the (multiplicative) height of a rational number^[62] also called naive height
$h(p/q)=\log H(p/q)$ 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

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

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

Fractals/Mathematics/Numbers