Fractals/Mathematics/Numbers
"Many questions concerning (discrete) dynamical systems are of a number theoretic or combinatorial nature." Christian Krattenthaler
Intro
[edit | edit source]- 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
[edit | edit source]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 ) :
- reducible form
- in explicit normalized form ( only when denominator is odd ):[4]
- 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)
- ratio = fraction ( Finite continued fraction ) = rational number ( the irrationality measure of any rational number is 1)[3]
- decimal floating point number [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)
- 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
[edit | edit source]
way to represent real numbers using a finite number of bits
[edit | edit source]- IEEE floating point
- fixed point
- unums (universal numbers) by John Gustafson
expansion/representation
[edit | edit source]- finite = terminating
- infinite = non-terminating
- periodic = infite repeating
- preperiodic = eventually periodic
- non-periodic: binary numerals which neither terminate nor recur represent irrational numbers
base
[edit | edit source]radix or base of a positional numeral system[14]
- 2 ( binary number)
- 8 ( octal number)
- 10 ( decimal umber)
- 16 ( hexadecimal)
form/notation
[edit | edit source]Notation[15]
- 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 measurment error: mean ± range [16], for example: 72.20 ± 0.02
- 1.23
- infinite sequence
- ratio of integers
- in lowest terms ( irreducible form ) :
- reducible form
- in explicit normalized form ( only when denominator is odd ):[17], for example :
- The explicit normalized form of formula for denominator of angle :
- continued fraction :
- scientific (exponential) form or notation: [18]
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
[edit | edit source]brackets with exponent ( superscript) denotes how many times the series repeats [19]
Trailing zeros
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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 :
Algorithms
[edit | edit source]- Find
- convert a number with a repeating fractional part
Reducing Fractions to Lowest Terms
[edit | edit source]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:
convert decimal fraction to binary
[edit | edit source]"... 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:
Repeating fractions :[37]
0.(567) = 567/999 = 189/333 = 63/111 0.(0011) = 0011 / 1111 =(in decimal) 3/15 = 1/5
Code
Convert binary fraction to decimal ratio
[edit | edit source]Geometric series
[edit | edit source](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 :[38]
For the infinite periodic binary fraction with empty or filled with zeros preperiodic part this formula is[39]
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
[edit | edit source]bc
[edit | edit source]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
[edit | edit source]- itoa
- snprintf[41]
- Binary integer constant
- Macro BOOST_BINARY
- gmp
- mandelbrot-symbolics lbrary by Claude Heiland-Allen
- Conversion numbers to binary representation by Wojciech Muła
itoa
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]# 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 ?
[edit | edit source]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
[edit | edit source]- types
- limits and overflow
Limit
[edit | edit source]/*
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
[edit | edit source]precision
[edit | edit source]- 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
[edit | edit source]"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
[edit | edit source]Binary
[edit | edit source]Numbers
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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
[edit | edit source]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 :
complex
[edit | edit source]- 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
[edit | edit source]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
[edit | edit source]- dumpfp: A Tool to Inspect Floating-Point Numbers by Joshua Haberman[58]
- float.exposed - and blog by Bartosz Ciechanowski
More
[edit | edit source]Cardinality
[edit | edit source]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
[edit | edit source]"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 :
- for the (multiplicative) height of a rational number[62] also called naive height
- 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
[edit | edit source]- paritition function : "partition numbers behave like fractals, possessing an infinitely-repeating structure" [64]
Random number
[edit | edit source]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
[edit | edit source]- generalisation : scalar / vector / tensor
- fields : scalar , vector, tensor
References
[edit | edit source]- ↑ exploring binary: nine-ways-to-display-a-floating-point-number
- ↑ wikipedia : Number base
- ↑ math.stackexchange question: are-there-real-numbers-that-are-neither-rational-nor-irrational
- ↑ HOW TO WORK WITH ONE-DI MENSIONAL QUADRATIC MAPS G. Pastor , M. Romera, G. Álvarez, and F. Montoya
- ↑ What Every Programmer Should Know About Floating-Point Arithmetic
- ↑ Stackoverflow : Why Are Floating Point Numbers Inaccurate?
- ↑ home school math : The fascinating irrational numbers
- ↑ Tutorial: Floating-Point Binary by Kip Irvine
- ↑ Dual Numbers & Automatic Differentiation
- ↑ videos: Imaginary Numbers are Real from Welch Labs
- ↑ math stackexchange question: is-there-a-third-dimension-of-numbers
- ↑ Beating Floating Point at its Own Game: Posit Arithmetic by John L. Gustafson , Isaac Yonemoto
- ↑ fractalforums.org : posits
- ↑ wikipedia: Radix
- ↑ Survey of Floating-Point Formats Robert Munafo's home pages on AWS © 1996-2022 Robert P. Munafo.
- ↑ Physics 132 Lab Manual: how-to-write-numbers-significant-figures by Brokk Toggerson and Aidan Philbin
- ↑ HOW TO WORK WITH ONE-DI MENSIONAL QUADRATIC MAPS G. Pastor , M. Romera, G. Álvarez, and F. Montoya
- ↑ calculatorsoup: scientific-notation-converter
- ↑ 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
- ↑ Converting fractions to strings of digits by Claude Heiland-Allen
- ↑ Rational Numbers From Their Decimal Expansion by William Stein
- ↑ basic-mathematics : converting-repeating-decimals-to-fractions
- ↑ Recurring decimal of a rational number by Yurii Lahodiuk
- ↑ quora : How-do-you-convert-repeating-fractions-to-different-bases
- ↑ knowledgedoor calculators: convert_a_ratio_of_integers
- ↑ R.Knott : Fractions – Decimals Calculator
- ↑ Base Number - Decimal Number Conversion
- ↑ Decimal to Floating-Point Converter By Rick Regan
- ↑ wolframalpha binary to decimal conversion
- ↑ stackoverflow question : best-algorithm-to-find-a-repeating-pattern
- ↑ stackoverflow questions : method-to-find-repeated-pattern-in-string-apart-from-regex?noredirect=1&lq=1
- ↑ stackoverflow question: finding-a-repeated-pattern-in-a-string
- ↑ stackoverflow question: finding-a-pattern-in-a-binary-string
- ↑ jsfiddle by Jan Turoń
- ↑ Virginia Tech Online CS module
- ↑ Stackoverflow : How do you convert a fraction to binary ?
- ↑ Converting a repeating binary number to decimal (express as a series?)
- ↑ wikipedia :Geometric_series
- ↑ stackoverflow question: converting-a-repeating-binary-number-to-decimal-express-as-a-series
- ↑ math.stackexchange question: find-a-fraction-given-the-repeating-binary-expansions
- ↑ Where is the itoa function in Linux?
- ↑ itoa with GCC by Stuart
- ↑ gcc - Binary-constants
- ↑ Programowanie_w_systemie_UNIX: GMP in polish wikibooks
- ↑ Converting fractions to strings of digits by Claude Heiland-Allen
- ↑ stackoverflow : python int to binary
- ↑ What Every Computer Scientist Should Know About Floating-Point Arithmetic by DAVID GOLDBERG
- ↑ wikipedia : Bounds checking
- ↑ stackoverflow questions : check-if-a-number-is-rational-in-python
- ↑ Joe McCullough : bitwise operators
- ↑ Stackoverflow : How do you express binary literals in Python?
- ↑ John D Cook : best-rational-approximation
- ↑ DISCOVERING EXACTLY WHEN A RATIONAL IS A BEST APPROXIMATE OF AN IRRATIONAL By KARI LOCK
- ↑ ams : The Most Irrational Number
- ↑ complex beauties : math-calendar
- ↑ David Bau : complex function viewer
- ↑ Complex Numbers in VBA by Pfadintegral
- ↑ dumpfp: A Tool to Inspect Floating-Point Numbers by Joshua Haberman
- ↑ home school math : The fascinating irrational numbers
- ↑ stackoverflow questions : irrational-number-check-function
- ↑ Introduction to Heights by Brian Lawrence
- ↑ sagemath : Rational.global_height
- ↑ Height Functions by Michael Tepper
- ↑ Fractal Structure to Partition Function.
See also
[edit | edit source]- 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