# Fractals/Mathematics/binary

Binary fraction in the theory of discrete dynamical systems ( number conversions and binary expansion of the decimal fraction )

# Notation
## general math notation

A binary fraction is a sum of negative powers of two
    $b_{0}.b_{1}b_{2}b_{3}\dots .=\sum _{i=0}^{\infty }b_{i}2^{-i}=b_{0}2^{0}+b_{1}2^{-1}+b_{2}2^{-2}+b_{3}2^{-3}+\dots$
    $b_{i}2^{-i}={\frac {b_{i}}{2^{i}}}$
Subscript number denotes number base ( radix of the positional numeral system)
    $0.1_{2}=0.5_{10}$
    $0.1_{10}=(0.0{\overline {0011}})_{2}$
Sometimes round brackets are used for greater clarity:
    $(0.1)_{10}=0.1_{10}$
Infinitely repeating part of binary expansion denoted by
* round brackets
* overline
infinite sequence is denoted by ellipsis ( = 3 dots)
    $0.10(00101010)=0.10{\overline {00101010}}=0.10001010100010101000101010001010100010101000101010\dots$
exponent ( superscript) with or without brackets:
* denotes how many times the series repeats
* will be used to indicate symbols (or groups of symbols) that are to be repeated a finite number of times
    $0.11(11101010)=0.11(11(10)^{3})$
    $0.((001)^{88}010)_{2}=0.(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010)$
    $0.(01^{3})=0.(0111)$
The leading zero is sometimes omitted
    $.1=0.1$
Trailing zeros to the right of a radix point (a small dot .), do not affect the value of a number and here will be omitted:
    $0.1(0)=0.1$

## Special notation

Special notation used in programs:
* program Mandel uses: 0p01 which in general math notation is 0.0(01). Generally apb = 0.a(b)
* program Book program uses: .0(01) which in general math notation is 0.0(01). Generally .a(b) = 0.a(b)

## preperiod and period

Preperiod is used in 2 meanings :
* T = preperiod of critical point $z_{cr}=0$
* t = preperiod of critical value $z_{cv}=c$
Note that :
Period p is the same for critical value and critical point
Preperiod:
* preperiod of critical value
** Wolf Jung uses : "... the usual convention is to use the preperiod of the critical value. This has the advantage, that the angles of the critical value have the same preperiod under doubling as the point, and the same angles are found in the parameter plane."
* preperiod of critical point
** Pastor uses preperiod of critical point : "all the Misiurewicz points are given with one unit more in their preperiods, therefore this $M_{2,1}$ is given as $M_{3,1}$"
** Demidov
** Claude Heiland-Allen

# Key words

* period
** smallest length of periodic part of binary expansion ( binary sequence )
** length of the orbit of the angle (decimal ratio) under doubling map = period under doubling map
* preperiod
** smallest length of preperiodic part of binary expansion ( binary sequence )

# Types of binary expansions

Binary expansion can be :
* finite - decimal ratio number with even denominator which is a power of 2. Note that it has also equal infinite preperiodic representation
* infinite
** periodic : preperiod = 0, period > 0, rational number with odd denominator which is not a power of 2
** preperiodic ( = eventually periodic) : preperiod > 0, period > 0, rational number with even denominator
** aperiodic : preperiod = 0, period = 0 ( or infinity), irrational number
If the decimal number is rational number check it's form. Is it is in the lowest terms ( irreducible = reduced = numerator and denominator are coprime)?

## How to find expansion type
# Description
## finite

A decimal ratio m/n has a finite binary representation if and only if it can be written as a fraction whose denominator n is a power of 2 ( also even):
${\frac {m}{n}}={\frac {m}{2^{t}}}$
for some integer m and some non-negative integer t."
In other words :
* fractions in binary arithmetic terminate only if 2 is the only prime factor in the denominator = dyadic rational.
* The dyadic rationals are precisely those numbers possessing finite binary expansions
It is because of binary fraction construction:
    $r=b_{0}.b_{1}b_{2}b_{3}\dots .=\sum _{i=0}^{\infty }{\frac {b_{i}}{2^{i}}}={\frac {b_{0}}{2^{0}}}+{\frac {b_{1}}{2^{1}}}+{\frac {b_{2}}{2^{2}}}+{\frac {b_{3}}{2^{3}}}+\dots$
Binary expansion of the unitary fraction:
$r={\frac {1}{2^{t}}}=0.\overbrace {b...b} ^{t-1}1=0.0^{t-1}1$
Binary expansion of unitary fractions 1/(2^t) :
| t | ${\frac {1}{2^{t}}}$ | binary expansion = $0.0^{t-1}1$ |
| 0 | 1/1 | 1.0 |
| 1 | 1/2 | 0.1 |
| 2 | 1/4 | 0.01 |
| 3 | 1/8 | 0.001 |
| 4 | 1/16 | 0.0001 |
| 5 | 1/32 | 0.00001 |
| 6 | 1/64 | 0.000001 |
| 7 | 1/128 | 0.0000001 |
| 8 | 1/256 | 0.00000001 |
| 9 | 1/512 | 0.000000001 |
| 10 | 1/1024 | 0.0000000001 |
| 34 | 1/17179869184 | 0.0000000000000000000000000000000001 |
This finite binary expansion has second equal representation: infinite and preperiodic ! Because this 2 representations have different preperiod and period then in the theory of discrete dynamical systems is better to use infinite version.

## infinite

* periodic : preperiod = 0, period > 0
* preperiodic : preperiod > 0, period > 0
* aperiodic : preperiod = 0, period = 0

### periodic

Here decimal number is a rational number with odd denominator.
There are 2 possible types ( complete classification):
* denominator is a odd prime (prime number other then 2)
* denominator is an odd composite number
There is special case which is not exclusive from th.e other two cases : denominator is integer one less than a power of two:
    $r={\frac {1}{2^{p}-1}}=0.((0)^{p-1}1)$
So binary fraction has:
* period = p
* preperiod = 0
Binary fraction 1/(2^p-1) has the same form in binary expansion as fraction 1/(2^p), but repeating.
Example 1/(2^5)=0.00001 and 1/(2^5-1)=0.(00001)
Binary expansion of unitary fractions 1/(2^p -1). Prime denominators are green, composite black :
| p | ${\frac {1}{2^{p}-1}}$ | factors(denominator) | binary expansion = $0.((0)^{p-1}1)$ |
| 2 | 1/3 | 3 | 0.(01) |
| 3 | 1/7 | 7 | 0.(001) |
| 4 | 1/15 | 3*5 | 0.(0001) |
| 5 | 1/31 | 31 | 0.(00001) |
| 6 | 1/63 | 3*3*7 | 0.(000001) |
| 7 | 1/127 | 127 | 0.(0000001) |
| 8 | 1/255 | 3*5*17 | 0.(00000001) |
| 9 | 1/511 | 7*73 | 0.(000000001) |
| 10 | 1/1023 | 3*11*31 | 0.(0000000001) |
| 34 | 1/17179869183 | 3*43691*131071 | 0.(0000000000000000000000000000000001) |

#### denominator is an odd prime

Rational number with odd prime denominator ( prime number other then 2 )
Period of binary expansion of reduced rational fraction m/n is equal to the multiplicative order of 2 modulo n:
$k=Period_{2}(m/n)={ord}_{n}(2)$
The longest possible period k is:
$k=n-1$
when 2 is a primitive root of denominator n :
Im Maxima CAS one can use :
* ifactors(n)
* zn_order(2,n)
* zn_primroot_p(2,n)
unitary fractions with prime number as an denominator. Longest possible period is red :
| ${\frac {m}{n}}$ | bin expansion | $k={period}_{2}$ |
| 1/3 | 0.(01) | 2 |
| 1/5 | 0.(0011) | 4 |
| 1/7 | 0.(001) | 3 |
| 1/11 | 0.(0001011101) | 10 |
| 1/13 | 0.(000100111011) | 12 |
| 1/17 | 0.(00001111) | 8 |
| 1/23 | 0.(00001011001) | 11 |
| 1/29 | 0.(0000100011010011110111001011) | 28 |
| 1/31 | 0.(00001) | 5 |
| 1/37 | 0.(000001101110101100111110010001010011) | 36 |
| 1/41 | 0.(00000110001111100111) | 20 |
| 1/43 | 0.(00000101111101) | 14 |
| 1/47 | 0.(00000101011100100110001) | 23 |
| 1/53 | 0.(0000010011010100100001110011111011001010110111100011) | 52 |
| 1/59 | 0.(0000010001010110110001111001011111011101010010011100001101) | 58 |
| 1/61 | 0.(000001000011001001011100010100111110111100110110100011101011) | 60 |
| 1/67 | 0.(000000111101001000100110001101010111111000010110111011001110010101) | 66 |
| 1/71 | 0.(00000011100110110000101011010001001) | 35 |
| 1/73 | 0.(000000111) | 9 |
| 1/79 | 0.(000000110011110110010001110100101010001) | 39 |
| 1/83 | 0.(000000110001010110010111001000011110110101111110011101010011010 0011011110000100101) | 82 |
| 1/89 | 0.(00000010111) | 11 |
| 1/97 | 0.(000000101010001110100000111111010101110001011111) | 48 |
| 1/9949 | 0. | 9948 |
1/9949 computed with the knowledgedoor calculator: convert_a_ratio_of_integers ( it allows for computations up to 100000 fractional digits in the new number):
Nonunitary fraction ( numerator is greater than 1 ) have :
* the same period k ( length of periodic part )
* if $k=\phi (n)$
** then periodic part cyclically shifted with respect to the corresponding unitary fraction
** else there are $\phi (n)/k$ different cycles ( periodic parts) all of length k
where
* $\phi$ is the Euler totient function. In Maxima CAS it is totient(n)

#### denominator is an odd composite number

Denominator is a composite number :
| ${\frac {m}{n}}$ | factors(n) | bin expansion | ${period}_{2}$ |
| 1/9 | 3*3 | 0.(000111) | 6 |
| 1/15 | 3*5 | 0.(0001) | 4 |
| 1/21 | 3*7 | 0.(000011) | 6 |
| 1/33 | 3*11 | 0.(0000011111) | 10 |
| 1/39 | 3*13 | 0.(000001101001) | 12 |
| 1/81 | 3^4 | 0.(000000110010100100010110000111111001101011011101001111) | 54 |
| 1/267 | 3*89 | 0.(0000000011110101011101) | 22 |
| 1/4369 | 17*257 | 0.(0000000000001111) | 16 |
period of binary expansion of ${\frac {m}{n}}$= period of the angle ${\frac {m}{n}}$ under doubling map
Hard case: 1/99007599 = 1 / (3*33002533), written as a binary fraction, have a period of nearly 50 million 0’s and 1’s
Tests:
* knowledgedoor calculator - "There are more than 100000 fractional digits in the new number. We are sorry, but we had to abort the calculation to control the loading on our server."
* In Maxima CAS : zn_order(2,99007599) = 33002532

### preperiodic

Here denominator n of fraction r
    $r={\frac {m}{n}}$
is even:
    $n=2^{t}q$
and q is odd.

#### Cases by preperiod and period

$\{t,p\}$ where t is preperiod and p period
* $\{1,1\}=0.0(1)=1/2=0.5$
* $\{1,2\}$
** $0.0(01)=1/6=2/12=0.1666666666666667$
** $0.0(10)=1/3=4/12=0.3333333333333333$
** $0.1(01)=2/3=8/12=0.6666666666666667$
** $0.1(10)=5/6=10/12=0.8333333333333333$
* $\{2,1\}$
** $0.00(1)=1/4=0.25$
** $0.10(1)=3/4=0.75$
* $\{2,2\}$
** $0.00(01)=1/12=0.0833333333333333$
** $0.01(10)=5/12=0.4166666666666667$
** $0.10(01)=7/12=0.5833333333333333$
* $\{4,4\}$
** $0.0110(1001)=99/240=0.4125$
** $0.1001(0110)=141/240=0.5875$
* $\{5,2\}$
** $0.01101(10)=41/96=0,42708333333333333$

#### Cases by q type

* q = 1, dyadic fractions. It is equal infinite representation of finite binary fraction. See black rows int the table below
* q is a prime number, See green rows int the table below
** q = 2^p - 1 then period = p
** if q is not integer one less the a power of two then period p = ( minimal length of repeating part) $p=\phi (q)$
* q is a composite number, period p "is the same as for the denominator q. It is either Phi(q) or a strict divisor of Phi(q) . You can determine it numerically when you double 1/q again and again until you get 1/q." ( Wolf Jung ). See red rows int the table below
Binary expansion of unitary fractions with even denominator 1/(2n) :
| k | ${\frac {1}{2k}}$ | factors(2k) = q*2^t | infinite binary expansion $=0.\overbrace {b...b} ^{t}(\overbrace {b...b} ^{p})$ | preperiod,period |
| 1 | 1/2 | 2 | 0.0(1) | 1,1 |
| 2 | 1/4 | 2^2 | 0.00(1) | 2,1 |
| 3 | 1/6 | 2*3 | 0.0(01) | 1,2 |
| 4 | 1/8 | 2^3 | 0.000(1) | 3,1 |
| 5 | 1/10 | 2*5 | 0.0(0011) | 1,4 |
| 6 | 1/12 | 2*2*3 | 0.00(01) | 2,2 |
| 7 | 1/14 | 2*7 | 0.0(001) | 1,3 |
| 8 | 1/16 | 2^4 | 0.0000(1) | |
| 9 | 1/18 | 2*9 | 0.0(000111) | 1,6 |
| 10 | 1/20 | 2*2*5 | 0.00(0011) | 2,4 |
| 11 | 1/22 | 2*11 | 0.0(0001011101) | 1,10 |
| 15 | 1/30 | 2*3*5 | 0.0(0001) | 1,4 |
| 21 | 1/42 | 2*3*7 | 0.0(000011) | 1,6 |
| 27 | 1/54 | 2*3*9 | 0.0(000010010111101101) | 1,18 |
| 33 | 1/66 | 2*3*11 | 0.0(0000011111) | 1,10 |
| 54 | 1/108 | 2*2*3*9 | 0.00(000010010111101101) | 2,18 |
| 66 | 1/132 | 2*2*3*11 | 0.00(0000011111) | 2,10 |

##### q is an integer one less than the power of two

If q is an integer one less than the power of two ( even):
    $q=2^{p}-1$
then fraction r has a form:
    $r={\frac {m}{n}}={\frac {m}{2^{t}(2^{p}-1)}}=0.\overbrace {b...b} ^{t}(\overbrace {b...b} ^{p})$
has:
* period ( minimal length of repeating part) $p$
* preperiod ( length of non-repeating part) = t
Examples:
    ${\frac {1}{6}}={\frac {1}{2*3}}={\frac {1}{2(2^{2}-1)}}={0.0(01)}_{2}$
    ${\frac {7}{24}}={\frac {7}{2^{3}*3}}={\frac {7}{2^{3}(2^{2}-1)}}={0.010(01)}_{2}$
    ${\frac {9}{56}}={\frac {9}{2^{3}*7}}={\frac {9}{2^{3}(2^{3}-1)}}={0.001(010)}_{2}$
How to check if q is an integer one less than the power of two:
    $q=2^{k}-1$
    $q+1=2^{k}$
    $k={log}_{2}(q+1)$
In Maxima cas one can use function
If k is near integer ( fractional part is 0 or near 0)
* then $q=2^{k}-1$ and use above method
* else use below method

##### q is a prime number

If $q\neq 2^{k}-1$then fraction r
    $r={\frac {m}{n}}={\frac {m}{2^{t}q}}=0.\overbrace {b...b} ^{t}(\overbrace {b...b} ^{k})$
where :
* q is odd
* $\phi$ is the Euler totient function. In Maxima CAS it is totient(n)
has:
* period ( minimal length of repeating part) $k=\phi (q)$
* preperiod ( length of non-repeating part) = t
Examples:
    ${\frac {9}{20}}={\frac {9}{2^{2}5}}={0.01(1100)}_{2}$ here period $=\phi (5)=4$ and preperiod t = 1
    ${0.1416}_{10}={\frac {1416}{10000}}={\frac {117}{1250}}={\frac {117}{2(5^{4})}}={0.0(00101111111011000101011011010101110011111010101011001101100111101000001111100100001001011010111011100110001100011111100010100000100100000010110111100000000011010001101101110001011101011000111000100001100101100101001010111101001111000011011000010001001101000000010011101010010010101000110000010101010011001001100001011111000001101111011010010100010001100111001110000001110101111101101111110100100001111111110010111001001000111010001010011100011101111001101001101011010100001011000011110010011110111011)}_{2}$
here:
* period $=\phi (5^{4})=500$
* preperiod = 1

### aperiodic

If expansion is:
* infinite
* aperiodic ( period = 0 or infinity )
then it is an irrational number.
Irrational number features (not a complete classification) :
* normality - normal number in wikipedia / non-normal number ( sequence)
* linearizability: not linearizable / linearizable, check Brjuno condition
* Transcendental(non-algebraic) /algebraic
Applications
* internal angle

#### normal

Random sequences
* pseudorandom
* true random
Examples from Random Number Generator:
* 10 bytes: 10001000111010111111001111010111101000110001010110010010011111100101110101011001
* 15 bytes: 100110100001100101010100001010100110110000100111001100110001101100101111100001001111101001011011101011110100011011000000
* 20 bytes: 1010100101110110001100111101101001001100001011111011001010000100100110010000001010110000101100100111111111011000001001110110000110111011100001101110001000101110
Binary Champernowne constant
C2 = 110111001011101111000100110101011110011011110111110000100011001010011101001010110110101111

#### non-normal

Binary Liouville's constant (binary Liouville number):
$L=\sum _{n=1}^{\infty }2^{-n!}={\frac {1}{2^{1!}}}+{\frac {1}{2^{2!}}}+\cdots +{\frac {1}{2^{n!}}}+\cdots ={\frac {1}{2}}+{\frac {1}{2^{2}}}+{\frac {1}{2^{6}}}+\cdots +{\frac {1}{2^{n!}}}+\cdots =0.11000100000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001\ldots$
where:
* ! denotes factorial
* $\sum$ denotes sum of infinite series
So binary digits on position:
$1,2,6,24,120,720,5040,40320,362880,39916800,479001600\ldots$
are 1, others are 0.
the Thue–Morse sequence = binary expansion of Prouhet–Thue–Morse constant:
    $\tau =\sum _{i=0}^{\infty }{\frac {t_{i}}{2^{i+1}}}={0.01101001100101101001011001101001\ldots }_{2}$
where:
* $t_{i}$ is the ith element of the Prouhet–Thue–Morse sequence
It obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far.
Define a sequence of strings of 0 and 1 as follows:
    $t_{0}=0$
    $t_{n+1}=t_{n}{\overline {t_{n}}}$
where:
* ${\overline {t}}$ means change all the 0’s in x to 1’s and vice-versa
First 5 sequences
    $t_{0}=0$
    $t_{1}=01$
    $t_{2}=0110$
    $t_{3}=01101001$
    $t_{4}=0110100110010110$
The Rabbit constant (The limiting Rabbit Sequence) given by the continued fraction
$R=[0,2^{F_{0}},2^{F_{1}},2^{F_{2}},2^{F_{3}},...]=0.1011010110110...$
where:
* $F_{n}$ are Fibonacci numbers with $F_{0}$ taken as 0

# Multiple representations

The representations are:
* equal in general meaning, because give the same decimal fraction
* nonequal here, because have different period/preperiod of binary fraction, so describe different dynamical system

## Periodic

periodic angles have multiple possible ways of writing down, one of which is special in having the smallest period
$1/3=.0101010101010101...=.(01)=.(0101)=.(010101)=...$
So "period" really means minimal possible period
Orbit of 1/3 under doubling map is periodic {1/3 , 2/3 }

## preperiodic

preperiodic angles have multiple possible ways of writing down ( infinite forms), one of which is special in having the smallest preperiod:

## finite

* every nonzero terminating binary number ( = rational number with odd denominator which is a power of 2 ) has two equal representations
* The binary expansions of dyadic rationals are not unique; there is one finite and one infinite representation of each dyadic rational other than 0 (ignoring terminal 0s). For example, 0.11$_2$ = 0.10111...$_2$, giving two different representations for 3/4. The dyadic rationals are the only numbers whose binary expansions are not unique.
$0.0^{t-1}1_{2}=0.0^{t}(1)_{2}$
Binary expansion of unitary fractions 1/(2^p) :
| t | decimal ${\frac {1}{2^{t}}}$ | binary finite = $0.0^{t-1}1$ | binary infinite = $0.0^{t}(1)$ |
| 0 | 1/1 | 1.0 | 0.(1) |
| 1 | 1/2 | 0.1 | 0.0(1) |
| 2 | 1/4 | 0.01 | 0.00(1) |
| 3 | 1/8 | 0.001 | 0.000(1) |
| 4 | 1/16 | 0.0001 | 0.0000(1) |
| 5 | 1/32 | 0.00001 | 0.00000(1) |
| 6 | 1/64 | 0.000001 | 0.000000(1) |
| 7 | 1/128 | 0.0000001 | 0.0000000(1) |
| 8 | 1/256 | 0.00000001 | 0.00000000(1) |
| 9 | 1/512 | 0.000000001 | 0.000000000(1) |
| 10 | 1/1024 | 0.0000000001 | 0.0000000000(1) |
| 34 | 1/17179869184 | 0.0000000000000000000000000000000001 | 0.0000000000000000000000000000000000(1) |
Examples:
* "The angle 1/4 or 01 has preperiod = 2 and period = 1." program from program Mandel. Here finite version is used

# Relation between binary numbers and the dynamical systems

Maps
* binary fraction , doubling map = left shift = Bernoulli shift
* complex number, complex quadratic map

## Doubling map

"The map $\lambda :\mathbb {R} /\mathbb {Z} \to J_{c}$ is called the Caratheodory semiconjugacy, with the associated identity $\lambda (2*t)=f_{c}(\lambda (t))$ in the degree 2 case. This identity allows us to easily track forward iteration of external rays and their landing points in $J_{c}$ by doubling the angle of their associated external rays modulo 1." Mary Wilkerson
Orbits of fraction under doubling map for :
* decimal rational fractions
** fractions with odd denominators are periodic
** fractions with even denominators are preperiodic
* decimal irrational fractions are non periodic and dense
the relationship between external angles and binary decomposition rendering for quadratic polynomials
* Removing the first bit is the same as doubling mod 1
when tracing external ray and passing dwell band ( boundary of escape time level set):
* inwards ( toward Julia/Mandelbrot set): add bit at the end
* outwards ( toward point at infinity): add the bit at the beginning
of binary expansion of external angle
* collecting bits outwards
* Binary decomposition of unrolled circle plane
* binary decomposition of dynamic plane for f(z) = z^2

## external angles

There are two kinds of rational angles ( decimal fractions):
* When the denominator is odd, the sequence of binary digits will be periodic, and the angle is periodic under doubling. The dynamic ray with this angle lands at a periodic point of the Julia set. The parameter ray lands at the root of a hyperbolic component.
* When the denominator is even, the sequence of binary digits will be preperiodic, and the angle is preperiodic under doubling. The dynamic ray lands at a preperiodic point of the Julia set, and the parameter ray lands at a Misiurewicz point.
Relation between point and external angle ?
Using binary decomposition:
* start with an unknown location and finds the external angles from the image
* start with angles and find the location that matches
The external arguments of the rays landing at z = −0.15255 + 1.03294i are:
$(\theta _{20}^{-},\theta _{20}^{+})=(0.{\overline {00110011001100110100}},0.{\overline {00110011001101000011}})$
where:
$\theta _{20}^{-}=0.{\overline {00110011001100110100}}_{2}=0.{\overline {20000095367522590181913549340772}}_{10}={\frac {209716}{1048575}}={\frac {209716}{2^{20}-1}}$

### collecting bits

Collecting bits of binary expansion of external angle when tracing externla rays on the binary decomposition of exterior of the Mandelbrot/Julia set
See also
* binary decomposition
** parameter plane
** dynamic plane
* reading bits from binary decomposition = collecting bits of external angle's binary expansion

### patterns

Patterns are sculpted in the intricate shape of the boundary the Mandelbrot set when zooming ( especially deep zooming). Zooming are reflected in the complex dynamics, in particular in the binary expansions of the pairs of external rays landing on the cusp of each baby Mandelbrot set copy at the centre of each phase.

# Conversions

* Convert binary fraction to decimal ratio
* convert decimal fraction to binary
* knowledgedoor calculators

## Can all decimal fractions be converted exactly to binary?

Not all. Only those for which denominator is a power of 2 ( finite ) have exact decimal representation. "In every other case, there will be an error in the representation. The error's magnitude depends on the number of digits used to represent it."

## decimal to binary
# See also

* Bit numbering in wikipedia
* Bit_manipulation in wikipedia
* ENT A Pseudorandom Number Sequence Test Program in C by John Walker
* Continued fraction representation of real number

# References

* ↑ Number of Decimal Digits In a Binary Fraction By Rick Regan
* ↑ Subscript_and_superscript in wikipedia
* ↑ Radix or base in wikipedia
* ↑ 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
* ↑ math.stackexchange question: is-it-an-abuse-of-notation-to-omit-the-leading-zero-in-a-decimal-less-than-1
* ↑ Trailing_zero in wikipedia
* ↑ G.Pastor, M. Romera, G. Alvarez, J. Nunez, D. Arroyo, F. Montoya, "Operating with External Arguments of Douady and Hubbard", Discrete Dynamics in Nature and Society, vol. 2007, Article ID 045920, 17 pages, 2007
* ↑ libretexts : The Mandelbrot and Julia sets Anatomy (Demidov)
* ↑ ibiblio
* ↑ irreducible in wikipedia
* ↑ Coprime integers in wikipedia
* ↑ math stackexchange question: number-with-finite-binary-representation-and-infinite-decimal-representation
* ↑ wikipedia: Dyadic rational
* ↑ Binary expasnion by John McIntosh
* ↑ Primitive root modulo n in wikipedia
* ↑ Number Theory in Science and Communication With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity by Manfred R. Schroeder
* ↑ Number Theory in Science and Communication With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity by Manfred R. Schroeder
* ↑ oeis : sequence A119919
* ↑ Chaotic Dynamics Fractals, Tilings, and Substitutions by Geoffrey R. Goodson,, Cambridge University Press 2017 , page 185
* ↑ scholarpedia : Linearization
* ↑ wikipedia: Random_sequence
* ↑ wikipedia: Pseudorandom_binary_sequence
* ↑ dice-o-matic : an automatic dice roller
* ↑ Champernowne constant in wikipedia
* ↑ oeis wik Champernowne_constant
* ↑ Liouville_number in wikipedia
* ↑ The Ubiquitous Thue-Morse Sequence by Jeffrey Shallit
* ↑ fractalforums.org: binary-fraction
* ↑ wikipedia: 0.999...
* ↑ doubling map by Mark McClure
* ↑ Václav Kučera: The Bernoulli shift as a basic chaotic dynamical system
* ↑ fractalforums.com : binary-decomposition-and-external-angles
* ↑ mandel - software for real and complex dynamics by Wolf Jung
* ↑ fractalforums.org : reading-off-external-angles-from-binary-decomposition-images by Claude Heiland-Allen
* ↑ deviantart fractalmonster journal : Hunting-for-Period-3
* ↑ deviantart fractalmonster journal : External-Rays-to-the-Mandelbrot-set
* ↑ A Method to Solve the Limitations in Drawing External Rays of the Mandelbrot Set M. Romera, G. Pastor, A. B. Orue, A. Martin, M.-F. Danca, and F. Montoya
* ↑ binary-fraction calculator by Davide Borchia
* Binary Numbers by Jamie King on YT