# Fractals/Mathematics/binary

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

# Notation

A binary fraction is a sum of negative powers of two[1]

${\displaystyle b_{0}.b_{1}b_{2}b_{3}\dots .=\sum _{i=0}^{\infty }b_{i}2^{-i}=b_{0}2^{0}+b_{1}2^{-1}+b_{2}2^{-2}+b_{3}2^{-3}+\dots }$
${\displaystyle b_{i}2^{-i}={\frac {b_{i}}{2^{i}}}}$

Subscript[2] number denotes number base ( radix of the positional numeral system) [3]

${\displaystyle 0.1_{2}=0.5_{10}}$
${\displaystyle 0.1_{10}=(0.0{\overline {0011}})_{2}}$

Sometimes round brackets are used for greater clarity:

${\displaystyle (0.1)_{10}=0.1_{10}}$

Infinitely repeating part of binary expansion denoted by

• round brackets
• overline

infinite sequence is denoted by ellipsis ( = 3 dots)

${\displaystyle 0.10(00101010)=0.10{\overline {00101010}}=0.10001010100010101000101010001010100010101000101010\dots }$

exponent ( superscript) with or without brackets:

• denotes how many times the series repeats [4]
• will be used to indicate symbols (or groups of symbols) that are to be repeated a finite number of times

${\displaystyle 0.11(11101010)=0.11(11(10)^{3})}$
${\displaystyle 0.((001)^{88}010)_{2}=0.(001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001001010)}$
${\displaystyle 0.(01^{3})=0.(0111)}$

The leading zero is sometimes omitted [5]

${\displaystyle .1=0.1}$

Trailing zeros[6] to the right of a radix point (a small dot .), do not affect the value of a number and here will be omitted:

${\displaystyle 0.1(0)=0.1}$

Special notation used in programs:

# Key words

• period
• smallest length of periodic part of binary expansion ( binary sequence )
• lentgh 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 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[7] = reduced = numerator and denominator are coprime[8])?

## 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):

${\displaystyle {\frac {m}{n}}={\frac {m}{2^{t}}}}$

for some integer m and some non-negative integer t."[9]

In other words :

• fractions in binary arithmetic terminate only if 2 is the only prime factor in the denominator = dyadic rational.[10]
• The dyadic rationals are precisely those numbers possessing finite binary expansions

It is because of binary fraction construction:

${\displaystyle r=b_{0}.b_{1}b_{2}b_{3}\dots .=\sum _{i=0}^{\infty }{\frac {b_{i}}{2^{i}}}={\frac {b_{0}}{2^{0}}}+{\frac {b_{1}}{2^{1}}}+{\frac {b_{2}}{2^{2}}}+{\frac {b_{3}}{2^{3}}}+\dots }$

Binary expansion of the unitary fraction:

${\displaystyle r={\frac {1}{2^{t}}}=0.\overbrace {b...b} ^{t-1}1=0.0^{t-1}1}$

Binary expansion of unitary fractions 1/(2^t)
t ${\displaystyle {\frac {1}{2^{t}}}}$  binary expansion = ${\displaystyle 0.0^{t-1}1}$
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

Here decimal number is a rational number with odd denominator.

There are 2 possible types ( complete classification):

There is special case which is not exclusive from th.e other two cases : denominator is integer one less than a power of two:

${\displaystyle r={\frac {1}{2^{p}-1}}=0.((0)^{p-1}1)}$

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)[11]

Binary expansion of unitary fractions 1/(2^p -1). Prime denominators are green, composite black
p ${\displaystyle {\frac {1}{2^{p}-1}}}$  factors(denominator) binary expansion = ${\displaystyle 0.((0)^{p-1}1)}$
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:

${\displaystyle k=Period_{2}(m/n)={ord}_{n}(2)}$

The longest possible period k is:

${\displaystyle k=n-1}$

when 2 is a primitive root[12] 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
${\displaystyle {\frac {m}{n}}}$  bin expansion ${\displaystyle k={period}_{2}}$
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):

 1/9949 = 0.(0.000000000000011010010110010100100110010000110010100010010100000 000011000101100111011010011110111101111011000001010110000010111001 010000111100110101000010000011010101010000101010101101101011111001 000001101101111011000111111011101000000010110101001001011101100111 000011011011011011111001100010101001110100110111110000100111001101 101110001001111100011111001101100100010001100100110000110111010001 010100101101010000101110000000011110011101110011110100001111011010 011011101011001000011100100011111100100100111110011100110001111100 011011111010110001101100110010101010100010111110110100101010001011 000110100101111111011111111000110010111001010111100010011010001011 100111100001111001001111101101110010000100010000100010111001000011 110001101010101110111010111011111111100000101101011111100010100100 000011111111010000001111100010101010101001100100011001110011101111 010011001110100100011111111110111110001000001100100000010110000001 101010001101111111000010001111101011101110010100101001100011100101 000111000110000110101100111111011011010110111101010110110010101001 101110010010001011011101101001100001100001010111011111000111011001 000010101111110010111011011011010010000001001010111011011110100100 110011101111101101100100111001000110001111110000101010100000100000 101110101110100101100001110110110001100111110110011110101011110011 101011001011101111010110100001010111000100110001000100011100011111 000000011001000111010001101000011110000000001010101101000100010111 100010110100100001111100001000001010000010010000000110000100101001 001111110100010111101001011010000111000101101100010110101010110101 000110001010110100011110101001010101100101010000001001110001110010 001001001100101110110000001110111011001001001010101011000000100111 111011110101001101111111011100100110000000010100100101011100000101 111001000111011110110011101000010011010011000110010101100001100011 000000111000011001110010000101111001111100001011011100110100110100 111000001010111101100010010100011010101111000001100001100100101010 010001101100001011001001000100000101011011011110010111101000100101 011010011100011111110101000101110000011110001010000011000100110010 101101110101110001011001011100010001011010111000011111100010111110 011010010011110110100000010101001100111101100100110010100000101010 100111000110010011111000001001101110011111010110100111111100101001 111010101000101001000111100101011001001101011100110111010010111110 000110100011000111000011101000100111000011110101110010001110001000 111010100111011010000100100111100110011011000101010000010110111100 111100011100010101001000000001011000111011010101100001001000101010 100011110011100001010011010111101000001011000100000111111001100100 010011001110001010001001101010010111110111011001111110000010000001 010001100001000011101110010111111100010110001001111001001100011010 111111011111011110011100100100110001010001100111101001010011100001 100000100010110010011110001100100001001010101110010011011010100000 100111010100010011101111000110000011011010001100110110100100110111 000010100000001001101011001100100100000011001010100011100110010110 001001000100011111110001110010111101111001010111111100110000100000 001101110010101011110010000001110010011100111101011110001100111011 100001000010111001101011010011001001101000010100000111110010111110 101110000100100101111101000001110010110111010011110010110011001100 010011100101001101101011101011110110100011100111111111100010010110 111000110100111101000111001001011001011111100100001101011101010001 101001010010101100110011111001100101111100100111011100100010101101 100010000000101001111111100100110100111110110000100010101011111000 100111010111100110100001101010110101100000100001001001000100111010 001000000111100100001010001010011111000100100000100110011111100111 000101111001100001110101001000001110100100000101101000101001100001 111011101101110011101101101001110101010010000110111011110011111110 111100011110110011001101111100111110100000000100101111000000101100 111000000001000101001010100110000100011100000100101010000100100001 000000110101111011101100001010010100010111011100001110111100110010 100011111101011001101011000101111110011110000000111111011001101101 100100000100011001100110100100001000111011011100000110101101110100 001000000000001001111000010111101110010110010010111100110111100000 001001010000110110001111011100111001110001000100000010001000101011 110010110110011111000110001001111111110010000000001001000011101011 000101001001110001010111110010111000001000011111011100011000110101 001010010010010011101100100111111101011110100111010001110101101001 001010011101110101011101101000101100110100101110010010100101110011 111110000111110010001010000001011011011001011011011100101110001111 010011000001011001010101101011110101101110111011010110010101110101 010011110000010101000110010111111111101000111100011101111110100001 010011110001111110011111101010011000101100000110100111001110100010 110110100101101011101111001001010110001100110001101000101011001011 010101000000001100110000110011111110100010001000011110100111101100 001011111101110000101110100111111111111100101101001101011011001101 111001101011101101011111111100111010011000100101100001000010000100 111110101001111101000110101111000011001010111101111100101010101111 010101010010010100000110111110010010000100111000000100010111111101 001010110110100010011000111100100100100100000110011101010110001011 001000001111011000110010010001110110000011100000110010011011101110 011011001111001000101110101011010010101111010001111111100001100010 001100001011110000100101100100010100110111100011011100000011011011 000001100011001110000011100100000101001110010011001101010101011101 000001001011010101110100111001011010000000100000000111001101000110 101000011101100101110100011000011110000110110000010010001101111011 101111011101000110111100001110010101010001000101000100000000011111 010010100000011101011011111100000000101111110000011101010101010110 011011100110001100010000101100110001011011100000000001000001110111 110011011111101001111110010101110010000000111101110000010100010001 101011010110011100011010111000111001111001010011000000100100101001 000010101001001101010110010001101101110100100010010110011110011110 101000100000111000100110111101010000001101000100100100101101111110 110101000100100001011011001100010000010010011011000110111001110000 001111010101011111011111010001010001011010011110001001001110011000 001001100001010100001100010100110100010000101001011110101000111011 001110111011100011100000111111100110111000101110010111100001111111 110101010010111011101000011101001011011110000011110111110101111101 101111111001111011010110110000001011101000010110100101111000111010 010011101001010101001010111001110101001011100001010110101010011010 101111110110001110001101110110110011010001001111110001000100110110 110101010100111111011000000100001010110010000000100011011001111111 101011011010100011111010000110111000100001001100010111101100101100 111001101010011110011100111111000111100110001101111010000110000011 110100100011001011001011000111110101000010011101101011100101010000 111110011110011011010101101110010011110100110110111011111010100100 100001101000010111011010100101100011100000001010111010001111100001 110101111100111011001101010010001010001110100110100011101110100101 000111100000011101000001100101101100001001011111101010110011000010 011011001101011111010101011000111001101100000111110110010001100000 101001011000000011010110000101010111010110111000011010100110110010 100011001000101101000001111001011100111000111100010111011000111100 001010001101110001110111000101011000100101111011011000011001100100 111010101111101001000011000011100011101010110111111110100111000100 101010011110110111010101011100001100011110101100101000010111110100 111011111000000110011011101100110001110101110110010101101000001000 100110000001111101111110101110011110111100010001101000000011101001 110110000110110011100101000000100000100001100011011011001110101110 011000010110101100011110011111011101001101100001110011011110110101 010001101100100101011111011000101011101100010000111001111100100101 110011001001011011001000111101011111110110010100110011011011111100 110101011100011001101001110110111011100000001110001101000010000110 101000000011001111011111110010001101010100001101111110001101100011 000010100001110011000100011110111101000110010100101100110110010111 101011111000001101000001010001111011011010000010111110001101001000 101100001101001100110011101100011010110010010100010100001001011100 011000000000011101101001000111001011000010111000110110100110100000 011011110010100010101110010110101101010011001100000110011010000011 011000100011011101010010011101111111010110000000011011001011000001 001111011101010100000111011000101000011001011110010101001010011111 011110110110111011000101110111111000011011110101110101100000111011 011111011001100000011000111010000110011110001010110111110001011011 111010010111010110011110000100010010001100010010010110001010101101 111001000100001100000001000011100001001100110010000011000001011111 111011010000111111010011000111111110111010110101011001111011100011 111011010101111011011110111111001010000100010011110101101011101000 100011110001000011001101011100000010100110010100111010000001100001 111111000000100110010010011011111011100110011001011011110111000100 100011111001010010001011110111111111110110000111101000010001101001 101101000011001000011111110110101111001001110000100011000110001110 111011111101110111010100001101001001100000111001110110000000001101 111111110110111100010100111010110110001110101000001101000111110111 100000100011100111001010110101101101101100010011011000000010100001 011000101110001010010110110101100010001010100010010111010011001011 010001101101011010001100000001111000001101110101111110100100100110 100100100011010001110000101100111110100110101010010100001010010001 000100101001101010001010101100001111101010111001101000000000010111 000011100010000001011110101100001110000001100000010101100111010011 111001011000110001011101001001011010010100010000110110101001110011 001110010111010100110100101010111111110011001111001100000001011101 110111100001011000010011110100000010001111010001011)


Nonunitary fraction ( numerator is greater then 1 ) have :[13]

• the same period k ( length of periodic part )
• if ${\displaystyle k=\phi (n)}$
• then periodic part cyclically shifted with respect to the corresponding unitary fraction
• else there are ${\displaystyle \phi (n)/k}$  different cycles ( periodic parts) all of length k

where

• ${\displaystyle \phi }$  is the Euler totient function. In Maxima CAS it is totient(n)

#### denominator is an odd composite number

Denominator is a composite number
${\displaystyle {\frac {m}{n}}}$  factors(n) bin expansion ${\displaystyle {period}_{2}}$
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= period 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[14]

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

${\displaystyle r={\frac {m}{n}}}$

is even:

${\displaystyle n=2^{t}q}$

and q is odd.

#### Cases by preperiod and period

${\displaystyle \{t,p\}}$  where t is preperiod and p period

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

#### Cases by q type

• q = 1, then one uses 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) ${\displaystyle p=\phi (q)}$
• 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 ${\displaystyle {\frac {1}{2k}}}$  factors(2k) = q*2^t infinite binary expansion ${\displaystyle =0.\overbrace {b...b} ^{t}(\overbrace {b...b} ^{p})}$  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):

${\displaystyle q=2^{p}-1}$

then fraction r has a form:

${\displaystyle r={\frac {m}{n}}={\frac {m}{2^{t}(2^{p}-1)}}=0.\overbrace {b...b} ^{t}(\overbrace {b...b} ^{p})}$

has:

• period ( minimal length of repeating part) ${\displaystyle p}$
• preperiod ( length of non-repeating part) = t

Examples:

${\displaystyle {\frac {1}{6}}={\frac {1}{2*3}}={\frac {1}{2(2^{2}-1)}}={0.0(01)}_{2}}$
${\displaystyle {\frac {7}{24}}={\frac {7}{2^{3}*3}}={\frac {7}{2^{3}(2^{2}-1)}}={0.010(01)}_{2}}$ [15]
${\displaystyle {\frac {9}{56}}={\frac {9}{2^{3}*7}}={\frac {9}{2^{3}(2^{3}-1)}}={0.001(010)}_{2}}$

How to check if q is an integer one less than the power of two:

${\displaystyle q=2^{k}-1}$
${\displaystyle q+1=2^{k}}$
${\displaystyle k={log}_{2}(q+1)}$

In Maxima cas one can use function

   Give_k(q):=float(log(q+1)/log(2));


If k is near integer ( fractional part is 0 or near 0)

• then ${\displaystyle q=2^{k}-1}$  and use above method
• else use below method
##### q is a prime number

If ${\displaystyle q\neq 2^{k}-1}$ then fraction r[16]

${\displaystyle r={\frac {m}{n}}={\frac {m}{2^{t}q}}=0.\overbrace {b...b} ^{t}(\overbrace {b...b} ^{k})}$

where :

• q is odd
• ${\displaystyle \phi }$  is the Euler totient function. In Maxima CAS it is totient(n)

has:

• period ( minimal length of repeating part) ${\displaystyle k=\phi (q)}$
• preperiod ( length of non-repeating part) = t

Examples:

${\displaystyle {\frac {9}{20}}={\frac {9}{2^{2}5}}={0.01(1100)}_{2}}$  here period ${\displaystyle =\phi (5)=4}$  and preperiod t = 1
${\displaystyle {0.1416}_{10}={\frac {1416}{10000}}={\frac {117}{1250}}={\frac {117}{2(5^{4})}}={0.0(00101111111011000101011011010101110011111010101011001101100111101000001111100100001001011010111011100110001100011111100010100000100100000010110111100000000011010001101101110001011101011000111000100001100101100101001010111101001111000011011000010001001101000000010011101010010010101000110000010101010011001001100001011111000001101111011010010100010001100111001110000001110101111101101111110100100001111111110010111001001000111010001010011100011101111001101001101011010100001011000011110010011110111011)}_{2}}$

here:

• period ${\displaystyle =\phi (5^{4})=500}$
• 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:[17] not linearizable / linearizable, check Brjuno condition
• Transcendental(non-algebraic) /algebraic

Applications

#### normal

Random sequences[18]

Examples from Random Number Generator:

• 10 bytes: 10001000111010111111001111010111101000110001010110010010011111100101110101011001
• 15 bytes: 100110100001100101010100001010100110110000100111001100110001101100101111100001001111101001011011101011110100011011000000
• 20 bytes: 1010100101110110001100111101101001001100001011111011001010000100100110010000001010110000101100100111111111011000001001110110000110111011100001101110001000101110

Binary Champernowne constant[21] C2 = 110111001011101111000100110101011110011011110111110000100011001010011101001010110110101111[22]

#### non-normal

Binary Liouville's constant (binary Liouville number[23]):

${\displaystyle L=\sum _{n=1}^{\infty }2^{-n!}={\frac {1}{2^{1!}}}+{\frac {1}{2^{2!}}}+\cdots +{\frac {1}{2^{n!}}}+\cdots ={\frac {1}{2}}+{\frac {1}{2^{2}}}+{\frac {1}{2^{6}}}+\cdots +{\frac {1}{2^{n!}}}+\cdots =0.11000100000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001\ldots }$

where:

• ! denotes factorial
• ${\displaystyle \sum }$  denotes sum of infinite series

So binary digits on position:

${\displaystyle 1,2,6,24,120,720,5040,40320,362880,39916800,479001600\ldots }$

are 1, others are 0.

the Thue–Morse sequence = binary expansion of Prouhet–Thue–Morse constant:

${\displaystyle \tau =\sum _{i=0}^{\infty }{\frac {t_{i}}{2^{i+1}}}={0.01101001100101101001011001101001\ldots }_{2}}$

where:

• ${\displaystyle t_{i}}$  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:[24]

${\displaystyle t_{0}=0}$
${\displaystyle t_{n+1}=t_{n}{\overline {t_{n}}}}$

where:

• ${\displaystyle {\overline {t}}}$  means change all the 0’s in x to 1’s and vice-versa

First 5 sequences

${\displaystyle t_{0}=0}$
${\displaystyle t_{1}=01}$
${\displaystyle t_{2}=0110}$
${\displaystyle t_{3}=01101001}$
${\displaystyle t_{4}=0110100110010110}$

The Rabbit constant (The limiting Rabbit Sequence) given by the continued fraction

${\displaystyle R=[0,2^{F_{0}},2^{F_{1}},2^{F_{2}},2^{F_{3}},...]=0.1011010110110...}$

where:

• ${\displaystyle F_{n}}$  are Fibonacci numbers with ${\displaystyle F_{0}}$  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[25]

## Periodic

periodic angles have multiple possible ways of writing down, one of which is special in having the smallest period

${\displaystyle 1/3=.0101010101010101...=.(01)=.(0101)=.(010101)=...}$

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:

${\displaystyle {\frac {1}{6}}=0.0(01)=0.00(10)=0.001(01)=...}$

${\displaystyle {\frac {3}{14}}=0.0(011)=0.00(110)}$

${\displaystyle 0.0{\overline {0110}}=0.{\overline {0011}}}$


## finite

• every nonzero terminating binary number ( = rational number with odd denominator which is a power of 2 ) has two equal representations[26]
• 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, 0.112 = 0.10111...2, giving two different representations for 3/4. The dyadic rationals are the only numbers whose binary expansions are not unique.

${\displaystyle 0.0^{t-1}1_{2}=0.0^{t}(1)_{2}}$

Binary expansion of unitary fractions 1/(2^p)
t decimal ${\displaystyle {\frac {1}{2^{t}}}}$  binary finite = ${\displaystyle 0.0^{t-1}1}$  binary infinite = ${\displaystyle 0.0^{t}(1)}$
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

# Binary expansion and the dynamical systems

Maps

## Doubling map

"The map ${\displaystyle \lambda :\mathbb {R} /\mathbb {Z} \to J_{c}}$  is called the Caratheodory semiconjugacy, with the associated identity ${\displaystyle \lambda (2*t)=f_{c}(\lambda (t))}$  in the degree 2 case. This identity allows us to easily track forward iteration of external rays and their landing points in ${\displaystyle J_{c}}$  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 external angle 11/31 turns is 5-periodic under doubling modulo 1. Its binary expansion is given by 11/31 = .01011, since 11 = 0*16 + 1*8 + 0*4 + 1*2 + 1*1 and 31 = 25 - 1. Doubling means that the digits are shifted to the left, e.g., 2*11/31 = 22/31 = .10110.

There are two kinds of rational angles:
* 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. ( from program Mandel by Wolf Jung)


## external angles

Using binary decomposition:

• start with an unknown location and finds the external angles from the image[28]
• start with angles and find the location that matches[29][30]

Examples from the parameter plane and Mandelbrot set:

• The northwest external angle is 3/8
• The north external angle is 1/4[31]
• The northeast external angle is 1/8[32]
• The west external angle is 1/2
• The east external angle is 0
• The southwest external angle is 5/8
• The south external angle is 3/4
• The southeast external angle is 7/8,

The external arguments of the rays landing at z = −0.15255 + 1.03294i are:[33]

${\displaystyle (\theta _{20}^{-},\theta _{20}^{+})=(0.{\overline {00110011001100110100}},0.{\overline {00110011001101000011}})}$

where:

${\displaystyle \theta _{20}^{-}=0.{\overline {00110011001100110100}}_{2}=0.{\overline {20000095367522590181913549340772}}_{10}={\frac {209716}{1048575}}={\frac {209716}{2^{20}-1}}}$

# Conversions

## 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." [34]