Continued Fractions John D Barrow
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Continued FractionsJohn D Barrow
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Headline in Prairie Life
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DecimalsDecimals
= 3.141592…
= i ai 10-i
= (ai) = (3,1,4,1,5,9,2,…)
But rational fractions like 1/3 = 0.33333..do not have finite decimal expansions
Why choose base 10?
Hidden structure?
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x2 – bx – 1 = 0
x = b + 1/x
Substitute for x on the RH side
x = b + 1/(b +1/x)x = b + 1/(b +1/x)
Do it again…and again…
b = 1 gives the golden mean b = 1 gives the golden mean x = x = = ½(1 + = ½(1 + 5) = 1·6180339887..5) = 1·6180339887..
A Different Way of Writing NumbersA Different Way of Writing Numbers
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William BrounckerWilliam Brouncker
First President of the Royal SocietyFirst President of the Royal Society
Introduced the ‘staircase’ notationIntroduced the ‘staircase’ notation
(1620-84)
John Wallis(1616-1703)
by using Wallis’ product formula for
Wallis: ‘continued fraction’ (1653-5)
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Euler’s FormulaEuler’s Formula
Log{(1+i)/(1-i)} = i/2
i = -1
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Avoiding the Typesetter’s Avoiding the Typesetter’s NightmareNightmare
x [a0 ; a1, a2, ……]
cfe of x
Rational numbers have finite cfes Take the shortest of the two
possibilities for the last digit eg ½ = [0;2] not [0;1,1]
Irrational numbers have a (unique) infinite cfes
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Pi and e
= [3;7,15,1,292,1,1,3,1,14,2…..]
e = 2.718…. = [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,….]Cotes (1714) = [1;1,1,1,1,1,1,1,……..] golden ratio
2 = [1;2,2,2,2,2,2,2,2,2,2,….] 3 = [1;1,2,1,2,1,2,1,2,1,2,1,.]
‘Noble’ numbers end in an infinite sequence of 1’s
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Plot of the cfe digits of
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Rational Approximations for Irrational NumbersRational Approximations for Irrational Numbers
Ending an infinite cfe at some point creates a rational approximation for an irrational number
= [3;7,15,1,292,1,1,…]
Creates the first 7 rational approximations for labelled pn/qn
3, 22/7, 333/106, 355/113, 103993/33102, 104384/33215, 3, 22/7, 333/106, 355/113, 103993/33102, 104384/33215, 208341/66317,…208341/66317,…
A large number (eg 292) in the cfe expansion creates a very good approx
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Truncating the decimal expn of gives 31415/1000 and 314/100
The denominators of 314/100 and 333/106 are almost the same,
but the error in the approximation 314/100 is 19 times as large as the error in the cfe approx 333/106.
As an approximation to , [3; 7, 15, 1] is more than one hundred times more accurate than 3.1416.
Better than DecimalsBetter than Decimals
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= (2143/22)1/4 is good to 3 parts in 104 !
Ramanujan knew that 4 = [97;2,2,3,1,16539,1,…]Note that the 431st digit of is 20776
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Minding your p’s and q’sMinding your p’s and q’sAs n increases the rational approximations to any irrational number, x, get better and better
x – pn/qn 0
In the limit the best possible rational approx is
x – p/q <1/(q25)The golden ratio is the most irrational number: it lies farthest from a rational approximation 1/(q25)Approximants are 5/3, 8/5, 13/8, 21/13,…They all run close to this boundary
qk > 2(k-1)/2
Same is true for all (a + b)/(c + d) with ad – bc = + 1
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The ratio of the numbers of teeth on two cogs governs their speed ratio. Mesh a 10-tooth with a a 50 tooth and the 10-tooth will rotate 5 times quicker (in the opposite direction). What if we want one to rotate 2 times faster than the other. No ratio will do it exactly. Cfe rational approximations to 2 are 3/2, 7/5, 17/12, 41/29, 99/70,…3/2, 7/5, 17/12, 41/29, 99/70,… So we could have 7 teeth on one and 5 on the other (too few for good meshing though) so use 70 and 50. If we can use 99 and 70 then the error is only 0.007%
Getting Your Teeth Into GearsGetting Your Teeth Into Gears
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Scale ModelsScale Modelsof of
the Solar Systemthe Solar System
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In 1682 Christian Huygens used 29.46 yrs for Saturn’s orbit around Sun (now 29.43)
Model solar system needs two gears with P and Q teeth: P/Q 29.46Needs smallish values of P and Q (between 20 and 220) for cutting
Find cfe of 29.46. Read off first few rational approximations29/1, 59/2, 206/7,..then simulate Saturn’s motion relative to Earth
by making one gear with 7 teeth and one with 206
Gears Without Tears
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Carl Friedrich GaussCarl Friedrich Gauss
(1777-1855)
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Probability and Continued Probability and Continued FractionsFractions
Any infinite list of numbers defines a unique real number by its cfe
There can’t be a general frequency distribution for the cfe There can’t be a general frequency distribution for the cfe of all numbersof all numbers
But for almost everyalmost every real number there is !
The probability of the appearance of the digit k in the cfe of almost every number isP(k) = ln[ 1 + 1/k(k + 2) ]/ln[2]P(k) = ln[ 1 + 1/k(k + 2) ]/ln[2]
P(1) = 0.41, P(2) = 0.17, P(3) = 0.09, P(4) = 0.06, P(5) = 0.04
P(k) 1/k2 as k ln(1+x) x
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Typical Continued FractionsTypical Continued FractionsArithmetic mean (average) value of the k’s is
k=1k=1 k P(k) k P(k) 1/ln[2] 1/ln[2]
k=1k=1 1/k 1/k
Geometric mean is finite and universal for a.e numberGeometric mean is finite and universal for a.e number
(k(k11........k........knn))1/n1/n K= 2.68545….. as n K= 2.68545….. as n
KK k=1k=1 {1+1/k(k+2)} {1+1/k(k+2)}ln(k)/ln(2)ln(k)/ln(2) : Khinchin’s constant : Khinchin’s constant
Captures the fact that the cfe entries are usually smallCaptures the fact that the cfe entries are usually smalle = 2.718..e = 2.718.. is an exception is an exception
(k(k11........k........knn))1/n1/n = [2 = [2N/3N/3(N/3)!](N/3)!]1/N1/N 0.6259N 0.6259N1/31/3
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= 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + .......
= 1 + (1/2 + 1/3) + (1/4 + 1/5 + 1/6 + 1/7) + (1/8 + 1/9 + 1/10 + 1/11 +…..1/15) +..
> 1/2 + (1/4 + 1/4) +(1/8 + 1/8 + 1/8 + 1/8) + (1/16 + 1/16 + 1/16 + 1/16 + ..+ 1/16 )
> 1/2 + 1/2 + 1/2 + 1/2 + …….
k=1k=11/k has an Infinite Sum1/k has an Infinite Sum
“Divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever”
Niels Abel
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Geometric Mean for the cfe Digits of Geometric Mean for the cfe Digits of
G Mean
k
K =2.68..
Aleksandr Khinchin1894-1959
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Geometric Means for Some Exceptional NumbersGeometric Means for Some Exceptional Numbers
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Cfe geometric means for , 2, , log(2), 21/3, 31/3
Slow Convergence to K-- with a pattern ?Slow Convergence to K-- with a pattern ?
Geo MeanGeo Mean
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LLéévy’s Constantvy’s Constant
Paul Lévy, 1886-1971
If x has a rational approx pn/qn aftern steps of the cfe, then for almost
every number
qqnn < exp[An] as n < exp[An] as n for some A>0 for some A>0
qn1/n L = 3.275… as n
LLfor cfe of for cfe of
3.275…
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A Strange SeriesA Strange SeriesWhat is the sum of this series??What is the sum of this series??
S(N) = p=1N 1/{p3sin2p}
(Pickover-Petit-McPhedran problem)
NN S(N)S(N)
2222 4.754104.75410
2626 4.757964.75796
2828 4.758734.75873
310310 4.806864.80686
313313 4.806974.80697
314314 4.806974.80697
355355 29.4 !!29.4 !!
Occasionally p Occasionally p q q so sin(n) so sin(n) 0 and S 0 and S This happens when This happens when pp/q is a rational approx to /q is a rational approx to
3/3/11, 22/, 22/77, 333/, 333/106106, 355/, 355/113113, , 103993/103993/3310233102, 104384/, 104384/3321533215, ,
208341/208341/6631766317,…,…
Dangerous values continue foreverDangerous values continue forever and diverge faster than 1/pand diverge faster than 1/p33
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Chaos in NumberlandChaos in NumberlandGenerate the cfe of
u = k + x = whole number + fractional part = [u] + x
= 3 + 0.141592.. = k1 + x1
k2 = [1/x1] = [7.0625459..] = 7
x2 = 0.0625459..
k3 = [1/x2] = [15.988488..] = 15
The fractional parts change from x1 x2 x3 ..chaotically. Small errors grow exponentially
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Gauss’s Probability DistributionGauss’s Probability Distribution
xxn+1 n+1 = 1/x= 1/xnn – [1/x – [1/xnn]]
As n the probability of outcome x tends to p(x) = 1/[(1+x)ln2] : p(x) = 1/[(1+x)ln2] : 00
11 p(x)dx = 1 p(x)dx = 1Error is < (0.7)n after n iterations
p(x)
x
In aLetter to Laplace
30th Jan 1812‘a curious problem’
that had occupied him for 12 years
Distribution of the fractional
parts
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xxn+1n+1 = 1/x = 1/xnn – [1/x – [1/xnn] = T(x] = T(xnn))
T(x)
x
n stepsn steps = = initialinitial exp[ht]: h = exp[ht]: h = 22/[6(ln2)/[6(ln2)22] ] 3.45 3.45
ldT/dxl = 1/x2 > 1
as 0 < x < 1
T(x) =1/x – kT(x) =1/x – k
(1-k)(1-k)-1-1<x<k<x<k-1-1
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The Mixmaster UniverseThe Mixmaster Universe
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u = 6.0229867.. = k + x = 6 + 0.0229867.. u 1/x = 1/0.0229867 = 43.503417 = 43 + 0.503417 u 1/0.503417 = 1.9864248 = 1 + 0.9864248Next cycles have 1, 72, 1 and 5 oscillations respectively
The Continued-Fraction UniverseThe Continued-Fraction Universe
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To be To be continued……continued……