Continued Fractions

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Continued Fractions John D Barrow
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Headline in Prairie Life
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Decimals

• ? 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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A Different Way of Writing Numbers
x2 bx 1 0 x b 1/x Substitute for x on
the RH side x b 1/(b 1/x) Do it againand
again
b 1 gives the golden mean x ? ½(1 ??5)
16180339887..
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William Brouncker First President of the Royal
Society Introduced the staircase notation
(1620-84)
by using Wallis product formula for ?
John Wallis
(1616-1703)
Wallis continued fraction (1653-5)
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Eulers Formula
Log(1i)/(1-i) i?/2 i ?-1
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Avoiding the Typesetters Nightmare
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 02 not 01,1 Irrational
numbers have a (unique) infinite cfes
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Pi and e
? 37,15,1,292,1,1,3,1,14,2..
e 2.718. 21,2,1,1,4,1,1,6,1,1,8,1,1,10,.
Cotes (1714)
? 11,1,1,1,1,1,1,.. golden ratio
?2 12,2,2,2,2,2,2,2,2,2,. ?3
11,2,1,2,1,2,1,2,1,2,1,.
Noble numbers end in an infinite sequence of 1s
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Plot of the cfe digits of
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Rational Approximations for Irrational Numbers
Ending an infinite cfe at some point creates a
rational approximation for an irrational number
? 37,15,1,292,1,1,
Creates the first 7 rational approximations for ?
labelled pn/qn 3, 22/7, 333/106, 355/113,
103993/33102, 104384/33215, 208341/66317,
A large number (eg 292) in the cfe expansion
creates a very good approx
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Better than Decimals
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.
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? (2143/22)1/4 is good to 3 parts in 104 !
Ramanujan knew that ?4 972,2,3,1,16539,1, No
te that the 431st digit of ? is 20776
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Minding your ps and qs

• As 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 ? lt1/(q2?5)

qk gt 2(k-1)/2
The golden ratio ? is the most irrational number
it lies farthest from a rational approximation
1/(q2?5) Approximants are 5/3, 8/5, 13/8,
21/13, They all run close to this boundary
Same is true for all (a? b?)/(c? d?) with ad
bc 1
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Getting Your Teeth Into Gears
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, 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
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Scale Models of the Solar System
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Gears Without Tears
In 1682 Christian Huygens used 29.46 yrs for
Saturns orbit around Sun (now 29.43) Model solar
system needs two gears with P and Q teeth P/Q ?
29.46 Needs smallish values of P and Q (between
20 and 220) for cutting Find cfe of 29.46. Read
off first few rational approximations 29/1, 59/2,
206/7,..then simulate Saturns motion relative to
Earth by making one gear with 7 teeth and one
with 206
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Carl Friedrich Gauss
(1777-1855)
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Probability and Continued Fractions
Any infinite list of numbers defines a unique
real number by its cfe ? There cant be a
general frequency distribution for the cfe of all
numbers But for almost every real number there
is ! The probability of the appearance of the
digit k in the cfe of almost every number is
P(k) ln 1 1/k(k 2) /ln2
P(1) 0.41, P(2) 0.17, P(3) 0.09, P(4)
0.06, P(5) 0.04
ln(1x) ? x
P(k) ? 1/k2 as k ??
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Typical Continued Fractions
Arithmetic mean (average) value of the ks is
??k1 k P(k) ? 1/ln2 ? ??k1 1/k ? ?
Geometric mean is finite and universal for a.e
number (k1........kn)1/n ? K 2.68545.. as n ?
? K? ??k1 11/k(k2)ln(k)/ln(2) Khinchins
constant Captures the fact that the cfe entries
are usually small
e 2.718.. is an exception (k1........kn)1/n
2N/3(N/3)!1/N ? 0.6259N1/3 ? ?
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??k11/k has an Infinite Sum
? 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) .. ? gt 1/2 (1/4 1/4)
(1/8 1/8 1/8 1/8) (1/16 1/16 1/16
1/16 .. 1/16 ) ? gt 1/2 1/2 1/2 1/2
. ? ?
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 ?
G Mean
K 2.68..
k
Aleksandr Khinchin 1894-1959
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Geometric Means for Some Exceptional Numbers
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Slow Convergence to K-- with a pattern ?
Geo Mean
Cfe geometric means for ?, 2?, ?, log(2), 21/3,
31/3
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Lévys Constant
If x has a rational approx pn/qn after n steps of
the cfe, then for almost every number qn lt
expAn as n ? ? for some Agt0
qn1/n ? L 3.275 as n ? ?
Paul Lévy, 1886-1971
L for cfe of ?
3.275
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A Strange Series
What is the sum of this series?? S(N) ?p1N
1/p3sin2p
(Pickover-Petit- McPhedran problem)
N S(N)
22 4.75410
26 4.75796
28 4.75873
310 4.80686
313 4.80697
314 4.80697
355 29.4 !!
Occasionally p ? q? so sin(n) ? 0 and S ? This
happens when p/q is a rational approx to ? 3/1,
22/7, 333/106, 355/113, 103993/33102,
104384/33215, 208341/66317, Dangerous values
continue forever and diverge faster than 1/p3
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Chaos in Numberland

• Generate 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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Gausss Probability Distribution
xn1 1/xn 1/xn
As n ? ? the probability of outcome x tends to
p(x) 1/(1x)ln2 ?01 p(x)dx 1 Error is
lt (0.7)n after n iterations
p(x)
In a Letter to Laplace 30th Jan 1812 a curious
problem that had occupied him for 12 years
Distribution of the fractional parts
x
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xn1 1/xn 1/xn T(xn)
T(x)
T(x) 1/x k (1-k)-1ltxltk-1
ldT/dxl 1/x2 gt 1 as 0 lt x lt 1
x
?n steps ?initial ? expht h ?2/6(ln2)2 ?
3.45
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The Mixmaster Universe
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The Continued-Fraction Universe
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 0.9864248 Next
cycles have 1, 72, 1 and 5 oscillations
respectively
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To be continued
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