Continued FractionsJohn D Barrow

Headline in Prairie Life

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?

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

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)

Euler’s FormulaEuler’s Formula

Log{(1+i)/(1-i)} = i/2

i = -1

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

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

Plot of the cfe digits of

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

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

= (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

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

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

Scale ModelsScale Modelsof of

the Solar Systemthe Solar System

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

Carl Friedrich GaussCarl Friedrich Gauss

(1777-1855)

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

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

= 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

Geometric Mean for the cfe Digits of Geometric Mean for the cfe Digits of

G Mean

k

K =2.68..

Aleksandr Khinchin1894-1959

Geometric Means for Some Exceptional NumbersGeometric Means for Some Exceptional Numbers

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

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…

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

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

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

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

The Mixmaster UniverseThe Mixmaster 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 + 0.9864248Next cycles have 1, 72, 1 and 5 oscillations respectively

The Continued-Fraction UniverseThe Continued-Fraction Universe

To be To be continued……continued……