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Continued Fractions John D Barrow

Continued Fractions John D Barrow. Headline in Prairie Life. Decimals.  = 3.141592… =  i a i 10 -i = (a i ) = (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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Continued Fractions John D Barrow

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  1. Continued Fractions John D Barrow

  2. Headline in Prairie Life

  3. 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?

  4. 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 again…and again… b = 1 gives the golden mean x =  = ½(1 + 5) = 1·6180339887..

  5. William Brouncker First President of the Royal Society Introduced the ‘staircase’ notation (1620-84) by using Wallis’ product formula for  John Wallis Wallis: ‘continued fraction’ (1653-5) (1616-1703)

  6. Euler’s Formula Log{(1+i)/(1-i)} = i/2 i = -1

  7. Avoiding the Typesetter’s 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 = [0;2] not [0;1,1] Irrational numbers have a (unique) infinite cfes

  8. 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

  9. Plot of the cfe digits of

  10. Rational 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, 208341/66317,… A large number (eg 292) in the cfe expansion creates a very good approx

  11. 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.

  12. = (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

  13. Minding your p’s and q’s • 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  <1/(q25) qk> 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

  14. 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%

  15. Scale Models of the Solar System

  16. Gears Without Tears 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  29.46 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 Saturn’s motion relative to Earth by making one gear with 7 teeth and one with 206

  17. Carl Friedrich Gauss (1777-1855)

  18. Probability and Continued Fractions Any infinite list of numbers defines a unique real number by its cfe  There can’t 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) ]/ln[2] P(1) = 0.41, P(2) = 0.17, P(3) = 0.09, P(4) = 0.06, P(5) = 0.04 ln(1+x)  x P(k)  1/k2 as k 

  19. Typical Continued Fractions Arithmetic mean (average) value of the k’s is k=1 k P(k)  1/ln[2]  k=1 1/k   Geometric mean is finite and universal for a.e number (k1........kn)1/n K= 2.68545….. as n   K k=1 {1+1/k(k+2)}ln(k)/ln(2) : Khinchin’s 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  

  20. k=11/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) +..  > 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 + …….   “Divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever” Niels Abel

  21. Geometric Mean for the cfe Digits of  G Mean K =2.68.. k Aleksandr Khinchin 1894-1959

  22. Geometric Means for Some Exceptional Numbers

  23. Slow Convergence to K-- with a pattern ? Geo Mean Cfe geometric means for , 2, , log(2), 21/3, 31/3

  24. Lévy’s Constant If x has a rational approx pn/qn after n steps of the cfe, then for almost every number qn < exp[An] as n   for some A>0 qn1/n L = 3.275… as n   Paul Lévy, 1886-1971 L for cfe of  3.275…

  25. A Strange Series What is the sum of this series?? S(N) = p=1N 1/{p3sin2p} (Pickover-Petit- McPhedran problem) 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

  26. 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

  27. Gauss’s Probability Distribution xn+1 = 1/xn – [1/xn] As n   the probability of outcome x tends to p(x) = 1/[(1+x)ln2] : 01 p(x)dx = 1 Error is < (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

  28. xn+1 = 1/xn – [1/xn] = T(xn) T(x) T(x) =1/x – k (1-k)-1<x<k-1 ldT/dxl = 1/x2 > 1 as 0 < x < 1 x n steps = initial  exp[ht]: h = 2/[6(ln2)2]  3.45

  29. The Mixmaster Universe

  30. 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

  31. To be continued……

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