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)
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
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
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
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
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
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