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Recurrences and Continued Fractions Great Theoretical Ideas In Computer Science V. Adamchik D. Sleator CS 15-251 Spring 2010 Lecture 8 Feb 04, 2010 Carnegie Mellon University

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Solve in integers x 1 + x 2 +…+ x 5 = 40 x k r k; y k =x k – k y 1 + y 2 +…+ y 5 = 25 y k ≥ 0;

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Solve in integers x + y + z = 11 x r 0; 0 y 3; z r 0 [X 11 ]

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Find the number of ways to partition the integer n 3 = 1+1+1 = 1+2 4 = 1+1+1+1 = 1+1+2 = 1+3 = 2+2 Partitions

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Plan Review: Sat, 6-8pm in 2315 DH Exam: Mon in recitations Characteristic Equations Golden Ratio Continued Fractions Applied Combinatorics, by Alan Tucker The Divine Proportion, by H. E. Huntley

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Leonardo Fibonacci In 1202, Fibonacci has become interested in rabbits and what they do

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The rabbit reproduction model A rabbit lives forever The population starts as a single newborn pair Every month, each productive pair begets a new pair which will become productive after 2 months old F n = # of rabbit pairs at the beginning of the n th month month1234567 rabbits11235813

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What is a closed form formula for F n ? F 0 =0, F 1 =1, F n =F n-1 +F n-2 for n ≥ 2 We won’t be using GFs!

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This lecture has explicit mathematical content that can be shocking to some students. WARNING!!!!

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Characteristic Equation F n = F n-1 + F n-2 Consider solutions of the form: F n = n for some (unknown) constant ≠ 0 must satisfy: n - n-1 - n-2 = 0

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iff iff n-2 ( 2 - - 1) = 0 iff 2 - - 1 = 0 Characteristic equation = , or = -1/ (“phi”) is the golden ratio

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So for all these values of the inductive condition is satisfied: 2 - - 1= 0 Do any of them happen to satisfy the base condition as well? F 0 =0, F 1 =1 a,b a n + b (-1/ ) n satisfies the inductive condition

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Adjust a and b to fit the base conditions. n=0: a + b = 0 n=1: a 1 + b (-1/ ) 1 = 1

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Leonhard Euler (1765)

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Fibonacci Power Series

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Fibonacci Bamboozlement 8 8 13 5

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Cassini’s Identity F n+1 F n-1 - F n 2 = (-1) n We dissect F n xF n square and rearrange pieces into F n+1 xF n-1 square

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How to convert kilometers into miles? Heads-on

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50 km = 34 + 13 + 3 Magic conversion F 8 + F 6 + F 3 = 31 miles 50 = F 9 + F 7 + F 4

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From the previous lecture

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Characteristic Equation a, b

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Characteristic Equation

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Theorem. Let be a root of multiplicity p of the characteristic equation. Then are all solutions.

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The theorem says that x n =n is a solution. This can be easily verified: n = 2 n - n

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From the previous lecture: Rogue Recurrence Characteristic equation:

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General solution:

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The Golden Ratio “Some other majors have their mystery numbers, like and e. In Computer Science the mystery number is , the Golden Ratio.” – S.Rudich

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Golden Ratio -Divine Proportion Ratio obtained when you divide a line segment into two unequal parts such that the ratio of the whole to the larger part is the same as the ratio of the larger to the smaller. A B C

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Golden Ratio - the divine proportion = 1.6180339887498948482045… “Phi” is named after the Greek sculptor Phidias

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Aesthetics plays a central role in renaissance art and architecture. After measuring the dimensions of pictures, cards, books, snuff boxes, writing paper, windows, and such, psychologist Gustav Fechner claimed that the preferred rectangle had sides in the golden ratio (1871).

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Which is the most attractive rectangle?

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Golden Rectangle 1

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The Golden Ratio

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Divina Proportione Luca Pacioli (1509) Pacioli devoted an entire book to the marvelous properties of . The book was illustrated by a friend of his named: Leonardo Da Vinci

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Table of contents The first considerable effect The second essential effect The third singular effect The fourth ineffable effect The fifth admirable effect The sixth inexpressible effect The seventh inestimable effect The ninth most excellent effect The twelfth incomparable effect The thirteenth most distinguished effect

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Table of contents “For the sake of salvation, the list must end here” – Luca Pacioli

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Divina Proportione Luca Pacioli (1509) "Ninth Most Excellent Effect" two diagonals of a regular pentagon divide each other in the Divine Proportion. C B A

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Expanding Recursively

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A (Simple) Continued Fraction Is Any Expression Of The Form: where a, b, c, … are whole numbers.

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A Continued Fraction can have a finite or infinite number of terms. We also denote this fraction by [a,b,c,d,e,f,…]

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Continued Fraction Representation = [1,1,1,1,1,0,0,0,…]

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Recursively Defined Form For CF

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Proposition: Any finite continued fraction evaluates to a rational. Converse: Any rational has a finite continued fraction representation.

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Euclid’s GCD = Continued Fraction Euclid(A,B) = Euclid(B, A mod B) Stop when B=0

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Euclid’s GCD = Continued Fraction

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A Pattern for Let r 1 = [1,0,0,0,…] = 1 r 2 = [1,1,0,0,0,…] = 2/1 r 2 = [1,1,0,0,0,…] = 2/1 r 3 = [1,1,1,0,0,0…] = 3/2 r 3 = [1,1,1,0,0,0…] = 3/2 r 4 = [1,1,1,1,0,0,0…] = 5/3 r 4 = [1,1,1,1,0,0,0…] = 5/3 and so on. Theorem: = F n /F n-1 when n ->

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Divine Proportion

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Quadratic Equations X 2 – 3x – 1 = 0 X 2 = 3X + 1 X = 3 + 1/X X = 3 + 1/X = 3 + 1/[3 + 1/X] = …

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A Periodic CF

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A period-2 CF

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Proposition: Any quadratic solution has a periodic continued fraction. Converse: Any periodic continued fraction is the solution of a quadratic equation

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What about those non-periodic continued fractions? What is the continued fraction expansion for ?

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Euclid’s GCD = Continued Fraction ??

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What is the pattern?

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Every irrational number greater than 1 is the limit of a unique infinite continued fraction.

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Every irrational number greater than 1 is the limit of a unique infinite continued fraction. Suppose that y>1 is irrational and y = [b 0 ;b 1,b 2,...]. where y 1 > 1. Thus, Similarly, for all b k.

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To calculate the continued fraction for a real number y, set y 0 =y and then

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What a cool representation! Finite CF: Rationals Periodic CF: Quadratic roots And some numbers reveal hidden regularity.

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Let us embark now on approximations

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CF for Approximations

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We say that an positive irrational number y is approximable by rationals to order n if there exist a positive constant c and infinitely many rationals p/q with q > 0 such that We will see that algebraic numbers are not approximable to arbitrarily high order.

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Positive rational numbers are approximable to order 1 Let y = a/b be a rational number, with gcd(a,b) = 1. By Euclid’s algorithm we can find p 0,q 0 such that p 0 b − q 0 a = 1. Then

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Positive rational numbers are approximable to order 1

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Similarly,

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Liouville’s number Transcendental number

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Liouville’s number and e are transcendental number What about e ?

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Apery’s constant irrational number transcendental number - ??? Riemann Zeta function

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Study Bee Review GCD algorithm Recurrences, Phi and CF Solving Recurrences

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