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Rational Walks RATIONAL WALKS A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Anthony Simon 2019 SIGNATURE PAGE THESIS: RATIONAL WALKS AUTHOR: Anthony Simon DATE SUBMITTED: Spring 2019 Department of Mathematics and Statistics Dr. Randall Swift Thesis Committee Chair Mathematics & Statistics Dr. Michael Green Mathematics & Statistics Dr. Alan Krinik Mathematics & Statistics ii ACKNOWLEDGMENTS I primarily thank all my roommates that I have lived with for the past years, there has been so much growth in everyone it encourages me to keep doing great things for the world. I also like to thank Dr. Swift for finding time to read my drafts, I know the transition from quarters to semesters for Cal Poly has been rough but, this is pivotal for the schools growth. iii ABSTRACT In this thesis a mapping from decimals :d1d2:::dn::: 2 [0; 1] into f0; 1; 2g is defined and described as a walk. The thesis explores visual representations of rational and irrational numbers as walks. Several interesting classes of real numbers are considered, including normal numbers and Khinchin's Constant. iv Contents Signature Page ii Acknowledgements iii Abstract iv List of Figures viii List of Tables viii Chapter 1 Introduction 1 1.1 Sets of Real Numbers .......................... 1 1.2 Ternary Strings ............................. 6 1.3 Rational Mapping . 7 1.3.1 Examples of Rational Maps .................. 8 Chapter 2 Arithmetic and Geometric Operations on Strings 12 2.1 Operations Arithmetic ......................... 12 2.2 Substrings . 19 2.3 Some Walks ............................... 22 2.4 String Analysis ............................. 24 v 2.5 Characterizations of Strings ...................... 27 Chapter 3 Analysis of Normal Strings 31 3.1 Normal Numbers ............................ 31 3.2 Normal Strings ............................. 33 Bibliography 36 Appendices 37 Appendix A Code 37 vi List of Figures 1.1 The Rational Map of :22439 ...................... 8 1.2 The Rational Map of :27388 ...................... 9 1.3 The Rational Map of :53850 ...................... 10 1.4 The Rational Map of :57845 ...................... 11 2.1 The Rational Map of 1 ....................... 21 27 [3;7] 2.2 The Rational Map of 1 ....................... 22 7 [1;30] p 2.3 The Rational Map of 2[1;100] ..................... 22 2.4 The Rational Map of π[1;100] ...................... 23 2.5 The Rational Map of π[1;1000] ...................... 23 2.6 The Rational Map of Champernowne(10)[1;48] ............ 28 2.7 The Rational Map of Khinchin's constant first 100 decimals . 30 2.8 The Rational Map of Khinchin's constant first 1,000 decimals . 30 3.1 The First 1,000 Steps of Champernowne(10) ............. 33 3.2 The First 100,000 Steps of Champernowne(10) . 33 3.3 The First 1000 Steps of the Copland-Erdos Constant . 34 3.4 The First 4000 Steps of the Copland-Erdos Constant . 34 3.5 The First 8000 Steps of the Copland-Erdos Constant . 35 3.6 The First 25000 Steps of the Copland-Erdos Constant . 35 vii List of Tables 2.1 2-step walks ............................... 18 viii Chapter 1 Introduction In this chapter the familiar sets of natural numbers, integers, rational numbers and irrational numbers are introduced. A class named the ternary class composed of equivalence classes named strings. Finally, there will be a definition of how to plot these strings. 1.1 Sets of Real Numbers Some sets of numbers that are used in this thesis are detailed here. Definition The set of natural numbers, N, is defined as N = f1; 2; 3; :::g: An alternative name for the natural numbers is the counting numbers. Notation: The natural numbers including zero will be denoted as, N0 = N [ f0g, and is sometimes called the whole numbers. Definition The set of integers, Z, is Z = f:::; −2; −1; 0; 1; 2; :::g: 1 Definition For n 2 N and a; b 2 Z, a and b are congruent modulo n, if their difference a − b is an integer multiple of n namely, if there is an integer k such that a − b = kn. This is denoted a ≡ b (mod n): Notation The set of integers modulo n, Zn, is defined as Zn = f0; 1; 2; :::; n − 1g; where n 2 N. In general this thesis focuses on the set Z3 = f0; 1; 2g. If Z2 = f0; 1g is the binary class, then Z3 is called the ternary class. Definition The set of rational numbers, Q is defined as follows, a = f ja; b 2 ; b =6 0; gcd(a; b) = 1g Q b Z In the definition the constraint gcd(a; b) = 1 gives that the representation of numbers in Q are in lowest form. 11 31 For instance, the number 0:5 could be expressed as 22 , 62 , or any other fraction but 1 with gcd(a; b) = 1, the only fraction that is appropriate is 2 . To define the mantissa of a number, there needs to be a definition of the floor function, since the only numbers of intrest are in the interval (0; 1). Definition The floor function, bxc is the greatest integer les than x. Examples bπc = 3 bec = 2 2 The mantissa is the positive fraction part of a number. Definition The mantissa of a real number x is x − bxc = 0:x1x2:::; where each xi 2 Zb, where b 2 N − f1g is a base. We consider a decimal expansion that either terminates or repeats. Example What is the fraction representation of the number :333:::? 3 3 :3333::: = + + ::: 10 100 1 X 3 = 10n+1 n=0 3 10 = 1 1 − 10 3 10 = ∗ 10 9 1 = 3 Lets generalize this idea from base 10 to base b, where b 2 N − f1g. Theorem Every finite repeating decimal number is a rational number. Proof. For an x 2 (0; 1) x = :d1d2:::dnd1d2:::(base b) where each di 2 Zb, where b 2 N − f1g. Since this number can be written in the 3 form x = :d1d2:::dn−1dnd1d2:::(base b) d d d d d d d d d = 1 +2 + ::: + n−1 +n +1 +2 + ::: + n−1 +n + 1 + ::: b b2 bn−1 bn bn+1 bn+2 b2n−1 b2 n b2n+1 d d d d d d = (1 +1 + 1 + :::) + ( 2 +2 + 2 + :::) + b bn+1 b2n+1 b2 bn+2 b2n+2 d d d d d d ::: +(n−1 + n−1 + n−1 + :::) + ( n + n +n + :::) bn−1 b2n−1 b3n−1 bn b2n b3 d 1 1 d 1 1 d 1 1 =1 (1 + + + :::) + 2 (1 + + + :::) + ::: +n (1 + + + :::) b bn b2n b2 bn b2n bn bn b2n Represented by using the sum of a geometric series 1 1 d d d d = (1 + + + :::)(1 + 2 + ::: + n−1 +n ) bn b2n b b2 bn−1 bn 1 d bn−1 + d bn−2 + ::: + d b + d = ( )( 1 2 n−1 n ) 1 n 1 − b n b bn d bn−1 + d bn−2 + ::: + d b + d d bn−1 + d bn−2 + ::: + d b + d = ( )( 1 2 n−1 n ) = 1 2 n−1 n n n n b − 1 b b − 1 n−1 n−2 n Since d1b + d2b + ::: + dn−1b + dn is an integer and b − 1 is also an integer then x is rational. □ 4 Example The number x 2 (0; 1) is a repeating base 6 decimal, for x = :1234512345:::(base 6) the fraction representation is this number. x = :12345:::12345:::(base 6) (1)64 + (2)63 + (3)62 + (4)6 + 5 = 65 − 1 1865 373 = = : 7775 1555 Finally the ternary string class will be defined. Definition Let T be the set T = f x j x = [δ1 δ2 :::]; δi = Z3g given, x = :d1d2::: each di 2 Zb, where b 2 N=f1g. Notation: A ternary string will be denoted in this thesis as string x. Example 21 Show 25 = [2 1]. 21 Write 25 as a decimal 21 = :84: 25 Now since d1 = 8 and d2 = 4, the corresponding δi's are, δ1 = 8 (mod 3) = 2 and 21 21 δ2 = 4 (mod 3) = 1. So the string of 25 is [2 1]. Thus 25 = [2 1]. Notation We denote finite repeating rational classes by [δ1 δ2 ::: δn] Examples Show 1=3 = [0] Begin with, converting 1=3 to it's decimal form 1 = :333::: 3 5 meaning each di = 3, thus each δi = 3 (mod 3) = 0. So, the string of 1=3 is [0 0 0 0 0 :::] = [0] 1 Thus 3 = [0]. 1.2 Ternary Strings In this section we consider the structure of the ternary class, T, this is done by defining size and length of a string. Definition The length of a string [d1 ::: dn] is l([d1 ::: dn]) = n: If a string does not terminate, it is said to be of unbounded length. Since 4 digits from Z10 map to 0 2 Z3, and 3 digits from Z10 map to 1; 2 2 Z3. Definition The size of the string x is defined as n0 n1+n2 s(x) = s([δ1...δn]) = (4) (3) ; where n0 is the number of δi's that are 0, n1 is the number of δi's that are 1 and n2 is the number of δi's that are 2. The size of a string is how many numbers from (0; 1) map to an individual string in T. Note: A string of unbounded length has unbounded size. Example Determine the size of each string [2 0 0 1], [2 0 2 0] and [0]. s([2 0 0 1]) = (4)2(3)1+1 = 144; 6 notice that s([1 0 2 0]) = 144 as well.
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