DOCSLIB.ORG

On H(X)-Fibonacci Octonion Polynomials, Ahmet Ipek & Kamil

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On H(X)-Fibonacci Octonion Polynomials, Ahmet Ipek & Kamil Alabama Journal of Mathematics ISSN 2373-0404 39 (2015) On h(x)-Fibonacci octonion polynomials Ahmet Ipek˙ Kamil Arı Karamanoglu˘ Mehmetbey University, Karamanoglu˘ Mehmetbey University, Science Faculty of Kamil Özdag,˘ Science Faculty of Kamil Özdag,˘ Department of Mathematics, Karaman, Turkey Department of Mathematics, Karaman, Turkey In this paper, we introduce h(x)-Fibonacci octonion polynomials that generalize both Cata- lan’s Fibonacci octonion polynomials and Byrd’s Fibonacci octonion polynomials and also k- Fibonacci octonion numbers that generalize Fibonacci octonion numbers. Also we derive the Binet formula and and generating function of h(x)-Fibonacci octonion polynomial sequence. Introduction and Lucas numbers (Tasci and Kilic (2004)). Kiliç et al. (2006) gave the Binet formula for the generalized Pell se- To investigate the normed division algebras is greatly a quence. important topic today. It is well known that the octonions The investigation of special number sequences over H and O are the nonassociative, noncommutative, normed division O which are not analogs of ones over R and C has attracted algebra over the real numbers R. Due to the nonassociative some recent attention (see, e.g.,Akyigit,˘ Kösal, and Tosun and noncommutativity, one cannot directly extend various re- (2013), Halici (2012), Halici (2013), Iyer (1969), A. F. Ho- sults on real, complex and quaternion numbers to octonions. radam (1963) and Keçilioglu˘ and Akkus (2015)). While ma- The book by Conway and Smith (n.d.) gives a great deal of jority of papers in the area are devoted to some Fibonacci- useful background on octonions, much of it based on the pa- type special number sequences over R and C, only few of per of Coxeter et al. (1946). Octonions made further appear- them deal with Fibonacci-type special number sequences ance ever since in associative algebras, analysis, topology, over H and O (see, e.g., Akyigit˘ et al. (2013), Halici (2012), and physics. Nowadays octonions play an important role in Halici (2013), Iyer (1969), A. F. Horadam (1963) and Keçil- computer science, quantum physics, signal and color image ioglu˘ and Akkus (2015)), notwithstanding the fact that there processing, and so on (e.g. Adler and Finkelstein (2008)). are a lot of papers on various types of Fibonacci-type spe- Many references may be given for Fibonacci-like se- cial number sequences over R and C (see, e.g., Cureg and quences and numbers and related issues. The readers are Mukherjea (2010), Falcon and Plaza (2007), A. Horadam referred to Koshy (n.d.) and Vajda (1989) for some other (1961), Kilic (2008), Öcal et al. (2005), Yang (2008)-Yilmaz sequences which can be obtained from the Fibonacci num- and Taskara (2013)). bers. One may find many applications of this type sequences In this paper, we introduce h(x)-Fibonacci octonion poly- of numbers in various branches of science like pure and nomials that generalize both Catalan’s Fibonacci octonion applied mathematics, in biology or in phyllotaxies, among polynomials Ψn(x) and Byrd’s Fibonacci octonion polyno- many others. Hence, many researches are performed on the mials and also k-Fibonacci octonion numbers that generalize various type of Fibonacci sequence; for example see Cureg Fibonacci octonion numbers. Also we derive the Binet for- and Mukherjea (2010), Falcon and Plaza (2007), A. Horadam mula and and generating function of h(x)-Fibonacci octonion (1961), Kilic (2008), Öcal, Tuglu, and Altini¸sik(2005), Yang polynomial sequence. (2008)-Yilmaz and Taskara (2013). The rest of the paper is structured as follows. Some pre- The connection between the Fibonacci-type numbers and liminaries which are required are given in Section 2. In Sec- the Golden Section is expressed by the well-known math- tion 3 we introduce Catalan, Byrid, Fibonacci octonion poly- ematical formula, so-called Binet’s formulas. Many refer- nomials and numbers and provide their some properties. Sec- ences may be given for Fibonacci-like sequences and num- tion 4 ends with our conclusion. bers and Binet’s formulas. By using a generating function Levesque (1985) gave a Binet formula for the Fibonacci se- Some Preliminaries quence. Kilic and Tasci (2006) gave the generalized Binet formulas for the generalized order-k Fibonacci (Er (1984)) We start this section by introducing some definitions and notations that will help us greatly in the statement of the re- sults. We begin this section by giving the following definition Corresponding Author Email: [email protected] for the classic Fibonacci sequence. The classic Fibonacci 1 2 IPEK˙ & ARI f g Fn n2N sequence is a very popular integer sequence. Catarino (2015) introduced h(x)-Fibonacci quaternion polynomials. The basic properties of Fibonacci quaternion Definition 1. The classic Fibonacci fF g sequence is de- n n2N numbers can be found in Akyigit˘ et al. (2013), Halici (2012), fined by Halici (2013), Iyer (1969) and A. F. Horadam (1963). The octonions in Clifford algebra C are a normed division algebra with eight dimensions over the real numbers larger 4 F0 = 0; F1 = 1 and Fn = Fn−1 + Fn−2 for n ≥ 2; (1) than the quaternions. The field O C of octonions th where Fn denotes the n classic Fibonacci number (Dunlap X7 (n.d.), Koshy (n.d.), Vajda (1989)). α = αses, ai(i = 0; 1; :::; 7) 2 R (4) The books written by Hoggat (1969), Koshy (n.d.) and s=0 Vajda (1989) collects and classifies many results dealing with is an eight-dimensional non-commutative and non- the these number sequences , most of them are obtained quite associative -field generated by eight base elements recently. R e ; e ; :::; e and e . The multiplication rules for the basis of Nalli and Haukkanen (2009) introduced the h(x)- 0 1 6 7 O are listed in the following table Fibonacci polynomials. Then, Ramírez (2014) introduced the convolved h(x)-Fibonacci polynomials and obtain new × 1 e1 e2 e3 e4 e5 e6 e7 identities. 1 1 e1 e2 e3 e4 e5 e6 e7 Definition 2. Let h(x) be a polynomial with real coefficients. e1 e1 −1 e3 −e2 e5 −e4 −e7 e6 1 e2 e2 −e3 −1 e1 e6 e7 −e4 −e5 The h(x)-Fibonacci polynomials fFh;n(x)gn=0 are defined by the recurrence relation e3 e3 e2 −e1 −1 e7 −e6 e5 −e4 e4 e4 −e5 −e6 −e7 −1 e1 e2 e3 Fh;n+1(x) = h(x)Fh;n(x) + Fh;n−1(x), n ≥ 1, (2) e5 e5 e4 −e7 e6 −e1 −1 −e3 e2 e6 e6 e7 e4 −e5 −e2 e3 −1 −e1 with initial conditions Fh;0(x) = 0,Fh;1(x) = 1 (Nalli and e7 e7 −e6 e5 e4 −e3 −e2 e1 −1 Haukkanen (2009)). Table 1. The multiplication table for the basis of O. For h(x) = x, it is obtained Catalan’s Fibonacci polynomi- (5) P7 als n(x), and for h(x) = 2x it is obtained Byrd’s Fibonacci The scalar and the vector part of a octonion α = αses 2 s=0 polynomials φn(x). If for k real number h(x) = k, it is ob- −! P7 tained the k-Fibonacci numbers Fk;n. For k = 1 and k = 2 O are denoted by S α = α0 and Vα = αses, respectively. it is obtained the usual Fibonacci numbers Fn and the Pell s=1 7 numbers P , respectively. P n Let α and β be two octonions such that α = αses and The quaternion, which is a type of hypercomplex num- s=0 bers, was formally introduced by Hamilton in 1843. P7 β = βses. We now are going to list some properties, with- s=0 Definition 3. The real quaternion is defined by out proofs, for α and β octonions. q = qr + qii + q j j + qkk P7 • Property 1. α ± β = (αs ± βs)es s=0 where qr; qi; q j and qk are real numbers and i; j and k are complex operators obeying the following rules • Property 2. αβ = S αS β+S αVβ+VαS β−Vα·Vβ+Vα×Vβ, 7 2 2 2 −! P i = j = k = −1; i j = − ji = k; jk = −k j = i; ki = −ik = j: where S α = α0 and Vα = αses. s=1 In recent years there has been a flurry of activity for doing P7 research with Fibonacci quaternion and octonion numbers. • Property 3. α = α0e0 − αses We can start from the Fibonacci quaternion numbers intro- s=1 duced by Horadam in 1963. • Property 4. (αβ) = βα. Definition 4. The Fibonacci quaternion numbers that are 7 th • α+α α−α P given for the n classic Fibonacci Fn number are defined Property 5. 2 = α0e0 and 2 = αkek: k=1 by the following recurrence relations: • Property 6. αα = αα Qn = Fn + iFn+1 + jFn+2 + kFn+3 (3) 7 • P 2 ± ± Property 7. αα = αk: where n = 0; 1; 2; :::(A. F. Horadam (1963)). k=0 FIBONACCI OCTONION POLYNOMIALS 3 p The norm of an octonion can be defined as kαk = αα. • If for k real number h(x) = k, it is obtained the k- Fibonacci numbers Fk;n from the h(x)-Fibonacci poly- • kαβk kαk kβk Property 8. = . nomials Fh;n(x), and thus for h(x) = k we obtain k-Fibonacci octonion numbers Q from the h(x)- P7 k;n • Property 9. k.α = (kαi) ei: Fibonacci octonion polynomials Qh;n(x). Also, for i=0 k = 1 and k = 2 we have the Fibonacci octonion P7 numbers Qn and the Pell octonion numbers Υn, respec- • Property 10. hα, βi = αiβi: i=0 tively. The Fibonacci octonion numbers are given by the follow- The ordinary generating function (OGF) of the sequence 1 ing definition.
Load more

Recommended publications
  • The Enigmatic Number E: a History in Verse and Its Uses in the Mathematics Classroom
    To appear in MAA Loci: Convergence The Enigmatic Number e: A History in Verse and Its Uses in the Mathematics Classroom Sarah Glaz Department of Mathematics University of Connecticut Storrs, CT 06269 [email protected] Introduction In this article we present a history of e in verse—an annotated poem: The Enigmatic Number e . The annotation consists of hyperlinks leading to biographies of the mathematicians appearing in the poem, and to explanations of the mathematical notions and ideas presented in the poem. The intention is to celebrate the history of this venerable number in verse, and to put the mathematical ideas connected with it in historical and artistic context. The poem may also be used by educators in any mathematics course in which the number e appears, and those are as varied as e's multifaceted history. The sections following the poem provide suggestions and resources for the use of the poem as a pedagogical tool in a variety of mathematics courses. They also place these suggestions in the context of other efforts made by educators in this direction by briefly outlining the uses of historical mathematical poems for teaching mathematics at high-school and college level. Historical Background The number e is a newcomer to the mathematical pantheon of numbers denoted by letters: it made several indirect appearances in the 17 th and 18 th centuries, and acquired its letter designation only in 1731. Our history of e starts with John Napier (1550-1617) who defined logarithms through a process called dynamical analogy [1]. Napier aimed to simplify multiplication (and in the same time also simplify division and exponentiation), by finding a model which transforms multiplication into addition.
    [Show full text]
  • Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS
    Irrational Numbers Unit 4 Lesson 6 IRRATIONAL NUMBERS Students will be able to: Understand the meanings of Irrational Numbers Key Vocabulary: • Irrational Numbers • Examples of Rational Numbers and Irrational Numbers • Decimal expansion of Irrational Numbers • Steps for representing Irrational Numbers on number line IRRATIONAL NUMBERS A rational number is a number that can be expressed as a ratio or we can say that written as a fraction. Every whole number is a rational number, because any whole number can be written as a fraction. Numbers that are not rational are called irrational numbers. An Irrational Number is a real number that cannot be written as a simple fraction or we can say cannot be written as a ratio of two integers. The set of real numbers consists of the union of the rational and irrational numbers. If a whole number is not a perfect square, then its square root is irrational. For example, 2 is not a perfect square, and √2 is irrational. EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Rational Number The number 7 is a rational number because it can be written as the 7 fraction . 1 The number 0.1111111….(1 is repeating) is also rational number 1 because it can be written as fraction . 9 EXAMPLES OF RATIONAL NUMBERS AND IRRATIONAL NUMBERS Examples of Irrational Numbers The square root of 2 is an irrational number because it cannot be written as a fraction √2 = 1.4142135…… Pi(휋) is also an irrational number. π = 3.1415926535897932384626433832795 (and more...) 22 The approx. value of = 3.1428571428571..
    [Show full text]
  • Hypercomplex Algebras and Their Application to the Mathematical
    Hypercomplex Algebras and their application to the mathematical formulation of Quantum Theory Torsten Hertig I1, Philip H¨ohmann II2, Ralf Otte I3 I tecData AG Bahnhofsstrasse 114, CH-9240 Uzwil, Schweiz 1 [email protected] 3 [email protected] II info-key GmbH & Co. KG Heinz-Fangman-Straße 2, DE-42287 Wuppertal, Deutschland 2 [email protected] March 31, 2014 Abstract Quantum theory (QT) which is one of the basic theories of physics, namely in terms of ERWIN SCHRODINGER¨ ’s 1926 wave functions in general requires the field C of the complex numbers to be formulated. However, even the complex-valued description soon turned out to be insufficient. Incorporating EINSTEIN’s theory of Special Relativity (SR) (SCHRODINGER¨ , OSKAR KLEIN, WALTER GORDON, 1926, PAUL DIRAC 1928) leads to an equation which requires some coefficients which can neither be real nor complex but rather must be hypercomplex. It is conventional to write down the DIRAC equation using pairwise anti-commuting matrices. However, a unitary ring of square matrices is a hypercomplex algebra by definition, namely an associative one. However, it is the algebraic properties of the elements and their relations to one another, rather than their precise form as matrices which is important. This encourages us to replace the matrix formulation by a more symbolic one of the single elements as linear combinations of some basis elements. In the case of the DIRAC equation, these elements are called biquaternions, also known as quaternions over the complex numbers. As an algebra over R, the biquaternions are eight-dimensional; as subalgebras, this algebra contains the division ring H of the quaternions at one hand and the algebra C ⊗ C of the bicomplex numbers at the other, the latter being commutative in contrast to H.
    [Show full text]
  • 0.999… = 1 an Infinitesimal Explanation Bryan Dawson
    0 1 2 0.9999999999999999 0.999… = 1 An Infinitesimal Explanation Bryan Dawson know the proofs, but I still don’t What exactly does that mean? Just as real num- believe it.” Those words were uttered bers have decimal expansions, with one digit for each to me by a very good undergraduate integer power of 10, so do hyperreal numbers. But the mathematics major regarding hyperreals contain “infinite integers,” so there are digits This fact is possibly the most-argued- representing not just (the 237th digit past “Iabout result of arithmetic, one that can evoke great the decimal point) and (the 12,598th digit), passion. But why? but also (the Yth digit past the decimal point), According to Robert Ely [2] (see also Tall and where is a negative infinite hyperreal integer. Vinner [4]), the answer for some students lies in their We have four 0s followed by a 1 in intuition about the infinitely small: While they may the fifth decimal place, and also where understand that the difference between and 1 is represents zeros, followed by a 1 in the Yth less than any positive real number, they still perceive a decimal place. (Since we’ll see later that not all infinite nonzero but infinitely small difference—an infinitesimal hyperreal integers are equal, a more precise, but also difference—between the two. And it’s not just uglier, notation would be students; most professional mathematicians have not or formally studied infinitesimals and their larger setting, the hyperreal numbers, and as a result sometimes Confused? Perhaps a little background information wonder .
    [Show full text]
  • 1.1 the Real Number System
    1.1 The Real Number System Types of Numbers: The following diagram shows the types of numbers that form the set of real numbers. Definitions 1. The natural numbers are the numbers used for counting. 1, 2, 3, 4, 5, . A natural number is a prime number if it is greater than 1 and its only factors are 1 and itself. A natural number is a composite number if it is greater than 1 and it is not prime. Example: 5, 7, 13,29, 31 are prime numbers. 8, 24, 33 are composite numbers. 2. The whole numbers are the natural numbers and zero. 0, 1, 2, 3, 4, 5, . 3. The integers are all the whole numbers and their additive inverses. No fractions or decimals. , -3, -2, -1, 0, 1, 2, 3, . An integer is even if it can be written in the form 2n , where n is an integer (if 2 is a factor). An integer is odd if it can be written in the form 2n −1, where n is an integer (if 2 is not a factor). Example: 2, 0, 8, -24 are even integers and 1, 57, -13 are odd integers. 4. The rational numbers are the numbers that can be written as the ratio of two integers. All rational numbers when written in their equivalent decimal form will have terminating or repeating decimals. 1 2 , 3.25, 0.8125252525 …, 0.6 , 2 ( = ) 5 1 1 5. The irrational numbers are any real numbers that can not be represented as the ratio of two integers.
    [Show full text]
  • Fibonacci and -Lucas Numbers
    Hindawi Publishing Corporation Chinese Journal of Mathematics Volume 2013, Article ID 107145, 7 pages http://dx.doi.org/10.1155/2013/107145 Research Article Incomplete -Fibonacci and -Lucas Numbers José L. Ramírez Instituto de Matematicas´ y sus Aplicaciones, Universidad Sergio Arboleda, Colombia Correspondence should be addressed to JoseL.Ram´ ´ırez; [email protected] Received 10 July 2013; Accepted 11 September 2013 Academic Editors: B. Li and W. van der Hoek Copyright © 2013 JoseL.Ram´ ´ırez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We define the incomplete k-Fibonacci and k-Lucas numbers; we study the recurrence relations and some properties of these numbers. 1. Introduction In [6],theauthorsdefinetheincompleteFibonacciandLucas -numbers. Also the authors define the incomplete bivariate Fibonacci numbers and their generalizations have many Fibonacci and Lucas -polynomials in [7]. interesting properties and applications to almost every field On the other hand, many kinds of generalizations of of science and art (e.g., see [1]). Fibonacci numbers are Fibonaccinumbershavebeenpresentedintheliterature.In defined by the recurrence relation particular, a generalization is the -Fibonacci Numbers. For any positive real number ,the-Fibonacci sequence, 0 =0, 1 =1, +1 = +−1,⩾1. (1) say {,}∈N, is defined recurrently by There exist a lot of properties about Fibonacci numbers. =0, =1, = + ,⩾1. In particular, there is a beautiful combinatorial identity to ,0 ,1 ,+1 , ,−1 (4) Fibonacci numbers [1] In [8], -Fibonacci numbers were found by studying ⌊(−1)/2⌋ −−1 = ∑ ( ). the recursive application of two geometrical transformations (2) =0 used in the four-triangle longest-edge (4TLE) partition.
    [Show full text]
  • Computability on the Real Numbers
    4. Computability on the Real Numbers Real numbers are the basic objects in analysis. For most non-mathematicians a real number is an infinite decimal fraction, for example π = 3•14159 ... Mathematicians prefer to define the real numbers axiomatically as follows: · (R, +, , 0, 1,<) is, up to isomorphism, the only Archimedean ordered field satisfying the axiom of continuity [Die60]. The set of real numbers can also be constructed in various ways, for example by means of Dedekind cuts or by completion of the (metric space of) rational numbers. We will neglect all foundational problems and assume that the real numbers form a well-defined set R with all the properties which are proved in analysis. We will denote the R real line topology, that is, the set of all open subsets of ,byτR. In Sect. 4.1 we introduce several representations of the real numbers, three of which (and the equivalent ones) will survive as useful. We introduce a rep- n n resentation ρ of R by generalizing the definition of the main representation ρ of the set R of real numbers. In Sect. 4.2 we discuss the computable real numbers. Sect. 4.3 is devoted to computable real functions. We show that many well known functions are computable, and we show that partial sum- mation of sequences is computable and that limit operator on sequences of real numbers is computable, if a modulus of convergence is given. We also prove a computability theorem for power series. Convention 4.0.1. We still assume that Σ is a fixed finite alphabet con- taining all the symbols we will need.
    [Show full text]
  • 1 the Real Number Line
    Unit 2 Real Number Line and Variables Lecture Notes Introductory Algebra Page 1 of 13 1 The Real Number Line There are many sets of numbers, but important ones in math and life sciences are the following • The integers Z = f:::; −4; −3; −2; −1; 0; 1; 2; 3; 4;:::g. • The positive integers, sometimes called natural numbers, N = f1; 2; 3; 4;:::g. p • Rational numbers are any number that can be expressed as a fraction where p and q 6= 0 are integers. Q q Note that every integer is a rational number! • An irrational number is a number that is not rational. p Examples of irrational numbers are 2 ∼ 1:41421 :::, e ∼ 2:71828 :::, π ∼ 3:14159265359 :::, and less familiar ones like Euler's constant γ ∼ 0:577215664901532 :::. The \:::" here represent that the number is a nonrepeating, nonterminating decimal. • The real numbers R contain all the integer number, rational numbers, and irrational numbers. The real numbers are usually presented as a real number line, which extends forever to the left and right. Note the real numbers are not bounded above or below. To indicate that the real number line extends forever in the positive direction, we use the terminology infinity, which is written as 1, and in the negative direction the real number line extends to minus infinity, which is denoted −∞. The symbol 1 is used to say that the real numbers extend without bound (for any real number, there is a larger real number and a smaller real number). Infinity is a somewhat tricky concept, and you will learn more about it in precalculus and calculus.
    [Show full text]
  • Real Number System Chart with Examples
    Real Number System Chart With Examples If eighteenth or sightless Rodney usually goffers his beguilers clamor uncomplainingly or elates apiece and fertilely, how ulnar is Obadiah? Is Wolfie bearable when Jameson impregnated repeatedly? Visional Sherman sometimes conning any judges idolised sloppily. In learning resources for numbers system real with examples of its multiplicative inverse of the specified by finding them. The same thing is true for operations in mathematics. Multiplication on the number line is subtler than addition. An algorithm is a recipe for computation. As illustrated below, there does not exist any real number that is neither rational nor irrational. Play a Live Game together or use Homework Mode. Why not create one? To convert a percent to a fraction, and an example. In a computation where more than one operation is involved, whole numbers, Whole numbers and Integers and glues or staples the shapes onto the notes page before adding examples and definitions. Use the properties of real numbers to rewrite and simplify each expression. Together they form the integers. The sum of a rational number and an irrational number is ____________. When evaluating a mathematical expression, could not survive without them. How much money does Fred keep? Monitor progress by class and share updates with parents. To picture to take this question did use as with real number system examples, hence they use cookies to use long division to use the associative property of its key is no cost to. Rational and irrational numbers are easy to explain. We use other bases all the time, and so on.
    [Show full text]
  • Alyssa Farrell
    Recursive Patterns and Polynomials Alyssa Farrell Iowa State University MSM Creative Component Summer 2006 Heather Thompson, Major Professor Irvin Hentzel, Major Professor Patricia Leigh, Committee Member Farrell 2 Patterns have been studied for centuries. Some of these patterns have been graced with names of the people who are most remembered for their work with them or related topics. Among these are the Fibonacci numbers, Lucas numbers, Pascal’s Triangle, Chebyshev’s Polynomials, and Cardan Polynomials. Patterns are used as building blocks in education from newborn babies and young children all the way through high school and beyond into post-secondary work. We were introduced as early as infancy to learning to recognize patterns, meaning both similarities and differences, when learning colors, shapes, the alphabet, and numbers. As we grew, we were introduced to new mathematical patterns like learning to count by 2’s, or 5’s. As we were more formally introduced to shapes, we learned to recognize that a square has four sides the same length, but unlike a kite, the square has four angles that are 90 degrees. Then we learned to add, subtract, multiply, and divide. All of these necessary operations were taught using a pattern-approach. Do you remember learning to read? We were taught to recognize patterns of letters which made certain sounds and that when certain sounds were strung together they made words. These patterns continued to be used as we developed higher in our math education. Teachers in junior high or middle school introduced us to an abstract idea of letting letters represent numbers as variables.
    [Show full text]
  • Infinitesimals
    Infinitesimals: History & Application Joel A. Tropp Plan II Honors Program, WCH 4.104, The University of Texas at Austin, Austin, TX 78712 Abstract. An infinitesimal is a number whose magnitude ex- ceeds zero but somehow fails to exceed any finite, positive num- ber. Although logically problematic, infinitesimals are extremely appealing for investigating continuous phenomena. They were used extensively by mathematicians until the late 19th century, at which point they were purged because they lacked a rigorous founda- tion. In 1960, the logician Abraham Robinson revived them by constructing a number system, the hyperreals, which contains in- finitesimals and infinitely large quantities. This thesis introduces Nonstandard Analysis (NSA), the set of techniques which Robinson invented. It contains a rigorous de- velopment of the hyperreals and shows how they can be used to prove the fundamental theorems of real analysis in a direct, natural way. (Incredibly, a great deal of the presentation echoes the work of Leibniz, which was performed in the 17th century.) NSA has also extended mathematics in directions which exceed the scope of this thesis. These investigations may eventually result in fruitful discoveries. Contents Introduction: Why Infinitesimals? vi Chapter 1. Historical Background 1 1.1. Overview 1 1.2. Origins 1 1.3. Continuity 3 1.4. Eudoxus and Archimedes 5 1.5. Apply when Necessary 7 1.6. Banished 10 1.7. Regained 12 1.8. The Future 13 Chapter 2. Rigorous Infinitesimals 15 2.1. Developing Nonstandard Analysis 15 2.2. Direct Ultrapower Construction of ∗R 17 2.3. Principles of NSA 28 2.4. Working with Hyperreals 32 Chapter 3.
    [Show full text]
  • Infinitesimals, and Sub-Infinities
    Gauge Institute Journal, Volume 6, No. 4, November 2010 H. Vic Dannon Infinitesimals H. Vic Dannon [email protected] May, 2010 Abstract Logicians never constructed Leibnitz infinitesimals and proved their being on a line. They only postulated it. We construct infinitesimals, and investigate their properties. Each real number α can be represented by a Cauchy sequence of rational numbers, (rrr123 , , ,...) so that rn → α . The constant sequence (ααα , , ,...) is a constant hyper-real. Any totally ordered set of positive, monotonically decreasing to zero sequences (ιιι123 , , ,...) constitutes a family of positive infinitesimal hyper-reals. The positive infinitesimals are smaller than any positive real number, yet strictly greater than zero. Their reciprocals ()111, , ,... are the positive infinite hyper- ιιι123 reals. The positive infinite hyper-reals are greater than any real number, yet strictly smaller than infinity. 1 Gauge Institute Journal, Volume 6, No. 4, November 2010 H. Vic Dannon The negative infinitesimals are greater than any negative real number, yet strictly smaller than zero. The negative infinite hyper-reals are smaller than any real number, yet strictly greater than −∞. The sum of a real number with an infinitesimal is a non- constant hyper-real. The Hyper-reals are the totality of constant hyper-reals, a family of positive and negative infinitesimals, their associated infinite hyper-reals, and non-constant hyper- reals. The hyper-reals are totally ordered, and aligned along a line: the Hyper-real Line. That line includes the real numbers separated by the non- constant hyper-reals. Each real number is the center of an interval of hyper-reals, that includes no other real number.
    [Show full text]