Constructions of the Real Numbers
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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. -
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.. -
MATH 162, SHEET 8: the REAL NUMBERS 8A. Construction of The
MATH 162, SHEET 8: THE REAL NUMBERS This sheet is concerned with proving that the continuum R is an ordered field. Addition and multiplication on R are defined in terms of addition and multiplication on Q, so we will use ⊕ and ⊗ for addition and multiplication of real numbers, to make sure that there is no confusion with + and · on Q: 8A. Construction of the real numbers via cuts, Addition and Order Definition 8.1. A subset A of Q is said to be a cut (or Dedekind cut) if it satisfies the following: (a) A 6= Ø and A 6= Q (b) If r 2 A and s 2 Q satisfies s < r; then s 2 A (c) If r 2 A then there is some s 2 Q with s > r; s 2 A: We denote the collection of all cuts by R. Definition 8.2. We define ⊕ on R as follows. Let A; B 2 R be Dedekind cuts. Define A ⊕ B = fa + b j a 2 A and b 2 Bg 0 = fx 2 Q j x < 0g 1 = fx 2 Q j x < 1g: Exercise 8.3. (a) Prove that A ⊕ B; 0; and 1 are all Dedekind cuts. (b) Prove that fx 2 Q j x ≤ 0g is not a Dedekind cut. (c) Prove that fx 2 Q j x < 0g [ fx 2 Q j x2 < 2g is a Dedekind cut. Hint: In Exercise 4.20 last quarter we showed that if x 2 Q; x ≥ 0; and x2 < 2; then there is some δ 2 Q; δ > 0 such that (x + δ)2 < 2: Exercise 8.4. -
Math 104: Introduction to Analysis
Math 104: Introduction to Analysis Contents 1 Lecture 1 3 1.1 The natural numbers . 3 1.2 Equivalence relations . 4 1.3 The integers . 5 1.4 The rational numbers . 6 2 Lecture 2 6 2.1 The real numbers by axioms . 6 2.2 The real numbers by Dedekind cuts . 7 2.3 Properties of R ................................................ 8 3 Lecture 3 9 3.1 Metric spaces . 9 3.2 Topological definitions . 10 3.3 Some topological fundamentals . 10 4 Lecture 4 11 4.1 Sequences and convergence . 11 4.2 Sequences in R ................................................ 12 4.3 Extended real numbers . 12 5 Lecture 5 13 5.1 Compactness . 13 6 Lecture 6 14 6.1 Compactness in Rk .............................................. 14 7 Lecture 7 15 7.1 Subsequences . 15 7.2 Cauchy sequences, complete metric spaces . 16 7.3 Aside: Construction of the real numbers by completion . 17 8 Lecture 8 18 8.1 Taking powers in the real numbers . 18 8.2 \Toolbox" sequences . 18 9 Lecture 9 19 9.1 Series . 19 9.2 Adding, regrouping series . 20 9.3 \Toolbox" series . 21 1 10 Lecture 10 22 10.1 Root and ratio tests . 22 10.2 Summation by parts, alternating series . 23 10.3 Absolute convergence, multiplying and rearranging series . 24 11 Lecture 11 26 11.1 Limits of functions . 26 11.2 Continuity . 26 12 Lecture 12 27 12.1 Properties of continuity . 27 13 Lecture 13 28 13.1 Uniform continuity . 28 13.2 The derivative . 29 14 Lecture 14 31 14.1 Mean value theorem . 31 15 Lecture 15 32 15.1 L'Hospital's Rule . -
HONOURS B.Sc., M.A. and M.MATH. EXAMINATION MATHEMATICS and STATISTICS Paper MT3600 Fundamentals of Pure Mathematics September 2007 Time Allowed : Two Hours
HONOURS B.Sc., M.A. AND M.MATH. EXAMINATION MATHEMATICS AND STATISTICS Paper MT3600 Fundamentals of Pure Mathematics September 2007 Time allowed : Two hours Attempt all FOUR questions [See over 2 1. Let X be a set, and let ≤ be a total order on X. We say that X is dense if for any two x; y 2 X with x < y there exists z 2 X such that x < z < y. In what follows ≤ always denotes the usual ordering on rational numbers. (i) For each of the following sets determine whether it is dense or not: (a) Q+ (positive rationals); (b) N (positive integers); (c) fx 2 Q : 3 ≤ x ≤ 4g; (d) fx 2 Q : 1 ≤ x ≤ 2g [ fx 2 Q : 3 ≤ x ≤ 4g. Justify your assertions. [4] (ii) Let a and b be rational numbers satisfying a < b. To which of the following a + 4b three sets does the number belong: A = fx 2 : x < ag, B = fx 2 : 5 Q Q a < x < bg or C = fx 2 Q : b < xg? Justify your answer. [1] (iii) Consider the set a A = f : a; n 2 ; n ≥ 0g: 5n Z Is A dense? Justify your answer. [2] 1 (iv) Prove that r = 1 is the only positive rational number such that r + is an r integer. [2] 1 (v) How many positive real numbers x are there such that x + is an integer? x Your answer should be one of: 0, 1, 2, 3, . , countably infinite, or uncountably infinite, and you should justify it. [3] 2. (i) Define what it means for a set A ⊆ Q to be a Dedekind cut. -
11 Construction of R
Math 361 Construction of R 11 Construction of R Theorem 11.1. There does not exist q Q such that q2 = 2. 2 Proof. Suppose that q Q satis¯es q2 = 2. Then we can suppose that q > 0 and express 2 q = a=b, where a; b ! are relatively prime. Since 2 a2 = 2 b2 we have that a2 = 2b2: It follows that 2 a; say a = 2c. Hence j 4c2 = 2b2 2c2 = 2b2 This means that 2 b, which contradicts the assumption that a and b are relatively prime. j Theorem 11.2. There exists r R such that r2 = 2. 2 Proof. Consider the continuous function f : [1; 2] R, de¯ned by f(x) = x2. Then f(1) = 1 and f(2) = 4. By the Intermediate Value !Theorem, there exists r [1; 2] such that r2 = 2. 2 Why is the Intermediate Value Theorem true? Intuitively because R \has no holes"... More precisely... De¯nition 11.3. Suppose that A R. r R is an upper bound of A i® a r for all a A. ² 2 · 2 A is bounded above i® there exists an upper bound for A. ² r R is a least upper bound of A i® the following conditions hold: ² 2 { r is an upper bound of A. { If s is an upper bound of A, then r s. · Axiom 11.4 (Completeness). If A R is nonempty and bounded above, then there exists a least upper bound of A. Deepish Fact Completeness Axiom Intermediate Value Theorem. ) 2006/10/25 1 Math 361 Construction of R Remark 11.5. -
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. -
Eudoxos and Dedekind: on the Ancient Greek Theory of Ratios and Its Relation to Modern Mathematics*
HOWARD STEIN EUDOXOS AND DEDEKIND: ON THE ANCIENT GREEK THEORY OF RATIOS AND ITS RELATION TO MODERN MATHEMATICS* 1. THE PHILOSOPHICAL GRAMMAR OF THE CATEGORY OF QUANTITY According to Aristotle, the objects studied by mathematics have no independent existence, but are separated in thought from the substrate in which they exist, and treated as separable - i.e., are "abstracted" by the mathematician. I In particular, numerical attributives or predicates (which answer the question 'how many?') have for "substrate" multi- tudes with a designated unit. 'How many pairs of socks?' has a different answer from 'how many socks?'. (Cf. Metaph. XIV i 1088a5ff.: "One la signifies that it is a measure of a multitude, and number lb that it is a measured multitude and a multitude of measures".) It is reasonable to see in this notion of a "measured multitude" or a "multitude of mea- sures" just that of a (finite) set: the measures or units are what we should call the elements of the set; the requirement that such units be distinguished is precisely what differentiates a set from a mere accumulation or mass. There is perhaps some ambiguity in the quoted passage: the statement, "Number signifies that it is a measured multi- tude", might be taken either to identify numbers with finite sets, or to imply that the subjects numbers are predicated of are finite sets. Euclid's definition - "a number is a multitude composed of units" - points to the former reading (which implies, for example, that there are many two's - a particular knife and fork being one of them). -
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 . -
WHY a DEDEKIND CUT DOES NOT PRODUCE IRRATIONAL NUMBERS And, Open Intervals Are Not Sets
WHY A DEDEKIND CUT DOES NOT PRODUCE IRRATIONAL NUMBERS And, Open Intervals are not Sets Pravin K. Johri The theory of mathematics claims that the set of real numbers is uncountable while the set of rational numbers is countable. Almost all real numbers are supposed to be irrational but there are few examples of irrational numbers relative to the rational numbers. The reality does not match the theory. Real numbers satisfy the field axioms but the simple arithmetic in these axioms can only result in rational numbers. The Dedekind cut is one of the ways mathematics rationalizes the existence of irrational numbers. Excerpts from the Wikipedia page “Dedekind cut” A Dedekind cut is а method of construction of the real numbers. It is a partition of the rational numbers into two non-empty sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. If B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B. The countable partitions of the rational numbers cannot result in uncountable irrational numbers. Moreover, a known irrational number, or any real number for that matter, defines a Dedekind cut but it is not possible to go in the other direction and create a Dedekind cut which then produces an unknown irrational number. 1 Irrational Numbers There is an endless sequence of finite natural numbers 1, 2, 3 … based on the Peano axiom that if n is a natural number then so is n+1. -
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. -
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.