Best Rational Approximations of an Irrational Number
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VECTOR ALGEBRAIC THEORY of ARITHMETIC: Part 2
7/25/13 VECTOR ALGEBRAIC THEORY OF ARITHMETIC: Part 2 VECTOR ALGEBRAIC THEORY OF ARITHMETIC: Part 2 Clyde L. Greeno The American Institute for the Improvement of MAthematics LEarning and Instruction (a.k.a. The MALEI Mathematics Institute) P.O. Box 54845 Tulsa, OK 74154 [email protected] ABSTRACT: Virtually unrecognized by curricular educators is that the theory of Arabic/base-number arithmetic relies on equivalence classes of whole-scalar vectors. Sooner or later the recognition will catalyze a major re-formation of scholastic curricular mathematics – and also of the mathematics education of all teachers of mathematics. Part 1 of this survey appears in the 2005 Proceedings of the MAA’s OK-AR Section Meeting. It sketches the vector-algebraic foundations of the Hindu-Arabic arithmetic for whole-number calculations – and that theory easily generalizes to include all base-number systems. As promised in Part 1, this Part 2 sequel outlines that theory’s extension to cover the whole-scalar vector arithmetic of fractions – which includes the arithmetics of finite decimal-points and of all other n-al points. Of course, the extension to infinite-dimensional whole-scalar vectors leads off to the n-al arithmetics for the reals. SUMMARY OF PART 1. Part 1 opened with an alphabetic construction of the Peano line of simple Arabic digit-strings – devoid of any reference to numbers – and extended that construction to achieve an Arabic numbers kind of whole-numbers system. Part 1 concluded by disclosing how the whole-scalars, vector-algebraic structure which underlies the Arabic arithmetic for whole numbers can serve as a mathematical basis for greatly improving the instructional effectiveness of curricular education in the arithmetic of whole numbers. -
Rationale of the Chakravala Process of Jayadeva and Bhaskara Ii
HISTORIA MATHEMATICA 2 (1975) , 167-184 RATIONALE OF THE CHAKRAVALA PROCESS OF JAYADEVA AND BHASKARA II BY CLAS-OLOF SELENIUS UNIVERSITY OF UPPSALA SUMMARIES The old Indian chakravala method for solving the Bhaskara-Pell equation or varga-prakrti x 2- Dy 2 = 1 is investigated and explained in detail. Previous mis- conceptions are corrected, for example that chakravgla, the short cut method bhavana included, corresponds to the quick-method of Fermat. The chakravala process corresponds to a half-regular best approximating algorithm of minimal length which has several deep minimization properties. The periodically appearing quantities (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, etc.) are correctly understood only with the new theory. Den fornindiska metoden cakravala att l~sa Bhaskara- Pell-ekvationen eller varga-prakrti x 2 - Dy 2 = 1 detaljunders~ks och f~rklaras h~r. Tidigare missuppfatt- 0 ningar r~ttas, sasom att cakravala, genv~gsmetoden bhavana inbegripen, motsvarade Fermats snabbmetod. Cakravalaprocessen motsvarar en halvregelbunden b~st- approximerande algoritm av minimal l~ngd med flera djupt liggande minimeringsegenskaper. De periodvis upptr~dande storheterna (jyestha-m~la, kanistha-mula, ksepaka, kuttakara, os~) blir forstaellga0. 0 . f~rst genom den nya teorin. Die alte indische Methode cakrav~la zur Lbsung der Bhaskara-Pell-Gleichung oder varga-prakrti x 2 - Dy 2 = 1 wird hier im einzelnen untersucht und erkl~rt. Fr~here Missverst~ndnisse werden aufgekl~rt, z.B. dass cakrav~la, einschliesslich der Richtwegmethode bhavana, der Fermat- schen Schnellmethode entspreche. Der cakravala-Prozess entspricht einem halbregelm~ssigen bestapproximierenden Algorithmus von minimaler L~nge und mit mehreren tief- liegenden Minimierungseigenschaften. Die periodisch auftretenden Quantit~ten (jyestha-mfila, kanistha-mfila, ksepaka, kuttak~ra, usw.) werden erst durch die neue Theorie verst~ndlich. -
IJR-1, Mathematics for All ... Syed Samsul Alam
January 31, 2015 [IISRR-International Journal of Research ] MATHEMATICS FOR ALL AND FOREVER Prof. Syed Samsul Alam Former Vice-Chancellor Alaih University, Kolkata, India; Former Professor & Head, Department of Mathematics, IIT Kharagpur; Ch. Md Koya chair Professor, Mahatma Gandhi University, Kottayam, Kerala , Dr. S. N. Alam Assistant Professor, Department of Metallurgical and Materials Engineering, National Institute of Technology Rourkela, Rourkela, India This article briefly summarizes the journey of mathematics. The subject is expanding at a fast rate Abstract and it sometimes makes it essential to look back into the history of this marvelous subject. The pillars of this subject and their contributions have been briefly studied here. Since early civilization, mathematics has helped mankind solve very complicated problems. Mathematics has been a common language which has united mankind. Mathematics has been the heart of our education system right from the school level. Creating interest in this subject and making it friendlier to students’ right from early ages is essential. Understanding the subject as well as its history are both equally important. This article briefly discusses the ancient, the medieval, and the present age of mathematics and some notable mathematicians who belonged to these periods. Mathematics is the abstract study of different areas that include, but not limited to, numbers, 1.Introduction quantity, space, structure, and change. In other words, it is the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. Mathematicians seek out patterns and formulate new conjectures. They resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity. -
Mathematics Standards Review and Revision Committee
Proposed for SBE Adoption – 2019-12-31 – Page 1 Mathematics Standards Review and Revision Committee Chairperson Joanie Funderburk President Colorado Council of Teachers of Mathematics Members Lisa Bejarano Lanny Hass Teacher Principal Aspen Valley High School Thompson Valley High School Academy District 20 Thompson School District Michael Brom Ken Jensen Assessment and Accountability Teacher on Mathematics Instructional Coach Special Assignment Aurora Public Schools Lewis-Palmer School District 38 Lisa Rogers Ann Conaway Student Achievement Coordinator Teacher Fountain-Fort Carson School District 8 Palisade High School Mesa County Valley School District 51 David Sawtelle K-12 Mathematics Specialist Dennis DeBay Colorado Springs School District 11 Mathematics Education Faculty University of Colorado Denver T. Vail Shoultz-McCole Early Childhood Program Director Greg George Colorado Mesa University K-12 Mathematics Coordinator St. Vrain Valley School District Ann Summers K-12 Mathematics and Intervention Specialist Cassie Harrelson Littleton Public Schools Director of Professional Practice Colorado Education Association This document was updated in December 2019 to reflect typographic and other corrections made to Standards Online. State Board of Education and Colorado Department of Education Colorado State Board of Education CDE Standards and Instructional Support Office Angelika Schroeder (D, Chair) 2nd Congressional District Karol Gates Boulder Director Joyce Rankin (R, Vice Chair) Carla Aguilar, Ph.D. 3rd Congressional District Music Content Specialist Carbondale Ariana Antonio Steve Durham (R) Standards Project Manager 5th Congressional District Colorado Springs Joanna Bruno, Ph.D. Science Content Specialist Valentina (Val) Flores (D) 1st Congressional District Lourdes (Lulu) Buck Denver World Languages Content Specialist Jane Goff (D) Donna Goodwin, Ph.D. 7th Congressional District Visual Arts Content Specialist Arvada Stephanie Hartman, Ph.D. -
Quantum Algorithm for Solving Pell's Equation
CS682 Project Report Hallgren's Efficient Quantum Algorithm for Solving Pell's Equation by Ashish Dwivedi (17111261) under the guidance of Prof Rajat Mittal November 15, 2017 Abstract In this project we aim to study an excellent result on the quantum computation model. Hallgren in 2002 [Hal07] showed that a seemingly difficult problem in classical computa- tion model, solving Pell's equation, is efficiently solvable in quantum computation model. This project explains the idea of Hallgren's quantum algorithm to solve Pell's equation. Contents 0.1 Introduction..................................1 0.2 Background..................................1 0.3 Hallgren's periodic function.........................3 0.4 Quantum Algorithm to find irrational period on R .............4 0.4.1 Discretization of the periodic function...............4 0.4.2 Quantum Algorithm.........................5 0.4.3 Classical Post Processing.......................7 0.5 Summary...................................8 0.1 Introduction In this project we study one of those problems which have no known efficient solution on the classical computation model and are assumed hard to be solved efficiently, but in recent years efficient quantum algorithms have been found for them. One of these problems is the problem of finding solution to Pell's equation. Pell's equation are equations of the form x2 − dy2 = 1, where d is some given non- square positive integer and x and y are indeterminate. Clearly, (1; 0) is a solution of this equation which are called trivial solutions. We want some non-trivial integer solutions satisfying these equations. There is no efficient algorithm known for solving such equations on classical compu- tation model. But in 2002, Sean Hallgren [Hal07] gave an algorithm on quantum compu- tation model which takes time only polynomial in input size (log d). -
Directional Morphological Filtering
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 23, NO. 11, NOVEMBER 2001 1313 Directional Morphological Filtering Pierre Soille and Hugues Talbot AbstractÐWe show that a translation invariant implementation of min/max filters along a line segment of slope in the form of an irreducible fraction dy=dx can be achieved at the cost of 2 k min/max comparisons per image pixel, where k max jdxj; jdyj. Therefore, for a given slope, the computation time is constant and independent of the length of the line segment. We then present the notion of periodic moving histogram algorithm. This allows for a similar performance to be achieved in the more general case of rank filters and rank-based morphological filters. Applications to the filtering of thin nets and computation of both granulometries and orientation fields are detailed. Finally, two extensions are developed. The first deals with the decomposition of discrete disks and arbitrarily oriented discrete rectangles, while the second concerns min/max filters along gray tone periodic line segments. Index TermsÐImage analysis, mathematical morphology, rank filters, directional filters, periodic line, discrete geometry, granulometry, orientation field, radial decomposition. æ 1INTRODUCTION IMILAR to the perception of the orientation of line Section 3. Performance evaluation and comparison with Ssegments by the human brain [1], [2], computer image other approaches are carried out in Section 4. Applications processing of oriented image structures often requires a to the filtering of thin networks and the computation of bank of directional filters or template masks, each being linear granulometries and orientation fields are presented sensitive to a specific range of orientations [3], [4], [5], [6], in Section 5. -
Equation Solving in Indian Mathematics
U.U.D.M. Project Report 2018:27 Equation Solving in Indian Mathematics Rania Al Homsi Examensarbete i matematik, 15 hp Handledare: Veronica Crispin Quinonez Examinator: Martin Herschend Juni 2018 Department of Mathematics Uppsala University Equation Solving in Indian Mathematics Rania Al Homsi “We owe a lot to the ancient Indians teaching us how to count. Without which most modern scientific discoveries would have been impossible” Albert Einstein Sammanfattning Matematik i antika och medeltida Indien har påverkat utvecklingen av modern matematik signifi- kant. Vissa människor vet de matematiska prestationer som har sitt urspring i Indien och har haft djupgående inverkan på matematiska världen, medan andra gör det inte. Ekvationer var ett av de områden som indiska lärda var mycket intresserade av. Vad är de viktigaste indiska bidrag i mate- matik? Hur kunde de indiska matematikerna lösa matematiska problem samt ekvationer? Indiska matematiker uppfann geniala metoder för att hitta lösningar för ekvationer av första graden med en eller flera okända. De studerade också ekvationer av andra graden och hittade heltalslösningar för dem. Denna uppsats presenterar en litteraturstudie om indisk matematik. Den ger en kort översyn om ma- tematikens historia i Indien under många hundra år och handlar om de olika indiska metoderna för att lösa olika typer av ekvationer. Uppsatsen kommer att delas in i fyra avsnitt: 1) Kvadratisk och kubisk extraktion av Aryabhata 2) Kuttaka av Aryabhata för att lösa den linjära ekvationen på formen 푐 = 푎푥 + 푏푦 3) Bhavana-metoden av Brahmagupta för att lösa kvadratisk ekvation på formen 퐷푥2 + 1 = 푦2 4) Chakravala-metoden som är en annan metod av Bhaskara och Jayadeva för att lösa kvadratisk ekvation 퐷푥2 + 1 = 푦2. -
Map Math Instruction Sheet
Converting Units � � � from to do this Examples � � � milimeters meters 1,000mm = 1m An expanded mm m divide by 1000 4,321mm = 4.321m � � meters milimeters 1m = 1,000mm � tutorial on using m mm multiply by 1000 4.3m = 4,300mm map math � meters kilometers 1,000m = 1km � divide by 1000 � is available at m km 4,300m = 4.3km kilometers meters 1km = 1,000m � www.MapTools.com multiply by 1000 � km m 4.3km = 4,300m � inches feet 12 in. = 1 ft. � in. ft. divide by 12 48 in. = 4 ft. � � feet inches multiply by 12 1 ft. = 12 in. � ft. in. 4 ft. = 48 in. � � Map scales feet miles 5,280 ft. = 1 mi. ft. mi. divide by 5280 7,392 ft. = 1.4 mi. � grid tools, rulers, � � miles feet 1 mi. = 5,280 ft. and other tools mi. ft. multiply by 5280 1.4 mi. = 7,392 ft. � � for measuring inches milimeters 1 in. = 24.5mm � multiply by 24.5 map coordinates in. mm 12 in. = 294mm � � milimeters inches divide by 24.5 24.5mm = 1 in. � are available mm in. 294mm = 12 in. miles kilometers multiply by 1 mi. = 1.6093km � � mi. km 1.6093 5 mi. = 8.0465km kilometers miles divide by 1.6093 1.6093km = 1 mi. � km mi. 1km = 0.6213 mi. � Check with your Map Distance v.s. Terrain Distance � � local map store or visit Distances measured on a map assume a flat surface and do not account or the Map Math � additional distance introduced as you climb up and down over the terrain. -
Hardin Middle School Math Cheat Sheets
Name: __________________________________________ 7th Grade Math Teacher: ______________________________ 8th Grade Math Teacher: ______________________________ Hardin Middle School Math Cheat Sheets You will be given only one of these books. If you lose the book, it will cost $5 to replace it. Compiled by Shirk &Harrigan - Updated May 2013 Alphabetized Topics Pages Pages Area 32 Place Value 9, 10 Circumference 32 Properties 12 22, 23, Comparing 23, 26 Proportions 24, 25, 26 Pythagorean Congruent Figures 35 36 Theorem 14, 15, 41 Converting 23 R.A.C.E. Divisibility Rules 8 Range 11 37, 38, 25 Equations 39 Rates Flow Charts Ratios 25, 26 32, 33, 10 Formulas 34 Rounding 18, 19, 20, 21, 35 Fractions 22, 23, Scale Factor 24 Geometric Figures 30, 31 Similar Figures 35 Greatest Common 22 Slide Method 22 Factor (GF or GCD) Inequalities 40 Substitution 29 Integers 18, 19 Surface Area 33 Ladder Method 22 Symbols 5 Least Common Multiple 22 Triangles 30, 36 (LCM or LCD) Mean 11 Variables 29 Median 11 Vocabulary Words 43 Mode 11 Volume 34 Multiplication Table 6 Word Problems 41, 42 Order of 16, 17 Operations 23, 24, Percent 25, 26 Perimeter 32 Table of Contents Pages Cheat Sheets 5 – 42 Math Symbols 5 Multiplication Table 6 Types of Numbers 7 Divisibility Rules 8 Place Value 9 Rounding & Comparing 10 Measures of Central Tendency 11 Properties 12 Coordinate Graphing 13 Measurement Conversions 14 Metric Conversions 15 Order of Operations 16 – 17 Integers 18 – 19 Fraction Operations 20 – 21 Ladder/Slide Method 22 Converting Fractions, Decimals, & 23 Percents Cross Products 24 Ratios, Rates, & Proportions 25 Comparing with Ratios, Percents, 26 and Proportions Solving Percent Problems 27 – 28 Substitution & Variables 29 Geometric Figures 30 – 31 Area, Perimeter, Circumference 32 Surface Area 33 Volume 34 Congruent & Similar Figures 35 Pythagorean Theorem 36 Hands-On-Equation 37 Understanding Flow Charts 38 Solving Equations Mathematically 39 Inequalities 40 R.A.C.E. -
International Conference on the 900Th Birth Anniversary of Bhāskarācārya
International Conference on the 900th Birth Anniversary of Bhāskarācārya 19, 20, 21 September 2014 ABSTRACTS Vidya Prasarak Mandai Vishnu Nagar, Naupada, Thane 400602 2 I BHĀSKARĀCĀRYA’S LIFE AND TIMES A. P. JAMKHEDKAR, Mumbai. ‘Learning and Patronage in 12th -13th Century A.D.: Bhaskarācarya and the Śāndilya Family, A Case Study’ … … … … … 5 II BHASKARĀCĀRYA’S POETIC GENIUS Pierre-Sylvain FILLIOZAT, Paris. ‘The poetical face of the mathematical and astronomical works of Bhāskarācārya’ … … … … … … … … 7 K. S. BALASUBRAMANIAN, Chennai. ‘Bhāskarācārya as a Poet’ … … … … 8 GEETHAKUMARI, K. K., Calicut. ‘Contribution of Līlāvatī to Prosody’ … … … 9 Poonam GHAI, Dhampur. ‘Līlāvatī men Kāvya-saundarya’ … … … … … … 10 III THE LĪLĀVATĪ K. RAMASUBRAMANIAN, Mumbai. ‘The Līlā of the Līlāvatī: A Beautiful Blend of Arithmetic, Geometry and Poetry’ … … … … … … … … … 11 PADMAVATAMMA, Mysore. ‘A Comparative Study of Bhāskarācārya’s Līlāvatī and Mahāvīrācārya’s Gaṇitasārasaṁgraha’… … … … … … … … 11 K. RAMAKALYANI, Chennai. ‘Gaṇeśa Daivajña’s Upapattis on Līlāvatī’ … … 12 Anil NARAYANAN, N., Calicut. ‘Parameswara’s Unpublished Commentary on Lilavati: Perceptions and Problems’ … … … … … … … … … … … 13 P. RAJASEKHAR & LAKSHMI, V. Calicut. ‘Līlāvatī and Kerala School of Mathematics’ 14 N. K. Sundareswaran & P. M. Vrinda, Calicut. ‘Malayalam Commentaries on Līlāvatī – A Survey of Manuscripts in Kerala’ … … … … … … 15 Shrikrishna G. DANI, Mumbai. ‘Mensuration of Quadrilaterals in Lilavati’ … … 16 Takanori KUSUBA, Osaka. ‘Aṅkapāśa in the Līlāvatī’ … … … … … … … 17 Medha Srikanth LIMAYE, Mumbai. Use of Bhūta-Saṅkhyās (Object Numerals) in Līlāvatī of Bhāskarācārya’ … … … … … … … … … … 17 Sreeramula Rajeswara SARMA, Düsseldorf. ‘The Legend of Līlāvatī’ … … … 17 IV THE BĪJAGAṆITA Sita Sundar RAM, Chennai. ‘Bījagaṇita of Bhāskara’ … … … … … … … 19 K. VIDYUTA, Chennai. ‘Sūryaprakāśa of Sūryadāsa – a Review’ … … … … 20 Veena SHINDE-DEORE & Manisha M. ACHARYA, Mumbai. ‘Bhaskaracharya and Varga Prakriti: the equations of the type ax2 + b = cy2’ … … … … … 20 K. -
Objectives: Definition of a Rational Number1 Class Rational
Chulalongkorn University Name _____________________ International School of Engineering Student ID _________________ Department of Computer Engineering Station No. _________________ 2140105 Computer Programming Lab. Date ______________________ Lab 6 – Pre Midterm Objectives: • Learn to use Eclipse as Java integrated development environment. • Practice basic java programming. • Practice all topics since the beginning of the course. • Practice for midterm examination. Definition of a Rational number1 In mathematics, a rational number is a number which can be expressed as a ratio of two integers. Non‐integer rational numbers (commonly called fractions) are usually written as the fraction , where b is not zero. 3 2 1 Each rational number can be written in infinitely many forms, such as 6 4 2, but is said to be in simplest form is when a and b have no common divisors except 1. Every non‐zero rational number has exactly one simplest form of this type with a positive denominator. A fraction in this simplest form is said to be an irreducible fraction or a fraction in reduced form. Class Rational An object of this class will have two integer numbers: numerator and denominator (e.g. for an object represents will have a as its numerator and b as its denominator). Each object will store its value as an irreducible fraction (e.g. ) This class has three constructors: • Constructor that has no argument: create an object represents zero ( ) • Constructor that has two integer arguments: numerator and denominator, create a rational number object in reduced form. • Constructor that has one Rational argument: create a rational number object which it numerator and denominator equal to the argument. -
Diophantine Analysis
The Second International Conference “Problems of Cybernetics and Informatics” September 10-12, 2008, Baku, Azerbaijan. The plenary paper www.pci2008.science.az/5/01.pdf DIOPHANTINE ANALYSIS Djangir Babayev Cox Associates, Denver, CO, USA 4834 Macintosh Place, Boulder CO, USA, 80301 [email protected] Diophantus was a Greek Mathematician sometimes known as”the father of algebra”, who lived in 3rd century of AD (200-284). He is best known for his Arithmetica. Diophantus studied solving equations or systems of equations with numbers of equations less than number of variables for finding integer solutions and was first to try to develop algebraic notations. His works had enormous influence on the development of Number Theory. Modern Diophantine Analysis embraces mathematics of solving equations and problems in integer numbers. While its roots reach back to the third century, diophantine analysis continues to be an extremely active and powerful area of number theory. Many diophantine problems have simple formulations, but they can be extremely difficult to attack, and many open problems and conjectures remain. Division algorithm. If a>b>0, integers, then a=bq+r, where q and r are integers and b>r ≥ 0. Let gcd(a,b) be the greatest common divisor of integers a and b. Theorem 1. gcd(a,b) = gcd(b,r) P R O O F. Suppose gcd(a,b) =d. Then a=sd and b=td. r=a-bq=sd-bqd=(s-bq)d. This implies gcd(b,r)=d and completes the proof. ▓ Euclids algorithm for determining gcd(a,b) works based on gcd(a,b) = gcd(b,r) , and b=r q2 +r2 , r> r2 ≥ 0 by repeatedly applying the division algorithm to the last two remainders.