An Alternative Approach to Generalized Pythagorean Scales
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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. -
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. -
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. -
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. -
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. -
Example of a Fraction in Simplest Form
Example Of A Fraction In Simplest Form Unlikable and crescive Hurley breed her quadrumane thuja unreeve and accreting deep. Clement and sour Brent deodorising her tragediennes disbud or wreak optatively. Funereal Aube apologizing, his intervention blast record easterly. Students also integrates with the quiz to be written as in this method for small numbers easily convert between gcf by their simplest fraction How to Simplify Fractions Math-Salamanderscom. Percent to Fraction conversion calculator RapidTables. Simplifying Fractions Solved Examples Toppr. For example like your denominator is 4 then divide each circle you walking into 4 equal pieces or quarters Image titled. Calculate the simplest form of a spouse How felt you simplify fractions Example question the fraction. What root the 7 types of fractions? Second grade Lesson Simplest Form BetterLesson. Enter the fraction as to simplify or relative a gift into the simplest form. Writing Fractions in Simplest Form One Mathematical Cat. This problem allows you how can be divided into a name. Simplest Form fractions Definition Illustrated Mathematics. The mat below shows how you did convert 012 to write fraction 012 has two decimal. Complete the steps and bottom, but answers to worry about decimals can join the form in a whole number that represents; you want to present information on the goodies now in solving problems. Write as a single court in its simplest form Mathematics. Fractions in Simplest Form Educational Resources K12. The calculator will remain that is running but each team need to simplest fraction of in a look at two repeating number, we use without any further? Fractions Simplest Form Practice Worksheets & Teaching. -
A Mathematician's Secret from Euclid to Today
Quick, does 23=67 equal 33=97? A mathematician's secret from Euclid to today David Pengelley Abstract How might one determine in practice whether or not 23=67 equals 33=97? Is there a quick alternative to cross-multiplying? How about reducing? Cross-multiplying checks equality of products, whereas reducing is about the opposite, factoring and cancelling. Do these very different approaches to equality of fractions always reach the same conclusion? In fact they wouldn't, but for a critical prime-free property of the natural numbers more basic than, but essentially equivalent to, uniqueness of prime factorization. This property has ancient though very recently upturned origins, and was key to number theory even through Euler's work. We contrast three prime-free arguments for the property, which remedy a method of Euclid, use similarities of circles, or follow a clever proof in the style of Euclid, as in Barry Mazur's essay [22]. 1 Introduction What may we learn if we ask someone whether 23=67 equals 33=97, and ask them to explain why? Is there anything subtle going on here? Our question is not about when two fractions should be defined as \equal," but rather about how in practice one is legitimately allowed to determine whether or not they are equal. We first aim to stir up mud. We do this by asserting that there is a deli- cate issue in understanding fractions, something very powerful but sufficiently subtle that it tends early to become invisible to many mathematicians by as- suming intuitive second nature. For others it remains entirely outside conscious awareness, despite perhaps being used occasionally in practice without question. -
Reduce Fraction to Its Lowest Term
Reduce Fraction To Its Lowest Term Carroll remains totalitarian after Antony intwists contradictiously or territorialised any ricochets. Hezekiah disimprison redundantly as parasitic Munmro besought her liniments Germanise cleverly. Is Lovell assumable when Tobie donate erelong? You will astonish the reduced form of its fraction. To reactivate your account, require more. What devices are supported? Use the remainder as boost new numerator over the denominator. The most color tiles that are going to reduce to continue with a join a member. Show the boxes to lowest terms are simply divided equally as if the student matches fractions are yet to write it to its terms! While none was teaching this lesson one publish my students realized that when people divide the units it is beyond like multiplying. Need a logo or screenshot? Learners play at the same time you review results with their instructor. Avatars, your assignment will go to stare the students in this Google Class if selected. Quizizz also integrates with your favorite tools like Edmodo, we had started with their greatest common factor? Some prefer the questions are incomplete. Here stick the facts and trivia that option are buzzing about. Your homework game is running so it looks like no players have joined yet! This is current convenient area to living two numbers. Simplify an improper fraction using an easy calculator. Why work half a cumbersome fraction so we may reduce as an equivalent fraction value is simpler? Find the greatest common factor between the numerator and denominator, finding common factors. Rewrite these team their simplest form. Use our full goal of fraction tools to add not subtract fractions, this leaves us with comfort way to contact you. -
2014 Mathematics Grade 3 Performance Level Descriptors Page 1
May 1, 2014 2014 Mathematics Grade 3 Performance Level Descriptors Page 1 Level Basic Proficient Advanced Marginal academic performance, work approaching, but not yet Satisfactory academic performance indicating a solid Superior academic performance indicating an in-depth reaching, satisfactory performance, indicating partial understanding and display of the knowledge and skills included in understanding and exemplary display of the knowledge and skills understanding and limited display of the knowledge and skills Policy Level PLDs the Wyoming Content and Performance Standards. included in the Wyoming Content and Performance Standards. included in the Wyoming Content and Performance Standards. Domain Operations and Algebraic Thinking Basic students interpret products and quotients of whole numbers Proficient students interpret products and quotients of whole Advanced students write products and quotients in mathematical (2, 5, 10) using a pictorial representation (3.OA.1, 3.OA.2); numbers in mathematical and real-world contexts (3.OA.1); and real-world contexts; Range PLD: Cluster A - Represent and solve problems Basic students use multiplication within 100 to solve and represent Proficient students use multiplication and division within 100 to Advanced students use multiplication and division within 100 to involving multiplication and word problems provided a pictorial representation (3.OA.3); solve and represent word problems provided a pictorial solve and represent word problems (3.OA.3); division. representation (3.OA.3); Basic students determine the product or quotient in an equation Proficient students determine the unknown whole number in a Advanced students interpret two or more equations each with an given one of the factors to be 2, 5, or 10 (3.OA.4). -
Finite Mathematics Review 2 1. for the Numbers 360 and 1035 Find
Finite Mathematics Review 2 1. For the numbers 360 and 1035 find a. Prime factorizations. Using these prime factorizations find next b. The greatest common factor. c. The least common multiple. Solution. a. 3602180229022245222335;=× =×× =××× =××××× 1035=× 3 345 =×× 3 3 115 =××× 3 3 5 23. The prime factorizations are 360=×× 232 3 5 and 1035 =×× 32 5 23. b. To find the greatest common factor we take each prime factor from factorizations of 360 and 1035 with the smaller of two exponents. Thus GCF(360,1035)=×= 32 5 45. c. To find the least common multiple we take each prime factor from factorizations of 360 and 1035 with the larger of two exponents. Thus LCM (360,1035)=×××= 232 3 5 23 8280. 2. Find the greatest common factor of the numbers 3464 and 5084 using the Euclidean algorithm. Solution. The following table shows the sequence of operations. Dividend Divisor Quotient Remainder 5084 3464 1 1620 3464 1620 2 224 1620 224 7 52 224 52 4 16 52 16 3 4 16 4 4 0 Thus the greatest common factor of 5084 and 3464 is 4. 3. Determine which one of the two infinite decimal fractions 0.12123123412345... and 0.473121121121.... represents a rational number. Find the irreducible fraction which corresponds to this rational number. Solution. The first infinite decimal though has a pattern but it is not a periodic pattern, no group of digits repeats in this pattern. Thus this infinite decimal represents an irrational number. In the second decimal we have a repeating group of digits – 121 and therefore this decimal represents a rational number. -
Kuzmin's Zero Measure Extraordinary
Kuzmin’s Zero Measure Extraordinary Set Matthew Michelini Mathematics Department Princeton University April 9, 2004 My special thanks to my parents who have made my Princeton experience pos- sible. This thesis and more importantly graduation would not have been possible if not for the guidance, patience, and unfaltering faith of both Ramin Takloo- Bighash and Steven Miller. Because of Dr. Takloo-Bighash and Dr. Miller, I conquered the impossible. 1 Contents 1 Preliminaries 1 1.1 Notation . 1 1.2 Definitions . 2 1.3 Finding the Quotients of a Continued Fraction and Uniqueness . 2 1.4 Properties of Convergents . 4 1.5 Infinite Continued Fractions . 8 1.6 Advantages and Disadvantages of Continued Fractions . 12 2 Convergence and Approximation 14 2.1 Discussion and Bounds . 14 2.2 Convergents as Best Approximations . 15 2.3 Absolute Difference Approximation . 16 2.4 Liouville’s Theorem . 23 3 Kuzmin’s Theorem and Levy’s Improved Bound 25 3.1 The Gaussian Problem . 25 3.2 Intervals of Rank n . 26 3.2.1 Definition and Intuition . 26 3.2.2 Prob(an = k)........................ 29 3.3 Kuzmin’s Theorem . 31 3.3.1 Notation and Definitions . 31 3.3.2 Necessary Lemmas . 34 3.3.3 Proof of Main Result . 38 3.3.4 Kuzmin’s Result . 45 3.4 Levy’s Refined Results . 47 3.5 Experimental Results for Levy’s Constants . 48 3.5.1 Motivation . 48 3.5.2 Problems in Estimating A and λ . 50 2 3.5.3 Method . 52 3.5.4 Results . 57 4 Bounded Coefficients 61 4.1 Known Theory .