Pythagorean Triples Before and After Pythagoras
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13-6 Success for English Language Learners
LESSON Success for English Language Learners 13-6 The Law of Cosines Steps for Success Step I In order to create interest in the lesson opener, point out the following to students. • Trapeze, trapezium (a geometric figure as well as a bone in the wrist), trapezoid (also a geometric figure and a bone in the wrist), and trapezius (a muscle in the back) all come from the same root. They come from the Greek word trapeza, or “four-legged table.” Step II Teach the lesson. • Have students derive the remaining formulas in the Law of Cosines that were not derived in the text. • Technically speaking, a knot is not a nautical mile. A knot is a speed of one nautical mile per hour. • Heron’s Formula is also known as Hero’s Formula. A Heronian triangle is a triangle having rational side lengths and a rational area. Step III Ask English Language Learners to complete the worksheet for this lesson. • Point out that Example 1A in the student textbook is supported by Problem 1 on the worksheet. Remind students that, for example, side a is opposite angle A, not adjacent to it. • Point out that Example 3 in the student textbook is supported by Problem 2 on the worksheet. • Think and Discuss supports the problems on the worksheet. Making Connections • Students comfortableᎏᎏ with matrices may wish to verify the following equation: If ᭝ ϭ ͙s ͑ s Ϫ a ͒ ͑ s Ϫ b ͒ ͑ s Ϫ c ͒ is the area of the triangle under consideration, then 2 Ϫ1 11 a 2 2 2 2 Ϫ 2 ͑ 4 ͒ ϭ a b c 1 1 1 b . -
Arxiv:Math/0701554V2 [Math.HO] 22 May 2007 Pythagorean Triples and a New Pythagorean Theorem
Pythagorean Triples and A New Pythagorean Theorem H. Lee Price and Frank Bernhart January 1, 2007 Abstract Given a right triangle and two inscribed squares, we show that the reciprocals of the hypotenuse and the sides of the squares satisfy an interesting Pythagorean equality. This gives new ways to obtain rational (integer) right triangles from a given one. 1. Harmonic and Symphonic Squares Consider an arbitrary triangle with altitude α corresponding to base β (see Figure 1a). Assuming that the base angles are acute, suppose that a square of side η is inscribed as shown in Figure 1b. arXiv:math/0701554v2 [math.HO] 22 May 2007 Figure 1: Triangle and inscribed square Then α,β,η form a harmonic sum, i.e. satisfy (1). The equivalent formula αβ η = α+β is also convenient, and is found in some geometry books. 1 1 1 + = (1) α β η 1 Figure 2: Three congruent harmonic squares Equation (1) remains valid if a base angle is a right angle, or is obtuse, save that in the last case the triangle base must be extended, and the square is not strictly inscribed ( Figures 2a, 2c ). Starting with the right angle case (Figure 2a) it is easy to see the inscribed square uniquely exists (bisect the right angle), and from similar triangles we have proportion (β η) : β = η : α − which leads to (1). The horizontal dashed lines are parallel, hence the three triangles of Figure 2 have the same base and altitude; and the three squares are congruent and unique. Clearly (1) applies to all cases. -
The Theorem of Pythagoras
The Most Famous Theorem The Theorem of Pythagoras feature Understanding History – Who, When and Where Does mathematics have a history? In this article the author studies the tangled and multi-layered past of a famous result through the lens of modern thinking, look- ing at contributions from schools of learning across the world, and connecting the mathematics recorded in archaeological finds with that taught in the classroom. Shashidhar Jagadeeshan magine, in our modern era, a very important theorem being Iattributed to a cult figure, a new age guru, who has collected a band of followers sworn to secrecy. The worldview of this cult includes number mysticism, vegetarianism and the transmigration of souls! One of the main preoccupations of the group is mathematics: however, all new discoveries are ascribed to the guru, and these new results are not to be shared with anyone outside the group. Moreover, they celebrate the discovery of a new result by sacrificing a hundred oxen! I wonder what would be the current scientific community’s reaction to such events. This is the legacy associated with the most ‘famous’ theorem of all times, the Pythagoras Theorem. In this article, we will go into the history of the theorem, explain difficulties historians have with dating and authorship and also speculate as to what might have led to the general statement and proof of the theorem. Vol. 1, No. 1, June 2012 | At Right Angles 5 Making sense of the history Why Pythagoras? Greek scholars seem to be in Often in the history of ideas, especially when there agreement that the first person to clearly state PT has been a discovery which has had a significant in all its generality, and attempt to establish its influence on mankind, there is this struggle to find truth by the use of rigorous logic (what we now call mathematical proof), was perhaps Pythagoras out who discovered it first. -
HISTORY of MATHEMATICS 1 Pythagorean Triples
HISTORY OF MATHEMATICS Summer 2010 Pythagoras and Fermat 1 Pythagorean Triples A Pythagorean triple is a triple of positive integers (a; b; c) such that a2 +b2 = c2. The simplest such triple is (3; 4; 5). Here are some facts about these triples. Since this is a math course, proofs are provided. I'll abbreviate Pythagorean triple to pt. 1. If (a; b; c) is a pt, then so is (ma; mb; mc) for any positive integer m (mul- tiples of pt's are pt's). In fact, (ma)2 + (mb)2 = m2a2 + m2b2 = m2(a2 + b2) = m2c2 = (mc)2: As an example, from the triple (3; 4; 5) we get the triples (6; 8; 10), (9; 12; 15), (12; 16; 20), etc. 2. If (a; b; c) is a pt and if the positive integer d divides a; b, and c, then (a=d; b=d; c=d) is a pt. In fact, ( ) ( ) ( ) a 2 b 2 a2 b2 a2 + b2 c62 c 2 + = + = = = : d d d2 d2 d2 d2 d Because of these two first properties, the basic pt's are those in which a; b; c do not have a common divisor other than 1. We make the following definition: A pt is said to be primitive, or to be a primitive Pythagorean triple if the greatest common divisor of a; b; c is one. Every Pythagorean triple is either primitive or a multiple of a primitive one. Let's continue with the properties 4. If (a; b; c) is a ppt, then c is odd and a; b have opposite parity; that is, one of a; b is odd, the other one even. -
Pythagorean Triples MAS 100 Concepts of Mathematics Sample Paper Jane D
Pythagorean Triples MAS 100 Concepts of Mathematics Sample Paper Jane D. Student September 2013 Everyone is familiar with the Pythagorean Theorem, which states that if a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then a2 +b2 = c2. Also familiar is the “3-4-5” right triangle. It is an example of three whole numbers that satisfy the equation in the theorem. Other such examples, from less familiar to downright obscure, are right triangles with side lengths 5-12-13, 35-12-37, and 45-28-53. Why are these interesting? Well, if you try to look for right triangles with whole number side lengths by trial and error, you will mostly be frustrated, since most of the time two whole number side lengths go with an irrational third side, like 1-1-√2. Is there a pattern to the whole number triples? If so, what is it? Problem A. Describe all possible triples (called Pythagorean triples) of whole numbers (a,b,c) which can occur as the lengths of the three sides of a right triangle. In other words, find all positive whole num- ber solutions to the equation a2 + b2 = c2. A complete solution was known to the Greeks by about 500 BCE ([3], p.37). Weisstein gives a thorough survey of solutions to this and related problems on his MathWorld website [4]. In this paper we describe a solution which uses a beautiful map called stereographic projection. Similar presentations can be found in [1] and [2]. -
The Brahmagupta Triangles Raymond A
The Brahmagupta Triangles Raymond A. Beauregard and E. R. Suryanarayan Ray Beauregard ([email protected]) received his Ph.D. at the University of New Hampshire in 1968, then joined the University of Rhode Island, where he is a professor of mathematics. He has published many articles in ring theory and two textbooks. Linear Algebra (written with John Fraleigh) is currently in its third edition. Besides babysitting for his grandchild Elyse, he enjoys sailing the New England coast on his sloop, Aleph One, and playing the piano. E. R. Suryanarayan ([email protected]) taught at universities in India before receiving his Ph.D. (1961) at the University of Michigan, under Nathaniel Coburn. He has been at the University of Rhode Island since 1960, where is a professor of mathematics. An author of more than 20 research articles in applied mathematics, crystallography, and the history of mathematics, he lists as his main hobbies music, languages, and aerobic walking. The study of geometric objects has been a catalyst in the development of number theory. For example, the figurate numbers (triangular, square, pentagonal, . ) were a source of many early results in this field [41.Measuring the length of a diagonal of a rectangle led to the problem of approxin~atingfi for a natural number N. The study of triangles has been of particular significance. Heron of Alexandria (c. A.D. 75)-gave the well-known formula for the area A of a triangle in terms of its sides: A = Js(s - a)(s- b)(s- c),where s = (a + b + c)/2 is the semiperimeter of the triangle having sides a,b, c [41.He illustrated this with a triangle whose sides are 13,14,15 and whose area is 84. -
Trees of Primitive Pythagorean Triples
http://user42.tuxfamily.org/triples/index.html Trees of Primitive Pythagorean Triples Kevin Ryde Draft 8, May 2020 Abstract All and only primitive Pythagorean triples are generated by three trees of Firstov, among which are the UAD tree of Berggren et al. and the Fibonacci boxes FB tree of Price and Firstov. Alternative proofs are oered here for the conditions on primitive Pythagorean triple preserving matrices and that there are only three trees with a xed set of matrices and single root. Some coordinate and area results are obtained for the UAD tree. Fur- ther trees with varying children are possible, such as ltering the Calkin- Wilf tree of rationals. Contents 1 Pythagorean Triples .......................... 2 1.1 Geometry .............................. 2 1.2 Primitive Pythagorean Triples .................... 3 2 UAD Tree ............................... 4 2.1 UAD Tree Row Totals ........................ 7 2.2 UAD Tree Iteration ......................... 9 2.3 UAD Tree Low to High ....................... 11 3 UArD Tree ............................... 12 3.1 UArD Tree Low to High ....................... 16 3.2 UArD Tree Row Area ........................ 17 3.3 UArD as Filtered Stern-Brocot ................... 19 4 FB Tree ................................ 20 5 UMT Tree ............................... 22 6 Triple Preserving Matrices ....................... 26 7 No Other Trees ............................ 30 8 Calkin-Wilf Tree Filtered ....................... 36 9 Parameter Variations ......................... 37 9.1 Parameter Dierence ........................ 37 9.2 Parameters Sum and Dierence ................... 38 References ................................. 40 Index ................................... 42 Copyright 2013, 2014, 2015, 2016, 2017, 2018, 2019 Kevin Ryde. Permission is granted for anyone to make a copy for the purpose of reading it. Permission is granted for anyone to make a full complete verbatim copy, nothing added, nothing removed, nothing overlaid, for any purpose. -
A New Approach to Generate All Pythagorean Triples
AIMS Mathematics, 4(2): 242–253. DOI:10.3934/math.2019.2.242 Received: 08 December 2018 Accepted: 14 February 2019 http://www.aimspress.com/journal/Math Published: 14 March 2019 Research article A new approach to generate all Pythagorean triples Anthony Overmars1, Lorenzo Ntogramatzidis2 and Sitalakshmi Venkatraman1;∗ 1 Department of Information Technology, Melbourne Polytechnic, VIC, Australia 2 Department of Mathematics and Statistics, Curtin University, Perth (WA), Australia * Correspondence: Email: [email protected]; Tel: +61892663143. Abstract: This paper revisits the topic of Pythagorean triples with a different perspective. While several methods have been explored to generate Pythagorean triples, none of them is complete in terms of generating all the triples without repetitions. Indeed, many existing methods concentrate on generating primitive triples but do not cater to non-primitives. By contrast, the approach presented in this paper to parameterise the Pythagorean triples generates all of the triples in a unique way, i.e., without repetitions. We also explore the relation of this new parameterisation with the Pythagorean family of odd triples and with the Platonic family of even triples. Keywords: Pythagorean triples; Diophantine equations; Euclidean triples; Platonic triples; right angle triangles Mathematics Subject Classification: 11A41, 11A51 1. Introduction The Pythogorean theorem states as follows: For any right-angled triangle, the square of the hypotenuse c equals the sum of squares of/on the two (shorter) legs lengths a and b, which is written as a2 + b2 = c2 and is among the Diophantine equations [1]. Several different proofs have been given for the Pythagorean theorem, see e.g., [2–4]. It was also shown that the converse of the theorem to be true. -
Clifford Algebras and Euclid's Parameterization of Pythagorean
Clifford Algebras and Euclid's Parameterization of Pythagorean Triples Jerzy Kocik Abstract. We show that the space of Euclid's parameters for Pythagorean triples is endowed with a natural symplectic structure and that it emerges as a spinor space of the Clifford algebra R21, whose minimal version may be conceptualized as a 4-dimensional real algebra of \kwaternions." We observe that this makes Euclid's parameterization the earliest appearance of the con- cept of spinors. We present an analogue of the \magic correspondence" for the spinor representation of Minkowski space and show how the Hall matrices fit into the scheme. The latter obtain an interesting and perhaps unexpected geometric meaning as certain symmetries of an Apollonian gasket. An exten- sion to more variables is proposed and explicit formulae for generating all Pythagorean quadruples, hexads, and decuples are provided. Keywords: Pythagorean triples, Euclid's parameterization, spinors, Clifford algebra, Minkowski space, pseudo-quaternions, modular group, Hall matrices, Apolloniam gasket, Lorentz group, Pythagorean quadruples and n-tuples. 1. Introduction In the common perception, Clifford algebras belong to modern mathematics. It may then be a point of surprise that the first appearance of spinors in mathematical literature is already more than two thousand years old! Euclid's parameters m and n of the Pythagorean triples, first described by Euclid in his Elements, deserve the name of Pythagorean spinors | as will be shown. The existence of Pythagorean triples, that is triples of natural numbers (a; b; c) satisfying a2 + b2 = c2 (1.1) has been known for thousands of years1. Euclid (ca 300 bce) provided a formula for finding Pythagorean triples from any two positive integers m and n, m > n, 1The cuneiform tablet known as Plimpton 322 from Mesopotamia enlists 15 Pythagorean triples and is dated for almost 2000 BCE. -
Problems 5475-5522 Ssma Mathematical Magazine
Problems Ted Eisenberg, Section Editor ********************************************************* This section of the Journal offers readers an opportunity to exchange interesting mathematical problems and solutions. Please send them to Ted Eisenberg, Department of Mathematics, Ben-Gurion University, Beer-Sheva, Israel or fax to: 972-86-477-648. Questions concerning proposals and/or solutions can be sent e-mail to <[email protected]>. Solutions to previously stated problems can be seen at <http://www.ssma.org/publications>. ||||||||||||||||||||{ Solutions to the problems stated in this issue should be posted before March 15, 2018 • 5475: Proposed by Kenneth Korbin, New York, NY p 8a + b = 14 ab − 48; < p Given positive integers a; b; c and d such that b + c = 14pbc − 48; :c + d = 14 cd − 48; with a < b < c < d. Express the values of b; c, and d in terms of a. • 5476: Proposed by Ed Gray, Highland Beach, FL Find all triangles with integer area and perimeter that are numerically equal. • 5477: Proposed by Daniel Sitaru, \Theodor Costescu" National Economic College, Drobeta Turnu-Sevrin, Meredinti, Romania Compute: p p p ! 1 − 1 + x2 3 1 + x2 · ::: · n 1 + x2 L = lim ln n + lim : n!1 x!0 x2 • 5478: Proposed by D. M. Btinetu-Giurgiu, \Matei Basarab" National Collge, Bucharest, Romania and Neculai Stanciu, \George Emil Palade" Secondary School, Buzu, Romania Compute: Z π=2 π π cos2 x sin x sin2 cos x + cos x sin2 sin x dx: 0 2 2 • 5479: Proposed by Jos´eLuis D´ıaz-Barrero, Barcelona Tech, Barcelona, Spain Let f : [0; 1] ! < be a continuous convex function. -
A Note on the Generation of Pythagorean Triples
MAT-KOL (Banja Luka) ISSN: 0354-6969 (p), ISSN: 1986-5228 (o) http://www.imvibl.org/dmbl/dmbl.htm XXIV(1)(2018), 41-51 DOI: 10.7251/MK1801041M A NOTE ON THE GENERATION OF PYTHAGOREAN TRIPLES Jovan Miki´c Abstract. We give a new method (up to our knowledge) for generating some primitive pythagorean triples. Our method is based on the construction of a new primitive pythagore-an triple from a predetermined primitive pythagorean triple. We generalize this idea and we construct a new pythagorean triple from a predetermined pythagorean n-tuple. 1. Introduction A pythagorean triple is a triple of positive integers (a; b; c) such that a2 + b2 = c2: Obviously, if k is a positive integer and (a; b; c) is a pythagorean triple, then (ka; kb; kc) is a pythagorean triple also. A pythagorean triple is called primitive, if the greatest common divisor of integers a, b, c is equal to 1. It is known, from ancient times, that there are infinitely many pythagorean triples. Euclid ([6]) gave formula for generating all primitive pythagorean triples. The formula states that integers (Eq:1) a = m2 − n2; b = 2mn; c = m2 + n2 form a primitive pythagorean triple, if and only if m and n are relatively prime positive integers with conditions m > n, where m and n are opposite parities. The Euclid formula has more symmetric variant [6]: m2 − n2 m2 + n2 (Eq:2) a = mn; b = ; c = ; 2 2 where m and n are odd, relatively prime positive integers such that m > n. 2010 Mathematics Subject Classification. Primary 11A99; Secondary 11A41, 11A05. -
Unifications of Pythagorean Triple Schema Emily Hammes
East Tennessee State University Digital Commons @ East Tennessee State University Undergraduate Honors Theses Student Works 5-2019 Unifications of Pythagorean Triple Schema Emily Hammes Follow this and additional works at: https://dc.etsu.edu/honors Part of the Algebraic Geometry Commons Recommended Citation Hammes, Emily, "Unifications of Pythagorean Triple Schema" (2019). Undergraduate Honors Theses. Paper 502. https://dc.etsu.edu/ honors/502 This Honors Thesis - Open Access is brought to you for free and open access by the Student Works at Digital Commons @ East Tennessee State University. It has been accepted for inclusion in Undergraduate Honors Theses by an authorized administrator of Digital Commons @ East Tennessee State University. For more information, please contact [email protected]. Unifications of Pythagorean Triple Schema April 18, 2019 Abstract Euclid's Method of finding Pythagorean triples is a commonly accepted and applied tech- nique. This study focuses on a myriad of other methods behind finding such Pythagorean triples. Specifically, we discover whether or not other ways of finding triples are special cases of Euclid's Method. Contents 1 Overview 2 2 Methods for Generating Pythagorean Triples 2 2.1 Classical Methods . .2 2.1.1 Pythagoras' Method, c. 540 BC [1] . .3 2.1.2 Plato's Method, c. 380 BC [1] [2] . .3 2.1.3 Euclid's Method, c. 300BC [3] [4] . .4 2.2 Post-Classical Methods . .4 2.2.1 Fibonacci's Method, c.1170 - c.1250 [5] . .4 2.2.2 Stifel's Method, 1544 [6] . .5 2.2.3 Ozanam's Method, 1694 [7] . .5 2.3 Modern Methods .