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Euclid's Elements
by Euclid Thomas L. HeathEuclid's Elements has stood for well over two millennia as the exemplar, not just of classical geometry, but of the axiomatic and deductive structure characteristic of all pure mathematics. Its range goes beyond what we think of as geometry, extending to the general theory of proportions, number theory, and an innovative and ingenious treatment of incommensurability. Euclid speaks to us with a voice as clear and universal as laughter. The simplicity, clarity and elegance of Euclid's proofs and the beautiful sequence of their unfoldings both delight and instruct. They provide a paradigm of what constructing a proof means and an education in how it is done. [This text is listed as an example that meets Common Core Standards in English language arts in grades 9-10 at http://www.corestandards.org.]
Euclid's Phaenomena: A Translation and Study of a Hellenistic Treatise in Spherical Astronomy (Routledge Revivals)
by J. L. Berggren R. S. ThomasOriginally published in 1996, this book contains a translation and study of Euclid's Phaenomena, a work which once formed part of the mathematical training of astronomers from Central Asia to Western Europe. Included is an introduction that sets Euclid's geometry of the celestial sphere, and its application to the astronomy of his day, into its historical context for readers not already familiar with it. So no knowledge of astronomy or advanced mathematics is necessary for an understanding of the work. The book shows mathematical astronomy shortly before the invention of trigonometry, which allowed the calculation of exact results and the subsequent composition of Ptolemy's Almagest. This work and the (roughly) contemporaneous treatises of Autolycus and Aristarchos form a corpus of the oldest extant works on mathematical astronomy. Together with Euclid's Optics one has the beginnings of the history of science as an application of mathematics.
Euclid: The Creation Of Mathematics
by Benno Artmann B. ArtmannEuclid presents the essential of mathematics in a manner which has set a high standard for more than 2000 years. This book, an explanation of the nature of mathematics from its most important early source, is for all lovers of mathematics with a solid background in high school geometry, whether they be students or university professors.
Euclidean Design Theory (SpringerBriefs in Statistics)
by Masanori Sawa Masatake Hirao Sanpei KageyamaThis book is the modern first treatment of experimental designs, providing a comprehensive introduction to the interrelationship between the theory of optimal designs and the theory of cubature formulas in numerical analysis. It also offers original new ideas for constructing optimal designs. The book opens with some basics on reproducing kernels, and builds up to more advanced topics, including bounds for the number of cubature formula points, equivalence theorems for statistical optimalities, and the Sobolev Theorem for the cubature formula. It concludes with a functional analytic generalization of the above classical results. Although it is intended for readers who are interested in recent advances in the construction theory of optimal experimental designs, the book is also useful for researchers seeking rich interactions between optimal experimental designs and various mathematical subjects such as spherical designs in combinatorics and cubature formulas in numerical analysis, both closely related to embeddings of classical finite-dimensional Banach spaces in functional analysis and Hilbert identities in elementary number theory. Moreover, it provides a novel communication platform for “design theorists” in a wide variety of research fields.
Euclidean Geometry and Transformations
by Clayton W. Dodge"A good textbook." - Mathematical Gazette. This introduction to Euclidean geometry emphasizes both the theory and the practical application of isometries and similarities to geometric transformations. Each chapter begins with an optional commentary on the history of geometry. Contents include modern elementary geometry, isometries and similarities in the plane, vectors and complex numbers in geometry, inversion, and isometries in space. Numerous exercises appear throughout the text, many of which have corresponding answers and hints at the back of the book. Prerequisites for this text, which is suitable for undergraduate courses, include high school algebra, geometry, and elementary trigonometry. 1972 edition.
Euclidean Geometry and its Subgeometries
by Edward John Specht Harold Trainer Jones Keith G. Calkins Donald H. RhoadsIn this monograph, the authors present a modern development of Euclidean geometry from independent axioms, using up-to-date language and providing detailed proofs. The axioms for incidence, betweenness, and plane separation are close to those of Hilbert. This is the only axiomatic treatment of Euclidean geometry that uses axioms not involving metric notions and that explores congruence and isometries by means of reflection mappings. The authors present thirteen axioms in sequence, proving as many theorems as possible at each stage and, in the process, building up subgeometries, most notably the Pasch and neutral geometries. Standard topics such as the congruence theorems for triangles, embedding the real numbers in a line, and coordinatization of the plane are included, as well as theorems of Pythagoras, Desargues, Pappas, Menelaus, and Ceva. The final chapter covers consistency and independence of axioms, as well as independence of definition properties. There are over 300 exercises; solutions to many of these, including all that are needed for this development, are available online at the homepage for the book at www. springer. com. Supplementary material is available online covering construction of complex numbers, arc length, the circular functions, angle measure, and the polygonal form of the Jordan Curve theorem. Euclidean Geometry and Its Subgeometries is intended for advanced students and mature mathematicians, but the proofs are thoroughly worked out to make it accessible to undergraduate students as well. It can be regarded as a completion, updating, and expansion of Hilbert's work, filling a gap in the existing literature.
Euclid’s Window: The Story of Geometry from Parallel Lines to Hyperspace
by Leonard MlodinowMlodinow brilliantly and delightfully leads us on a journey through five revolutions in geometry, from the Greek concept of parallel lines to the latest notions of hyperspace. Here is an altogether new, refreshing, alternative history of math revealing how simple questions anyone might ask about space -- in the living room or in some other galaxy -- have been the hidden engine of the highest achievements in science and technology. Based on Mlodinow's extensive historical research; his studies alongside colleagues such as Richard Feynman and Kip Thorne; and interviews with leading physicists and mathematicians such as Murray Gell-Mann, Edward Witten, and Brian Greene, Euclid's Window is an extraordinary blend of rigorous, authoritative investigation and accessible, good-humored storytelling that makes a stunningly original argument asserting the primacy of geometry. For those who have looked through Euclid's Window, no space, no thing, and no time will ever be quite the same.
Euler's Gem
by David S. RichesonLeonhard Euler's polyhedron formula describes the structure of many objects--from soccer balls and gemstones to Buckminster Fuller's buildings and giant all-carbon molecules. Yet Euler's formula is so simple it can be explained to a child. Euler's Gem tells the illuminating story of this indispensable mathematical idea. From ancient Greek geometry to today's cutting-edge research, Euler's Gem celebrates the discovery of Euler's beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. In 1750, Euler observed that any polyhedron composed of V vertices, E edges, and F faces satisfies the equation V-E+F=2. David Richeson tells how the Greeks missed the formula entirely; how Descartes almost discovered it but fell short; how nineteenth-century mathematicians widened the formula's scope in ways that Euler never envisioned by adapting it for use with doughnut shapes, smooth surfaces, and higher dimensional shapes; and how twentieth-century mathematicians discovered that every shape has its own Euler's formula. Using wonderful examples and numerous illustrations, Richeson presents the formula's many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who's who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem's development, Euler's Gem will fascinate every mathematics enthusiast.
Euler's Gem: The Polyhedron Formula and the Birth of Topology
by David S. RichesonHow a simple equation reshaped mathematicsLeonhard Euler’s polyhedron formula describes the structure of many objects—from soccer balls and gemstones to Buckminster Fuller’s buildings and giant all-carbon molecules. Yet Euler’s theorem is so simple it can be explained to a child. From ancient Greek geometry to today’s cutting-edge research, Euler’s Gem celebrates the discovery of Euler’s beloved polyhedron formula and its far-reaching impact on topology, the study of shapes. Using wonderful examples and numerous illustrations, David Richeson presents this mathematical idea’s many elegant and unexpected applications, such as showing why there is always some windless spot on earth, how to measure the acreage of a tree farm by counting trees, and how many crayons are needed to color any map. Filled with a who’s who of brilliant mathematicians who questioned, refined, and contributed to a remarkable theorem’s development, Euler’s Gem will fascinate every mathematics enthusiast. This paperback edition contains a new preface by the author.
Eulerian Numbers
by T. Kyle PetersenThis text presents the Eulerian numbers in the context of modern enumerative, algebraic, and geometric combinatorics. The book first studies Eulerian numbers from a purely combinatorial point of view, then embarks on a tour of how these numbers arise in the study of hyperplane arrangements, polytopes, and simplicial complexes. Some topics include a thorough discussion of gamma-nonnegativity and real-rootedness for Eulerian polynomials, as well as the weak order and the shard intersection order of the symmetric group. The book also includes a parallel story of Catalan combinatorics, wherein the Eulerian numbers are replaced with Narayana numbers. Again there is a progression from combinatorics to geometry, including discussion of the associahedron and the lattice of noncrossing partitions. The final chapters discuss how both the Eulerian and Narayana numbers have analogues in any finite Coxeter group, with many of the same enumerative and geometric properties. There are four supplemental chapters throughout, which survey more advanced topics, including some open problems in combinatorial topology. This textbook will serve a resource for experts in the field as well as for graduate students and others hoping to learn about these topics for the first time.
Eureka Math - A Story of Functions: Algebra I (9) Student Edition Book #1 (modules 1-3)
by Great MindsEureka Math helps students truly understand mathematics and connect it to the real world, preparing them to solve problems they have not encountered.
Eureka Math Algebra I Study Guide
by Great MindsThe Eureka Math curriculum provides detailed daily lessons and assessments to support teachers in integrating the Common Core State Standards for Mathematics (CCSSM) into their instruction. The companion guides to Eureka Math gather the key components of the curriculum for each grade into a single location. Both users and non-users of Eureka Math can benefit equally from the content presented. The CCSSM require careful study. A thorough study of the Guidebooks is a professional development experience in itself as users come to better understand the standards and the associated content. Each book includes narratives that provide educators with an overview of what students learn throughout the year, information on alignment to the instructional shifts and the standards, design of curricular components, and descriptions of mathematical models. The Guidebooks can serve as either a self-study professional development resource or as the basis for a deep group study of the standards for a particular grade. For teachers who are either brand new to the classroom or to the Eureka Math curriculum, the Grade Level Guidebooks introduce them not only to Eureka Math but also to the content of the grade level in a way they will find manageable and useful. Teachers already familiar with the curriculum will also find this resource valuable as it allows for a meaningful study of the grade level content in a way that highlights the coherence between modules and topics. The Guidebooks allow teachers to obtain a firm grasp on what it is that students should master during the year.
Eureka Math Curriculum Guide
by Great MindsEureka Math is a comprehensive, content-rich PreK-12 curriculum that follows the focus and coherence of the Common Core State Standards in Mathematics (CCSSM) and carefully sequences the mathematical progressions into expertly crafted instructional modules. The companion Study Guides to Eureka Math gather the key components of the curriculum for each grade into a single location, unpacking the standards in detail so that both users and non-users of Eureka Math can benefit equally from the content presented. Each of the Eureka Math Curriculum Study Guides includes narratives that provide educators with an overview of what students should be learning throughout the year, information on alignment to the instructional shifts and the standards, design of curricular components, approaches to differentiated instruction, and descriptions of mathematical models. The Study Guides can serve as either a self-study professional development resource or as the basis for a deep group study of the standards for a particular grade. For teachers who are new to the classroom or the standards, the Study Guides introduce them not only to Eureka Math but also to the content of the grade level in a way they will find manageable and useful. Teachers familiar with the Eureka Math curriculum will also find this resource valuable as it allows for a meaningful study of the grade level content in a way that highlights the coherence between modules and topics. The Study Guides allow teachers to obtain a firm grasp on what it is that students should master during the year. The Eureka Math Curriculum Study Guide, Grade 2 provides an overview of all of the Grade 2 modules, including Sums and Differences to 20; Addition and Subtraction of Length Units; Place Value, Counting, and Comparison of Numbers to 1,000; Addition and Subtraction Within 200 with Word Problems to 100; Addition and Subtraction Within 1,000 with Word Problems to 100; Foundations of Multiplication and Division; Problem Solving with Length, Money, and Data; and Time, Shapes, and Fractions as Equal Parts of Shapes.
Eureka Math Geometry Study Guide
by Great MindsThe Eureka Math curriculum provides detailed daily lessons and assessments to support teachers in integrating the Common Core State Standards for Mathematics (CCSSM) into their instruction. The companion guides to Eureka Math gather the key components of the curriculum for each grade into a single location. Both users and non-users of Eureka Math can benefit equally from the content presented. The CCSSM require careful study. A thorough study of the Guidebooks is a professional development experience in itself as users come to better understand the standards and the associated content. Each book includes narratives that provide educators with an overview of what students learn throughout the year, information on alignment to the instructional shifts and the standards, design of curricular components, and descriptions of mathematical models. The Guidebooks can serve as either a self-study professional development resource or as the basis for a deep group study of the standards for a particular grade. For teachers who are either brand new to the classroom or to the Eureka Math curriculum, the Grade Level Guidebooks introduce them not only to Eureka Math but also to the content of the grade level in a way they will find manageable and useful. Teachers already familiar with the curriculum will also find this resource valuable as it allows for a meaningful study of the grade level content in a way that highlights the coherence between modules and topics. The Guidebooks allow teachers to obtain a firm grasp on what it is that students should master during the year.
Eureka Math Grade 6 Study Guide
by Great MindsEureka Math is a comprehensive, content-rich PreK-12 curriculum that follows the focus and coherence of the Common Core State Standards in Mathematics (CCSSM) and carefully sequences the mathematical progressions into expertly crafted instructional modules. The companion Study Guides to Eureka Math gather the key components of the curriculum for each grade into a single location, unpacking the standards in detail so that both users and non-users of Eureka Math can benefit equally from the content presented. Each of the Eureka Math Curriculum Study Guides includes narratives that provide educators with an overview of what students should be learning throughout the year, information on alignment to the instructional shifts and the standards, design of curricular components, approaches to differentiated instruction, and descriptions of mathematical models. The Study Guides can serve as either a self-study professional development resource or as the basis for a deep group study of the standards for a particular grade. For teachers who are new to the classroom or the standards, the Study Guides introduce them not only to Eureka Math but also to the content of the grade level in a way they will find manageable and useful. Teachers familiar with the Eureka Math curriculum will also find this resource valuable as it allows for a meaningful study of the grade level content in a way that highlights the coherence between modules and topics. The Study Guides allow teachers to obtain a firm grasp on what it is that students should master during the year. The Eureka Math Curriculum Study Guide, Grade 6 provides an overview of all of the Grade 6 modules, including Ratios and Unit Rates; Arithmetic Operations Including Dividing by a Fraction; Rational Numbers; Expressions and Equations; Area, Surface Area, and Volume Problems; Statistics.
Eureka Math Grade 7 Study Guide
by Great MindsEureka Math is a comprehensive, content-rich PreK-12 curriculum that follows the focus and coherence of the Common Core State Standards in Mathematics (CCSSM) and carefully sequences the mathematical progressions into expertly crafted instructional modules. The companion Study Guides to Eureka Math gather the key components of the curriculum for each grade into a single location, unpacking the standards in detail so that both users and non-users of Eureka Math can benefit equally from the content presented. Each of the Eureka Math Curriculum Study Guides includes narratives that provide educators with an overview of what students should be learning throughout the year, information on alignment to the instructional shifts and the standards, design of curricular components, approaches to differentiated instruction, and descriptions of mathematical models. The Study Guides can serve as either a self-study professional development resource or as the basis for a deep group study of the standards for a particular grade. For teachers who are new to the classroom or the standards, the Study Guides introduce them not only to Eureka Math but also to the content of the grade level in a way they will find manageable and useful. Teachers familiar with the Eureka Math curriculum will also find this resource valuable as it allows for a meaningful study of the grade level content in a way that highlights the coherence between modules and topics. The Study Guides allow teachers to obtain a firm grasp on what it is that students should master during the year. The Eureka Math Curriculum Study Guide, Grade 7 provides an overview of all of the Grade 7 modules, including Ratios and Proportional Relationships; Rational Numbers; Expressions and Equations; Percent and Proportional Relationships; Statistics and Probability; Geometry.
Eureka Math Practice: Grade 3 Fluency, Modules 5-7
by Great MindsEureka Math fluency activities provide differentiated practice through a variety of formats--some are conducted orally, some use manipulatives, others use a personal whiteboard, and still others use a handout and paper-and-pencil format. Eureka Math Practice provides each student with the printed fluency exercises for his or her grade level.
Eureka Math Study Guide
by Great MindsEureka Math is a comprehensive, content-rich PreK-12 curriculum that follows the focus and coherence of the Common Core State Standards in Mathematics (CCSSM) and carefully sequences the mathematical progressions into expertly crafted instructional modules. The companion Study Guides to Eureka Math gather the key components of the curriculum for each grade into a single location, unpacking the standards in detail so that both users and non-users of Eureka Math can benefit equally from the content presented. Each of the Eureka Math Curriculum Study Guides includes narratives that provide educators with an overview of what students should be learning throughout the year, information on alignment to the instructional shifts and the standards, design of curricular components, approaches to differentiated instruction, and descriptions of mathematical models. The Study Guides can serve as either a self-study professional development resource or as the basis for a deep group study of the standards for a particular grade. For teachers who are new to the classroom or the standards, the Study Guides introduce them not only to Eureka Math but also to the content of the grade level in a way they will find manageable and useful. Teachers familiar with the Eureka Math curriculum will also find this resource valuable as it allows for a meaningful study of the grade level content in a way that highlights the coherence between modules and topics. The Study Guides allow teachers to obtain a firm grasp on what it is that students should master during the year. The Eureka Math Curriculum Study Guide, Grade 1 provides an overview of all of the Grade 1 modules, including Sums and Differences to 10; Introduction to Place Value Through Addition and Subtraction Within 20; Ordering and Comparing Length Measurements as Numbers; Place Value, Comparison, Addition and Subtraction to 40; Identifying, Composing, and Partitioning Shapes; and Place Value, Comparison, Addition and Subtraction to 100.
Eureka Math Study Guide: A Story of Units, Grade 3 (Common Core Mathematics)
by Great MindsEureka Math is a comprehensive, content-rich PreK–12 curriculum that follows the focus and coherence of the Common Core State Standards in Mathematics (CCSSM) and carefully sequences the mathematical progressions into expertly crafted instructional modules. The companion Study Guides to Eureka Math gather the key components of the curriculum for each grade into a single location, unpacking the standards in detail so that both users and non-users of Eureka Math can benefit equally from the content presented. <P><P> Each of the Eureka Math Curriculum Study Guides includes narratives that provide educators with an overview of what students should be learning throughout the year, information on alignment to the instructional shifts and the standards, design of curricular components, approaches to differentiated instruction, and descriptions of mathematical models. The Study Guides can serve as either a self-study professional development resource or as the basis for a deep group study of the standards for a particular grade. For teachers who are new to the classroom or the standards, the Study Guides introduce them not only to Eureka Math but also to the content of the grade level in a way they will find manageable and useful. Teachers familiar with the Eureka Math curriculum will also find this resource valuable as it allows for a meaningful study of the grade level content in a way that highlights the coherence between modules and topics. The Study Guides allow teachers to obtain a firm grasp on what it is that students should master during the year.<P> The Eureka Math Curriculum Study Guide, Grade 3 provides an overview of all of the Grade 3 modules, including Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10; Place Value and Problem Solving with Units of Measure; Multiplication and Division with Units of 0, 1, 6–9, and Multiples of 10; Multiplication and Area; Fractions as Numbers on the Number Line; and Collecting and Displaying Data.
Eureka Math Study Guide: A Story of Units, Grade 5 (Common Core Mathematics)
by Great MindsEureka Math is a comprehensive, content-rich PreK–12 curriculum that follows the focus and coherence of the Common Core State Standards in Mathematics (CCSSM) and carefully sequences the mathematical progressions into expertly crafted instructional modules. The companion Study Guides to Eureka Math gather the key components of the curriculum for each grade into a single location, unpacking the standards in detail so that both users and non-users of Eureka Math can benefit equally from the content presented. <P><P> Each of the Eureka Math Curriculum Study Guides includes narratives that provide educators with an overview of what students should be learning throughout the year, information on alignment to the instructional shifts and the standards, design of curricular components, approaches to differentiated instruction, and descriptions of mathematical models. The Study Guides can serve as either a self-study professional development resource or as the basis for a deep group study of the standards for a particular grade. For teachers who are new to the classroom or the standards, the Study Guides introduce them not only to Eureka Math but also to the content of the grade level in a way they will find manageable and useful. Teachers familiar with the Eureka Math curriculum will also find this resource valuable as it allows for a meaningful study of the grade level content in a way that highlights the coherence between modules and topics. The Study Guides allow teachers to obtain a firm grasp on what it is that students should master during the year.<P> The Eureka Math Curriculum Study Guide, Grade 5 provides an overview of all of the Grade 5 modules, including Place Value and Decimal Fractions; Multi-Digit Whole Number and Decimal Fraction Operations; Addition and Subtraction of Fractions; Multiplication and Division of Fractions and Decimal Fractions; Addition and Multiplication with Volume and Areal; Problem Solving with the Coordinate Plane.
