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IN THE BEGINNING…(Grassmann's Linear Algebra)Grassmann is considered to be the "father" of linear algebraDeveloped the idea of a linear algebra in which the symbols representing geometric objects can be manipulatedSeveral of his operations: the interior product, the exterior product, and the multiproduct
What's a Multiproduct Equation Look Like?δ1⊗δ2 + δ12 = 0The multiproduct has many uses, including scientific, mathematic, and industrialGot updated by William Clifford
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TO GRASSMAN'S EQUATIONδ1⊗δ2 + δ12 = 2kijThe 2kij is what's referred to as Kronecker's SymbolBoth of these equations are used for Quantum Theory Math
VECTOR SPACEAnother idea which is kind of tied with GrassmanVector Space refers to some set of vectors that contains the originIt is usually infiniteSubspace is a subset of vector space. It, of course, is also vector space
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Cholesky DecompositionAlgorithm developed by Arthur CayleyTakes a matrix and factors it into a triangular matrix times its transposeA=R'RUseful for matrix applicationsBecomes even more worthwhile in parallel
HOW TO USE LINEAR ALGEBRA FOR PDE'SYou can use matrices and vectors to solve partial differential equationsFor equations with lots of variables, you'll wind up with really sparse matricesHence, the project we've been working on all year
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From the developer: ""MoNooN Grapher 2DP is the simple software that draws and shows the graph of the 2-dimensional mathematical functions in polar coordinates.
""If you input equations, choose colors, and choose line types, you can see the graph. MoNooN Grapher 2.""
Requirements:
Windows 95/98/Me/NT/2000/XP, 8Simple Math Test for Windows 8 Simple Math Test for Windows 8 is a math test. It consists of four basics. There are addition, subtraction, division, and multiplication. It is a good application to test your understanding of these math basics. It has ten questions each round with four ch
Matrix Calculator Solve any system of linear equations using the augmented matrix method. The calculator can quickly solve equations by defining pivot points and performing row operations to obtain solutions. It is very useful for solving unknown voltages in electrical circ
WordSpell2 Tutor driven teaching tool designed to encourage students at all levels of education, to improve their USA or UK English spelling speaking and typing. Uses natural speech from your tutor, a friend, mum, dad, or your own voice. Learn any word by creating yo
ArcGIS Explorer ArcGIS Explorer is a free downloadable application that offers an easy way to access online GIS content and capabilities. With ArcGIS Explorer, you can connect to a variety of free, ready-to-use datasets hosted by ESRI. Combine these with your own local da
Machine Learning Framework The machine learning framework for Mathematica is a collection of powerful machine learning algorithms integrated into a framework for the main purpose of data analysis.
Fuzzy logic is one of its key techniques. The framework allows for combining differen
Infinity There are many things that require mathematical modeling: from exchange rates prediction to engineering and financial planning. Infinity is an innovative non-linear math application that allows you use complex math expressions within equations to describe | 677.169 | 1 |
Calculus Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"
M2O2C2 provides a first taste of multivariable differential calculus. By introducing the machinery of linear algebra, this course provides helpful tools for understanding the derivative of a function of many variables. | 677.169 | 1 |
ISBN 9789351418436
ISBN-10
935141843X
Binding
Paperback
Edition
7th
Number of Pages
1012 Pages
Language
(English)
Subject
Entrance Exam Preparation
To equally help the beginners as well as experts, the skills in mathematics series aims at helping the students take their knowledge and application for various mathematical concepts to a whole new level. The books in this series have been designed so as to work as elementary text books which will play a crucial role in building the concepts from scratch till the end and help in preparation for various competitive examinations.
This master piece on algebra takes an engaging and easily understandable approach to make students confident in tacking mathematics problems and is infused with the authors passion for teaching mathematics to the engineering aspirants. The book has been divided into 15 chapters in total namely complex numbers, equations, in equations and expressions, sequences and series, mathematical induction, permutations and combinations, binomial theorem, determinants, inequalities, probability, binomial theorem (Any Rational Index), exponential and logarithmic series, logarithms and their properties, partial functions, matrices and set theory, each containing definitions along with figures for better understanding of the topics. The contents of the book have been devised to cover the syllabic of and serve as a textbook for all engineering entrance examinations being conducted in India. The fact that algebra is the foundation for higher mathematics and the further work in mathematics, science and engineering all depends upon ones proper groundings in the algebraic methods has been kept in mind. Solved examples have been given in between each chapter for on spot assessment of the concepts discussed in the chapter. Exercises at the end of each chapter have been divided into three levels with various types of questions such as subjective questions, MCQs and fill in the blanks. A section containing previous years questions has also been included in the book to give the aspirants an insight into the question pattern and the level of the questions asked.
This is a must have textbook which starts from fundamentals and gradually builds your concepts up to the level required for engineering entrances and finally will place you among the toppers. TABLE OF CONTENTS : -
1. Complex Numbers
2. Equations, In Equations and Expression
3. Sequences and Series
4. Mathematical Induction
5. Permutations and Combinations
6. Binomial Theorem
7. Determinants
8. Inequalities
9. Probability
10. Binomial Theorem (Any Rational Index)
11. Exponential and Logarithmic Series
12. Logarithms & their Properties
13. Partial Fractions
14. Matrices
15. Set Theory | 677.169 | 1 |
Solving Equations and Inequalities Algebra Assessment
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0.02 MB | 5 pages
PRODUCT DESCRIPTION
This is an assessment designed for middle and high school Algebra I students. It covers TEKS A.5A, A.5B, and A.5C. Topics include writing and solving equations and inequalities, the distributive and commutative properties, and literal equations. Problem types include error analysis, straightforward solving, definition identification, and both mathematical and real world situations represented in picture and written form. The assessment is 14 questions long | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
Noriko is just getting started as a junior reporter for the Asagake Times. She wants to cover the hard-hitting issues, like world affairs and politics, but does she have the smarts for it? Thankfully, her overbearing and math-minded boss, Mr. Seki, is here to teach her how to analyze her stories with a mathematical eye.
In The Manga Guide to Calculus, you'll follow along with Noriko as she learns that calculus is more than just a class designed to weed out would-be science majors. You'll see that calculus is a useful way to understand the patterns in physics, economics, and the world around us, with help from real-world examples like probability, supply and demand curves, the economics of pollution, and the density of Shochu (a Japanese liquor).
Mr. Seki teaches Noriko how to:
Use differentiation to understand a function's rate of change
Apply the fundamental theorem of calculus, and grasp the relationship between a function's derivative and its integral
Integrate and differentiate trigonometric and other complicated functions
Use multivariate calculus and partial differentiation to deal with tricky functions
Use Taylor Expansions to accurately imitate difficult functions with polynomials
Whether you're struggling through a calculus course for the first time or you just need a painless refresher, you'll find what you're looking for in The Manga Guide to Calculus.
This EduManga book is a translation from a bestselling series in Japan, co-published with Ohmsha, Ltd. of Tokyo, Japan.
Synopsis
Calculus isn't just a required math class to weed out would-be science majors-it's a useful way to understand the patterns in physics, economics, and the natural world. With its distinctive mix of serious educational content and Japanese-style comics, The Manga Guide to Calculus will entertain you while it helps you understand the key concepts of calculus (and ace those exams).
Our story begins as Noriko, a recent liberal arts grad, arrives at a branch office of the Asagake Newspaper to start her career as a journalist. With the help of her overbearing and math-minded boss Kakeru, she's finally able to do some real reporting. But Noriko soon discovers the mathematical functions behind all the stories she struggles to cover.
Using real-world examples like probability, supply and demand, polluting companies, and shochu density, Kakeru explains to Noriko: Techniques of differentiation and integration How to integrate and differentiate trigonometric and other complicated functions Multivariate calculus and partial differentiation Taylor expansions
Reluctant calculus students of all abilities will enjoy following along with Noriko as she learns calculus from Kakeru's quirky stories and examples. This charming and easy-to-read guide also includes an appendix with answers to the book's many useful exercises.
This EduManga book is a translation from a bestselling series in Japan, co-published with Ohmsha, Ltd. of Tokyo, Japan.
Synopsis
Teaching calculus in an original and refreshing way, this guide combines Japanese-style manga cartoons with mathematical content as it follows the story of heroine Noriko. Noriko takes a job with a local newspaper and quickly befriends math whiz Kakeru, who wants to help her understand the practical uses of calculus in journalism.
About the Author
Hiroyuki Kojima received his PhD in Economics from the Graduate School of Economics, Faculty of Economics, at the University of Tokyo. He has worked as a lecturer and is now an associate professor in the Faculty of Economics at Teikyo University in Tokyo, Japan. While well-regarded as an economist, he is also active as an essayist and has published a wide range of books on mathematics and economics at the fundamental, practical, and academic levels. | 677.169 | 1 |
Michel Thomas Method : Maths Course
Description
You'll stick with it because you'll love it * Use the unique method perfected over fifty years by the celebrated psychologist and linguist Michel Thomas. * This method works with your brain, helping you to learn or brush up you Maths techniques in manageable, enjoyable steps by thinking out the answers for yourself. The new Michel Thomas Maths Course Maths is similar to languages inasmuch as people tend to think that they need to have an 'aptitude' for it, and that without this it will be a hard slog; the Michel Thomas Method will remove this mystique and will enable even the most maths-phobic student to succeed. The author of the Michel Thomas Method Maths Course, Paul Carson, has been true to the Michel Thomas principle of breaking the material to be taught down to its simplest components and then presenting these components in a meticulously structured way so that learning is gradual and apparently effortless. There is no chance for the student to get lost or left behind as it's impossible to go on unless each item being taught has been understood or 'internalised', and as Michel Thomas said, 'What you understand you know, and what you know you don't forget'.
The Michel Thomas Method has been proved to work with hundreds of thousands of satisfied customers who believed that learning a language would be stressful, hard work, or simply beyond them. Now we show that the Method can be applied to other areas with equal success, as Michel Thomas himself always insisted. Contents: 3 DVD-ROMs plus 64-page booklet giving the graphics from the DVD-ROMs, and track listings to help navigate the courseshow more
Review quote
"A great way to learn; it's fast and it lasts" -- The Daily Telegraph "Five minutes into the first CD, you already feel like you're winning." -- Time Out "Michel Thomas is a precious find indeed." -- The Guardian "Thomas makes it simple" -- Sunday Times "Michel's methods will teach you effectively and easily" -- Daily Star "Hugely inspiring" -- Red "Ideal for any business traveller who needs to be able to get around confidently." -- Sunday Businessshow more
About Paul Carson
Paul Carson has a 100% pass rate with his private GCSE students using his method and is now teaching in mainstream secondary school and adult education. He has learned several languages using Michel Thomas and is the perfect evangelist to take the Method forward.show more | 677.169 | 1 |
About this product
Description
Description
Basic math review included Text and workbook in one; includes 124 practice and lab activities -- from lab experiments to crossword puzzles and word searches Activities on perforated pages can be torn out and submitted to instructor Over 500 multiple-choice review questions Numerous illustrations reinforce learning Each chapter begins with an outline and chapter objectives and ends with a summary and multiple-choice review questions | 677.169 | 1 |
Summary and Info
This is an outstanding book on a field of mathematics which, although very accessible and widely applicable, is not considered as fundamental as I believe it should be. The author writes clearly, providing concise proofs with sound logic, good motivation for the material, discussion of historical development of the subject, and directions for future research. Diagrams are used very well in this text. The exercises are numerous and very illuminating, and very fun! The author provides an extensive bibliography and references results throughout the text.There is something very enticing about the way of thinking used in lattice theory, and in particular, the way Gratzer approaches the subject in this book. Like in most areas of mathematics, a given concept can be viewed in many different ways, and one can study these objects at progressively higher levels of abstraction and generality. What is most remarkable about lattice theory, however, is that these higher levels of abstraction and generality do not become overwhelmingly difficult to comprehend as they develop--something that unfortunately happens in many other areas of mathematics. In particular, I think that algebraic concepts such as congruence relations, equational varieties, and "freeness" are much easier to understand in the context of lattices than in other algebraic structures. However, this is also true of more general mathematical concepts such as the effect of weakening conditions on theorems, searching for counterexamples, finding equivalent formulations of a given condition, and studying properties preserved under maps. This is partly due to the fact that in lattices, much of what is going on can be easily drawn or visualized. For this reason, lattices provide an excellent framework for understanding many of the basic concepts that underly all areas of mathematics.Although the topic of this book is viewed as specialized and esoteric by some mathematicians, I believe that the material it contains is quite universal. Lattices appear in virtually every area of mathematics, and they are especially useful in universal algebra and combinatorics. While this book does not directly talk much about these applications, this book would certainly enrich the knowledge and understanding of people who work in those fields. I would recommend this book to anyone who is serious about algebra or combinatorics as well, as the ways of thinking developed by reading this book and working exercises will prove invaluable in these disciplines. This book also might be useful to beginning graduate students who want to develop their general mathematical maturity in a setting which can be a lot more fun and accessible than other areas of abstract math.As a final note, the binding on the hardcover edition is excellent. I rarely encounter books this well-bound, in a day an age when sometimes even hardback books start falling apart after moderate use.
More About the Author
George A. Grätzer (Hungarian: Grätzer György, born 2 August 1936 in Budapest) is a Hungarian-Canadian mathematician, specializing in lattice theory and universal algebra. | 677.169 | 1 |
Daily Math Review #3: Openers to review for standardized tests
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6.91 MB | 30 pages
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Review past topics and help students perform well on the standardized test and final exam! This is such a simple concept that makes such a huge difference! Replace your regular class openers with this activity. Each day for two weeks, students practice the same five types of problems in order to solidify that standard in their memories. Hand the day's Daily Math Review to your students as they walk in the door. After five to ten minutes, show the answer key (included in your purchase), which shows how to obtain the answer step-by-step. At the end of two weeks, you give a quiz (also included in your purchase) on the five problems. You will be shocked at how well they work on this! I don't even have to grade or collect the reviews because students know they must take it seriously in order to pass the two-sided, ten-question quiz at the end of two weeks. This has been a wonderful way for me to really get students to keep practicing the most important concepts of Algebra 1 so that my test scores are excellent at the end of the year. Try this and become a believer too!
Included in this file:
*9 Daily Math Reviews
*1 double-sided Quiz (3 versions)
*Answer key for all reviews and quizzes
Concepts Covered:
• Equations with variables on both sides
• Equations with combining like terms
• Graphing linear inequalities
• Plotting points on a coordinate plane
• Equations with a fraction coefficient
• Equations with distribution and combining like terms | 677.169 | 1 |
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Description
Elementary Linear Algebra develops and explains in careful detail the computational techniques and fundamental theoretical results central to a first course in linear algebra. This highly acclaimed text focuses on developing the abstract thinking essential for further mathematical study.
The authors give early, intensive attention to the skills necessary to make students comfortable with mathematical proofs. The text builds a gradual and smooth transition from computational results to general theory of abstract vector spaces. It also provides flexbile coverage of practical applications, exploring a comprehensive range of topics.
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Horizons Pre-Algebra Set
Give your child a smooth transition into advanced math with the Horizons Pre-Algebra Set! This popular Alpha Omega curriculum builds on basic math operations with hands-on lessons in basic algebra, trigonometry, geometry, and real-life applications. The set includes a full-color student workbook with 160 engaging lessons, a user-friendly teacher's guide, and a separate tests and resources book for evaluating student progress.
The course takes students from basic operations in whole numbers, decimals, fractions, percents, roots, and exponents and introduces them to math-building concepts in algebra, trigonometry, geometry, and exciting real-life applications.
Divided into 160 lessons, this Alpha Omega curriculum comes complete with one consumable student book, a student tests and resources book, and an easy-to-use teacher's guide.
Every block of ten lessons in this Horizons math course begins with a challenging set of problems that prepares students for standardized math testing and features personal interviews showing how individuals make use of math in their everyday lives.
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Alpha Omega Publications is a Christian publishing company that proudly provides academically rigorous, Christian curriculum for students in preschool through 12th grade. With print-based, computer-based, and online formats, our Bible-based curriculum includes Monarch, Ignitia, Switched-On Schoolhouse, LIFEPAC, Horizons, and The Weaver Curriculum. We also offer an accredited online education through Alpha Omega Academy. | 677.169 | 1 |
Linked Data Explorer
Condition the geometry of numerical algorithms
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""This book gathers threads that have evolved across different mathematical disciplines into seamless narrative. It deals with condition as a main aspect in the understanding of the performance ---regarding both stability and complexity---Àlgebra linemsmus d'error numèricaAnalyse Mathematics and Numerical Analysis Science and EngineeringConjunts convexDistribution (Probability theory estoconditionszahl optimizationATHEMATICS / Numerical Analysis of Computingumerische MathematOptimizationProbability Theory and Stochastic Processació | 677.169 | 1 |
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This course explains how you can use numbers to describe the natural world and make sense of everything from atoms to oceans. This practical, hands-on course will help you to start thinking like a scientist, by using numbers to describe and understand the natural world.
Learn basic engineering mathematics and how to apply basic mathematics to solve engineering problems. The goal of this mathematics course is to provide high school students and college freshmen an introduction to basic mathematics and especially show how mathematics is applied to solve fundamental engineering problems.
In this short course, you will see four lessons from the Quantway® course developed by the Carnegie Foundation. These lessons will develop your understanding of common numbers often found in the news, on advertisements, and online. | 677.169 | 1 |
As of 2015, the New York State Geometry curriculum is completely aligned with Common Core Standards. This up-to-date book will prepare students for the new Geometry (Common Core) Regents exam. It features:
The first two actual Regents exams administered for the updated Geometry RegentsUsed by more than half a million students each year! Actual Regents Exams develop your subject knowledge and test-taking skills and show you where you need more study. Answers to All Questions provide...
Description :
"The section on how to answer Part C questions and the glossary were adapted from Let's review: physics, by Miriam A. Lazar and Albert S. Tarendash, published by Barron's Educational Series, Inc., 199 | 677.169 | 1 |
Transformations Math Match Up {Algebra 2}
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1.92 MB | 3 pages of cards + answer key pages
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This activity has transformations of 15 functions represented as a graph, equation and description of the transformations. Equations include quadratic function, absolute value function and square root function.
An optional student sheet (2 to a page) is included for students to record their answers and make it easier on you to grade.
After students have matched the graphs and equations, I have students start quizzing each other using the cards, having them explain them out loud to each other. This keeps them working and practicing their math | 677.169 | 1 |
provides a seamless approach to numerical algorithms, modern programming techniques and parallel computing. These concepts and tools are usually taught serially across different courses and different textbooks, thus observing the connection between them. | 677.169 | 1 |
Paperback
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Overview
Describes and explains uses of equations, polynomials, the binomial formula, exponential functions, logarithms, and much more, with exercises and answers.
Over the years, Barron's popular and widely-used Easy Way books have proven themselves to be accessible self-teaching manuals. They have also found their way into many classrooms as valuable and easy-to-use textbook supplements. The titles cover a wide variety of both practical and academic topics, presenting fundamental subject matter so that it can be clearly understood and provide a foundation for more advanced study. Easy Way books fulfill many purposes. They help students improve their grades, serve as good test preparation review books, and provide readers working outside classroom settings with practical information on subjects that relate to their occupations and careers. All Easy Way books include review questions and mini-tests with answers. All new Easy Way editions feature type in two-colors, the second color used to highlight important study points and topic heads.
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Algebra the Easy Way (Baron's Educational Series) 2 out of 5based on
0 ratings.
4 reviews.
Jason_Nesquik
More than 1 year ago
I have the whole Easy way series. Sometimes, when a concept is new you want to approach it from an oblique. I literally went from being mystified in a few classes to being a straight A math student. That was 1992-1994. Twenty years later, as a transportation planner, I work with statistical analysis, trig, and concepts from both integral and differential calculus regularly. These made that possible. The books tell a kids fantasy story of a traveler who washes up on the shore of an unfamiliar land, and walks through the process of understanding the problems they need to solve. As an adult, I still find the characters and stories amusing and a relief from the ridiculously thick and boring textbooks companies are cranking out now. And yes. I did just fine on all the tests, and developed an understanding and appreciation for mathematics that drives my career today.
Anonymous
More than 1 year ago
I work in Finance. I sing and dance Algebra. I offered to tutor a friend in Algebra. My friend brought me this book to structure his learning. I just read the first chapter. What a disaster. How much garbage do you have to read through to get to the point? And the story has no real practical application. This is the Barney, the friendly purple dinosaur, of Algebra books. I would rather be on the Dr. Phil show with everyone who ever thought I needed counseling than read another chapter of this book. Just because the cover screams, "Two thumbs up!" or "Best Algebra book I ever read!" don't let it fool you. Save your money and move on.
Guest
More than 1 year ago
I bought this book to refresh my high school math skills when going back to college. I spent more time on line trying to find examples. This books tells stories between a professor and wizard to teach you math. If you dig that sort of thing and don't need step by step instructions on how to learn math skills long forgotten, then this book is for you.
Guest
More than 1 year ago
Maybe it was my mistake for not investigating this book a little more before I got, but this book is for kids. And beyond that there aren't nearly enough examples. It was written as a supplement to a high school Algebra class. If you want some hard core refresher material press back on our browser now and look for something else. The first of the chapters are so simple you could puke, and then tossed the book from a moving vehical. I'm going to buy Algebra Dymystified and I suggest you do the same. | 677.169 | 1 |
Product details
ISBN-13: 9780821802366
ISBN: 0821802364
Edition: 1st
Publication Date: 1994
Publisher: American Mathematical Society
AUTHOR
V. V. Prasolov
SUMMARY
There are a number of very good books available on linear algebra. From this one might deduce that the existing books contain all that one needs to know in the best possible form and that any new book would just repeat material in the old ones. However, new results in linear algebra appear constantly, as do new, simpler, and better proofs of old results. Many linear algebra results obtained in the past thirty years are accessible to undergraduate mathematics majors, but are usually ignored by textbooks. In addition, more than a few interesting old results are not covered in many books. In this book, Prasolov provides the basics of linear algebra, with an emphasis on new results and on nonstandard and interesting proofs. The book features about 230 problems with complete solutions. It would be a fine supplementary text for an undergraduate or graduate algebra course.V. V. Prasolov is the author of 'Problems and Theorems in Linear Algebra (Translations of Mathematical Monographs, Vol. 134)', published 1994 under ISBN 9780821802366 and ISBN 08218023 | 677.169 | 1 |
Prerequisites: MATH 111 or demonstrated competency through appropriate assessment or earning a grade of"C"or better in MATH 035 or MATH 043.The course gives a theoretical treatment of com- mon topics underlying an elementary mathematics curriculum. This course covers topics in elementary number theory.Students will be encouraged to explore,make and debate conjectures,build connec- tions among concepts,and solve problems from their explorations. The selection of topics presented in this course is based upon stan- dards and recommendations for the mathematical content knowl-
edge essential for prospective teachers made by the National
Council of Teachers of Mathematics.
MATH 128 Mathematics for Elementary Education II
3 Credits
Prerequisites: MATH 111 or demonstrated competency through appropriate assessment or earning a grade of"C"or better in MATH 035 or MATH 043.This course gives a theoretical treatment of com- mon topics underlying an elementary mathematics curriculum. This course covers algebraic equations,probability,and statisticsTeachers of Mathematics.
MATH 129 Mathematics for Elementary Education III
3 Credits
Prerequisites: MATH 111 or demonstrated competency through appropriate assessment or earning a grade of"C"or better in MATH 035 or MATH 043. The course gives a theoretical treatment of com- mon topics underlying an elementary mathematics curriculum. This course covers plane and solid geometry,and measurement Teachers of Mathematics. MATH 131 Algebra/Trigonometry I
3 Credits radicals,complex numbers,right triangle trigonometry,oblique tri- angles, vectors,and graphs of sine and cosine functions.First in a series of two courses of College Algebra/Trigonometry.
MATH 132 Algebra/Trigonometry II
3 Credits
Prerequisites: MATH 131.Continues study of algebra and trigonome- try including systems of equations,matrices,graphing of trigono- metric functions,trigonometric equations and identities,rectangular and polar coordinates,complex numbers,exponential and logarith- mic functions and conics.Second in a series of two courses of Collegeonometric functions,trigonometric identities and equations and complex numbers in rectangular and polar/trigonometric forms,rec- tangular and polar coordinates.A standard college trigonometryrad- icals, complex numbers,systems of equations,matrices,rational functions and exponential and logarithmic functions.MATH 136 and MATH 137 together comprise a standard two-semester college alge-
bra and trigonometry course.
MATH 137 Trigonometry with Analytic
Geometry
Transfer IN 3 Creditsono-
metric functions,trigonometric identities and equations and complex numbers in rectangular and polar/trigonometric forms,rectangular
and polar coordinates and conics.
MATH 141 Mathematics for Elementary Teachers
4 Credits
Prerequisites: Successful completion of MATH 111 or demonstrated competency through appropriate assessment or a grade of"C" or better in MATH 035 or MATH 043.An in-depth treatment of common topics underlying an elementary mathematics curriculum.Students in the course will gain an appreciation for mathematics and will add to their pedagogical expertise by gaining conceptual understanding of elementary mathematics through the use of selected modes,materi- als, and problem solving situations.The course is designed to connect knowledge of the real number system to other subjects.The selection of topics presented in this course is based upon standards and recom- mendations for the mathematical content knowledge essential for prospective teachers made by the National Council ofTeachers of Mathematics,the Mathematical Association of America,and the
Prerequisites: Demonstrated competenecy through appropriate assessment or MATH 131 and MATH 133 or MATH 136.An introducto- ry course in calculus.Fundamental concepts and operations of calculus including algebraic,exponential and logarithmic functions:limits,con- tinuity, derivatives,points-ofinflection,first-derivative test,concavity, second-derivative test,optimization,antiderivatives,integration by substitution,and elementary applications of the derivative and of the definite integral.
MATH 202 Brief Calculus II
Transfer IN 3 Credits
Prerequisites: MATH 201.Covers topics in elementary differential equations,calculus of functions of several variables and infinite series.
MATH 211 Calculus I
Transfer IN 4 Credits
Prerequisites:Demonstrated competency through appropriate assess- ment or MATH 131 and MATH 132 or MATH 133 and MATH 134 or MATH 136 and MATH 137.Reviews the concepts of exponential,loga- rithmic and inverse functions.Studies in depth the fundamental con- cepts and operations of calculus including limits,continuity,differenti- ation including implicit and logarithmic differentiation.Applies differ- | 677.169 | 1 |
Algebra Solving Quadratics Flip Book
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Solving Quadratic Equations Flip Book for Algebra and PreCalculus.
Great addition to your classes that cover Quadratic Equations. This easy to make, foldable, paper saving Flip Book helps your students review and enhances your teaching of the techniques needed to solve Quadratic Equations. The flip book can be incorporated in an interactive notebook and pulled out as a guideline to solving quadratic equations when students need help or used as a standalone. There are no complex solutions.
The foldable flip book includes Factoring (Zero Property), Square Root, Quadratic Formula, Graphing, and Completing the Square Methods. Examples of each are done out plus 14 practice problems. Great review before a test and for end of year review.
This flip book will measure 4 1/4" by 7" when printed. Three sheets of paper 8 1/2 by 11" will make two books. There is room for notes on the back.
Printing directions and full solutions key are included also. Double sided printing is required.
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Understand the base of the coding theory as an application of finite fields.
Demonstrate knowledge that the rational numbers and real numbers can be ordered and that the complex numbers cannot be ordered, but that any polynomial equation with real coefficients can be solved in the complex field.
Discuss the three major concrete models of Boolean algebra: the algebra of sets, the algebra of electrical circuits, and the algebra of logic.
Describe other applications of abstract algebra such as in avoiding problems of round off in computations.
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Description: In a Liberal Arts Math course, a common question students ask is, 'Why do I have to know this?' A Survey of Mathematics with Applications continues to be a best-seller because it shows students how we use mathematics in our daily lives and why this is important. The Ninth Edition further emphasizes this with the addition of new 'Why This Is Important' sections throughout the text. Real-life and up-to-date examples motivate the topics throughout, and a wide range of exercises help students to develop their problem-solving and critical thinking skills. Angel, Abbott, and Runde present the material in a way that is clear and accessible to non-math majors. The text includes a wide variety of math topics, with contents that are flexible for use in any one- or two-semester Liberal Arts Math | 677.169 | 1 |
How do mathematical equations translate into physical form? What formulas are needed to create a work of art? Students in A "Sine" of the Times chose their inspiration, developed designs... More > using oscillation graphs, and wrote reflections about how they learned to see math as well as themselves as mathematicians. With patterns as unique as their creators, this publication will inspire students and teachers to find creative applications for math in their own classrooms'Calculus...the way to do it' (140 pages) is a study of the calculus from basics up to GCE Advanced Level (years 12, 13 and 14). Section 1 is appropriate for students of GCSE Additional Mathematics... More > (year 12) and GCE Advanced Subsidiary Level (year 13). Section 2 is appropriate for GCE Advanced Level (year 14). It contains clearly explained teaching text, worked examples (in graduated order of difficulty), followed by exercises (with fully worked answers).< Less
"BASIC MATH. An Introduction to Calculus" is an easy way to learn to learn mathematics with four main chapters; the first is based on set theory, the numerical system and the real straight... More > with the Cartesian system from the plane and space. The second chapter shows applications of the theory of sets, permutations, combinations, relations and functions. The third chapter illustrates translations and functional models with the types of functions: real, polynomial, constant, linear, quadratic, exponential, logarithmic, trigonometric and inverse function. The fourth chapter develops equations and inequalities, along with linear or nonlinear systems of equations and inequalities. The fifth chapter concludes with solved recapitulation exercises.
This work is aimed at university students in traditional academic programs or distance education in economic, administrative, social and humanistic sciences.< Less
This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high... More > schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed. A brief tutorial on using Gnuplot to graph trigonometric functions is included.< Less
Beginning and Intermediate Algebra was designed to reduce textbook costs to students while not reducing the quality of materials. This text includes many detailed examples for each section along with... More > several problems for students to practice and master concepts. Complete answers are included for students to check work and receive immediate feedback on their progress.
Topics covered include: pre-algebra review, solving linear equations, graphing linear equations, inequalities, systems of linear equations, polynomials, factoring, rational expressions and equations, radicals, quadratics, and functions including exponential, logarithmic and trigonometric.
Each lesson also includes a World View Note which describes how the lesson fits into math history and into the world, including China, Russia, Central America, Persia, Ancient Babylon (present day Iraq) and | 677.169 | 1 |
Nahant GeometryKate C.
...Mathematics particularly is a subject where knowledge gaps in elementary arithmetic and algebra can cause cascading problems. A student who can easily factor a quadratic equation with multiple techniques may misunderstand "simpler" ideas, such as the meaning of negative exponents. A student who... | 677.169 | 1 |
A playful, irreverent counting book that celebrates the special things we know and love about Australia. Start with 'one little nipper', then count the pies, potaroos and blue wrens up to 'twelve kelpie legs' - in this friendly and amusing book you can count from one to a thrillion!. more...
This is the second revised edition of an introduction to contemporary relative homological algebra. It supplies important material essential to understand topics in algebra, algebraic geometry and algebraic topology. Each section comes with exercises providing practice problems for students as well as additional important results for specialists.... more...
A Concrete Approach to Abstract Algebra begins with a concrete and thorough examination of familiar objects like integers, rational numbers, real numbers, complex numbers, complex conjugation and polynomials, in this unique approach, the author builds upon these familar objects and then uses them to introduce and motivate advanced concepts in algebra... more...
Includes proof of van der Waerden's 1926 conjecture on permanents, Wilson's theorem on asymptotic existence, and other developments in combinatorics since 1967. Also covers coding theory and its important connection with designs, problems of enumeration, and partition. Presents fundamentals in addition to latest advances, with illustrative problemsHelp learners in grades 1–8 get it "write" with practical strategies to help them write and understand mathematics content. This resource is designed in an easy-to-use format providing detailed strategies, graphic organizers, and activities with classroom examples by grade ranges. Specific suggestions for differentiating instructionExpander families enjoy a wide range of applications in mathematics and computer science, and their study is a fascinating one in its own right. Expander Families and Cayley Graphs: A Beginner's Guide provides an introduction to the mathematical theory underlying these objects. more...
This book focuses on combinatorial problems in mathematical competitions. It provides basic knowledge on how to solve combinatorial problems in mathematical competitions, and also introduces important solutions to combinatorial problems and some typical problems with often-used solutions. Some enlightening and novel examples and exercises are well... more... | 677.169 | 1 |
Contains strategies for solving and provides completely worked-out solutions to all odd-numbered exercises within the text, the review sections, the True-False Quizzes, the Problem Solving sections, and to all the exercises in the Concept Checks, giving students a way to check their answers and ensure that they took the correct steps to arrive at the answer. This is the most popular supplement for students.This is the most popular supplement for students. | 677.169 | 1 |
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A proven motivator for readers of diverse mathematical backgrounds, this book explores mathematics within the context of real life using understandable, realistic applications consistent with the abilities of most readers.Graphing techniques are emphasized, including a thorough discussion of polynomial, rational, exponential, and logarithmic functions and conics. Chapter topics include Functions and Their Graphs; Trigonometric Functions; Analytic Trigonometry; Analytic Geometry; Exponential and Logarithmic Functions; and more.For anyone interested in trigonometry. | 677.169 | 1 |
DESCRIPTION: The author team of David Barton and Anna Cox has been joined by Philip Lloyd to provide a brand new, easy-to-use, write-on workbook. It contains approx 80 self-contained assignments linked to the Delta Mathematics textbook, making it easy for the teacher to give practice to students without having to devise new work. It contains a huge array of exercises that are linked to the corresponding Delta Mathematics. This makes it easy for teachers to choose, and students to remember, homework, extra practice and revision exercises that match what has been done in class. This workbook can be used alongside any mathematics textbook, and contains the worked examples, content summaries, and real-world, in-context problems, investigations and puzzles that David Barton is famous for - encouraging all students to gain and demonstrate a thorough understanding of mathematical concepts at NCEA Level 3 | 677.169 | 1 |
This textbook contains a collection of six high-quality chapters. The techniques are illustrated by a wide sample of applications. This book is devoted to Linear Mathematics by presenting problems in Applied Linear Algebra of general or special interest.
It introduces the basic notions of group theory by a thorough treatment of important examples, including complex numbers, modular arithmetic, symmetries, and permutations. Also included are applications to communications, cryptography, and coding theory.
This book is on modern algebra which treat various topics in commutative and noncommutative algebra and provide introductions to the theory of associative algebras, homological algebras, algebraic number theory, and algebraic geometry.
This book is on Preliminaries about the Integers, Polynomials, and Matrices; Vector Spaces over Q, R, and C; Inner-Product Spaces; Groups and Group Actions; Theory of a Single Linear Transformation; Multilinear Algebra; Advanced Group Theory; etc.
This book surveys fundamental algebraic structures and maps between these structures. Its techniques are used in many areas of mathematics, with applications to physics, engineering, and computer science as well.
This introduction to modern or abstract algebra addresses the conventional topics of groups, rings, and fields with symmetry as a unifying theme, while it introduces readers to the active practice of mathematics.
This is a book on linear algebra and matrix theory. It gives a self- contained treatment of linear algebra with many of its most important applications which does not neglect arbitrary fields of scalars and the proofs of the theorems.
An brief overview of pre-algebra and introductory algebra topics, written in a style meant to give you the basic facts in a way that is easy to understand, suitable for high-school Algebra I, or for anyone who wants to learn introductory algebra.
This book is an introduction to the basic concepts of linear algebra, along with an introduction to the techniques of formal mathematics. It begins with systems of equations and matrix algebra before moving into the advanced topics.
This book contains selected topics in linear algebra, which represent the recent contributions in the most famous and widely problems. It includes a wide range of theorems and applications in different branches of linear algebra, etc.
Focus on the mathematical framework that underlies linear systems arising in physics, engineering and applied mathematics - from the theory of linear transformation on finite dimensional vector space to the infinite dimensional vector spaces.
This book brings out how sets in algebraic structures can be used to construct the most generalized algebraic structures, like set linear algebra / vector space, set ideals in groups and rings and semigroups, and topological set vector spaces.
Super Linear Algebras are built using super matrices. These new structures can be applied to all fields in which linear algebras are used. Super characteristic values exist only when the related super matrices are super square diagonal super matrices.
This book is intended for researchers in applied mathematics and scientific computing as well as for practitioners interested in understanding the theory of numerical methods used for eigenvalue problems.
This unique text presents the new domain of consistent non-linear counterparts for all basic objects and tools of linear algebra, and develops an adequate calculus for solving non-linear algebraic and differential equations.
It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques.
TThis book is a standalone version of part of From Arithmetic To Differential Calculus (A2DC), a course of study developed to allow a significantly higher percentage of students to complete Differential Calculus in three semesters.
This book is to present a complete course on global analysis topics and establish some orbital applications of the integration on topological groups and their algebras to harmonic analysis and induced representations in representation theory.
You will find algebra exercises and problems, grouped by chapters, intended for higher grades in high schools or middle schools of general education. Its purpose is to facilitate training in mathematics for students in all high school categories. | 677.169 | 1 |
The Mutual Construction of Statistics and Society (Routledge
Students who have mastered the material are given "enrichment" opportunities, while those who have not mastered it receive additional instruction on the topic. This is the application deadline for most courses. Topics include: Riemannian geometry, Ricci flow, and geometric evolution. You will find on this website: Theory: theoretical explanation of how and why you should be teaching mental models. This WEBsite makes available for downloading a Mathwright Library Player which may be used as a viewer for the WorkBooks in this Mathwright Library.
Pages: 314
Publisher: Routledge; 1 edition (September 13, 2010)
ISBN: B0042FZYQO
Saxon Math Intermediate 5: Reteaching Masters 2008
Houghton Mifflin Mathmatics: Ways To Success Cd-Rom Lv 1
Learning ICT with Maths (Teaching ICT through the Primary Curriculum)
Teaching Language Arts, Math, and Science to Students with Significant Cognitive Disabilities
50 Math and Science Games for Leadership
It's Our Business: Level 5 (Mathematics Readers)
Teach the concept to the whole class, and then go around to each student to check for understanding. As students need help, help them individually. This also allows for the teacher to conduct error analysis and find out where students are making mistakes. Error correction is a great way for students to learn math Advanced Decimals (Straight Forward Math Series). This methodology was created by a Singaporean teacher called Hector Chee. Due to its practicality, the method was soon taught in all schools, starting from Primary One. This new method presented a challenge to parents who might have been taught using algebra or other math methods The Mutual Construction of Statistics and Society (Routledge Advances in Research Methods) online. The placement test from DISTAR Arithmetic I is illustrated. Within program assessments help determine the efficacy of instruction. Teachers may decide to repeat lessons to ensure firm responding (mastery) before moving on or accelerate students to higher lessons/levels based on their performance Holt McDougal Geometry: Practice and Problem Solving Workbook Teacher Guide. Retrieved from Hiebert, J., & Grouws, D Math Connects, Grade 4, Real-World Problem Solving Readers Package (On-Level) (ELEMENTARY MATH CONNECTS). Formal authority: Authoritative teachers incorporate the traditional lecture format and share many of the same characteristics as experts, but with less student interaction Math for All Participant Book (3-5). It includes defining the different figures, as well as describing their location and movement in space. Geometry concepts can be used in subjects such as reading and social studies, as well as math Summer Express Between PreK and Kindergarten. Observing students closely, experts would alert students of wrong moves, would point out reasons for wrong moves, and would demonstrate correct moves repeatedly. This modeling and coaching process between experts and students, often considered apprenticeship in nature, is the very foundation of the sociocultural approach that requires social participation and interaction of the whole classroom community download The Mutual Construction of Statistics and Society (Routledge Advances in Research Methods) pdf. Give them time to process the information before moving on to the next level. Able to use across grade levels, from early elementary through high school MathVIDS also has some excellent video clips of math being taught using the CRA method (Concrete, Representational, and Abstract) Saxon Math 1 Texas: Technology Pack.
ActivInspire® and Promethian® are registered trademarks of Promethean Limited. Apple, the Apple logo, and iPad are trademarks of Apple Inc., registered in the U. App store is a service mark of Apple Inc. ExamView® is a registered trademark of Turning Technologies, LLC Holt McDougal Larson Algebra 2: EasyPlanner DVD-ROM. Specific software provides students more focused feedback, allowing for the computer to tutor the student. These can be quite sophisticated, with the direction of the software path being determined by the student's correct or incorrect response Measurement, Grade 1 (Hands-On Mathematics). The information in the unit planning organizers can easily be placed into the curriculum model in used at the local level during the revision process. It is expected that local and/or regional curriculum development teams determine the "Big Ideas" and accompanying "Essential Questions" as they complete the units with critical vocabulary, suggested instructional strategies, activities and resources GED Test For Dummies with Online Practice.
A high school teacher in Marrero, La., demonstrates how she helps her students expand on their existing knowledge by assimilating new words. In this video, a 5th grade teacher in Bellevue, Wash., shows the strategies she uses to help students analyze texts in small and large discussion groups Teaching and Learning Geometry. Students may have poor decoding (reading) skills or expressive or receptive language difficulties MPJ's Ultimate Math Lessons. The thorough animated "whiteboard" explanations for every problem means students have the opportunity to get a better education than they could receive in any other way: With its slider-bar controller, you can fast-forward the explanation to the exact spot where you got stumped, then listen to the explanation Saxon Math Florida: Test Prep Teachbook. InstructionalDesign.org is designed to provide information about instructional design principles and how they relate to teaching and learning. Instructional design (or instructional systems design), is the analysis of learning needs and systematic develoment of instruction. Resources on this site were created by Greg Kearsley and Richard Culatta Effective instructional designers are also familiar with a wide range of educational technology that can be used for delivering learning experiences Dot-To-Dot, 1-100 (Classroom Helpers). The learners in this approach, manipulate concrete objects and/or perform activities to arrive at a conceptual understanding of phenomena, situation, or concept. The environment is a laboratory where the natural events/phenomena can be subjects of mathematical or scientific investigations The Amazing Mathematical Amusement Arcade. Interactive Note Taking (INT) begins with a skeleton of the class notes in the form of a workbook. The students then adds to this skeleton during class adding missing details and their own notes to what is already there. This allows students greater engagement with the material during class, a greater chance of comprehending the material there and an alternative to the text in which they have a degree of ownership Lifepac Mathematics 6th Grade.
Concrete and Virtual Manipulatives Research: The George Mason University Mathematics Education Center focuses on the study of concrete and virtual manipulatives. You will find a list of published articles and abstracts on this topic. The Critical Thinking Community comprises The Center for Critical Thinking and Moral Critique and the Foundation For Critical Thinking. "The work of the Foundation is to integrate the Center's research and theoretical developments, and to create events and resources designed to help educators improve their instruction" (Mission) Houghton Mifflin Harcourt Math West Virginia: Grab And Go Kit Games 1-10 Level 2. Combining direct instruction with cooperative learning procedures did not produce higher levels of achievement than cooperative learning alone Making Sense of Fractions. For example, they can survey what skills will be taught to students during the upcoming year and assess whether students have these skills or not on a teacher-developed pretest. Further, as teachers provide instruction in the classroom, they can assess key aspects of the lesson to determine if further instruction is needed or if they can proceed to the next lesson pdf. In that context, I think (good) teachers are generally well regarded and generally treated professionally. (Outside of teacher union shenanigans.) Our culture's perception of teachers is pretty much like our culture's perception of plumbers and accountants Selected Regular Lectures from the 12th International Congress on Mathematical Education. Is this k value positive or negative, and what does it tell you about dA/dt? for your value of k, solve this equation for A(t). Your solution in part b) will have an arbitrary constant in it. Calling that constant D, find its exact value. Using your final solution to part c), make a reasonable argument that A(t) is never larger than 5. Notice that the above is still not an easy problem it wasn't supposed to be Math in Focus: Singapore Math: Assessment. Representation: students use models, diagrams, and mathematical symbols in the process of learning mathematics. Further descriptions of each principle, content standard, and process standard can be located at the NCTM website by selecting the Standards and Focal Points section. As a member of NCTM, you will have access to the complete online document Saxon Math 4: Overhead Transparencies and Manipulatives Binder. The dimensions to evaluate the design of the app "include (C1) Navigation, (C2) Ease of Use, (C3) Customization, (C4) Aesthetics, (C5) Screen Design, (C6) Information Presentation, (C7) Media Integration, and (C8) Free of Distractors" (p. 125) Harcourt School Publishers Math: Intervention Start/Act Cd Package of 1 Grade 6. Mathematics: A Simple Tool for Geologists (2000, David Waltham, Blackwell Science) This book aimed at students teaches simple mathematics using geological examples to illustrate mathematical ideas. Quantitative Skills Assessment In Geoscience Courses ( This site may be offline. ) (Shah and West) A website at Columbia University with tips for teaching quantitative skills, a guide for assessing quantitative skills, and a survey of the skills in courses Cooperative Math Pre-k - 2 Revised Advanced Mathematics: Answers and Hints for Book 3 (School Mathematics Project Revised Advanced Mathematics) (Bk.3). | 677.169 | 1 |
Chamber's seven-figure mathematical tables consisting of
I brought forth destruction and chaos for the pleasure of the lower beings. I really got tired of dealing with those kinds of people and that was one of the reasons (along with simply getting busier here at Lamar) that made me decide to quit answering any email asking for help. Mathematics grade nine, GCF of 750 and 125, free worksheets for adding negative numbers 5th grade, factoring calculator trinomials, free printable mats sats papers key stage 2, algebra expression calculator, instructions for graphing systems of equations.
Pages: 454
Publisher: Chambers (1946)
ISBN: B0007JLHW4
Precalculus
Trigonometry A Mini Book Overview for High Schooler's and Others (Mini Book Series 5)
Vega Seven Place Logarithmic Tables of Numbers and Trigonometrical Functions
TRIGONOMETRY WITH TABLES (A Beka)
Freshman Mathematics
New Plane and Spherical Trigonometry
College Algebra and Trigonometry with Analytic Geometry
The student is expected to: (A) calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association; (B) compare and contrast association and causation in real-world problems; and (C) write, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems. (5) Linear functions, equations, and inequalities Trigonometry (SparkCharts). In Magical Girl Lyrical Nanoha A's, Suzuka is the only one amongst Nanoha's circle of school friends who is neither an elite mage in a setting where the construction of magical circles requires a good deal of math, nor an overachiever who gets top marks at everything. She thus has sub-par grades in mathematics, although her language grades are not much better Chamber's seven-figure mathematical tables consisting of logarithms of numbers 1 to 108000, trigonometrical, nautical, and other tables online. Topics include group algebras and modules, semisimple algebras and the theorem of Maschke; characters, character tables, orthogonality relations and calculation; and induced representations and characters Study and Solutions Guide for Algebra and Trigonometry, 6th Edition. It is ideal for children from 2 to 5 years old. Simple version of popular Tangram designed with kids in mind. Avoids unnecessary design and decoration to keep kids attention over important geometric concepts that the game develops. The pieces are handled naturally following the movement of fingers as much to move them or turn them both at once Trigonometry (Speedy Study Guides). I am looking forward to using this course again in a few years with our now-9th grader. I have been a huge fan of Alpha Omega products for the past 10 years, but please don't purchase the 12 grade mathematics Lifepacs for your child Plane Trigonometry 4TH Edition. Online Messaging System Our online messaging system is an effective tool to track your custom paper orders because it helps you to stay in contact with your essay writers 24/7. What can you expect from purchasing custom essay writing on our website Plane Trigonometry Made Plain With Logarithmic and Trigonometric Tables?
The edutainment short Donald Duck In Mathmagic Land plays this straight at the beginning, when Donald insists that math is for "eggheads". The Spirit of Adventure manages to convince him otherwise... by showing him how he can use it to shoot pool Elements of Geometry and Conic Sections. Free trial of ti 84 calculator, factoring quadratics equations, math worksheets slope fun printable, 4th grade regularfractions, free worksheets geometric mean. Ordering least to greatest, kids, how to solve dividing scientific notations, maths challenge book high school australia, free online algebra problem solver, factor by grouping volume worksheet, rational expressions free worksheets download. The method was first carried out on a large scale by another Dutchman, Willebrord Snell (1580?–1626), who surveyed a stretch of 130 km (80 miles) in Holland, using 33 triangles pdf. You will also have to find the composite inverse trig functions with non-special angles, which means that they are not found on the Unit Circle ELEMENTS OF PLANE AND SPHERICAL TRIGONOMETRY.
Seven Place Logarithmic Tables of Numbers and Trigonometrical Functions
A System of Geometry and Trigonometry: Together with a Treatise On Surveying; Teaching Various Ways of Taking the Survey of a Field; Also to Protract ... Or, an Accurate Method of Calculating the Ar
Maple for Trigonometry (Trade/Tech Mathematics S)
Mathematical Excursions to the World's Great Buildings
Trigonometry Success In 20 Minutes a Day
The Elements of Plane Trigonometry
By download. You do need to practice using the calculator EARLY ON and become confident about its operations College Trigonometry. These books are very charming, written in a conversational tone, with just a touch of dry humor. The name comes from the main character in the series, Fred, who learns about each book's topics thru practical, everyday experiences. All of these books are complete courses, with plenty of problems to work. Explore the website to learn more and see UCCP Open Access College Prep -- free online AP Calculus courses, includes Calculus AB and Calculus BC, use as prep for taking AP tests Mathematics Articles by Stan Brown -- online tutorials in Algebra, Trig, retired math teacher High School Mathematics Competition Seminar: trigonometric(Chinese Edition). Just enter the known angles and sides and all the others will be calculated for you. To see how the answers were arrived at just select the 'Workings' button and a page is shown with a comprehensive set of steps to show how each was calculated download Chamber's seven-figure mathematical tables consisting of logarithms of numbers 1 to 108000, trigonometrical, nautical, and other tables pdf. The instructions which come with your calculator usually give a number of examples and illustrate a variety of possibilities Trigonometry Mechanics: Trigonometry and Graphs Mechanics and Machine Elements. A Futurama episode plot involves a mind switching device that could only switch minds that haven't been switched before. Amy and the Professor try it first, but realize they can't switch back directly (since Amy and the Professor's minds have already been switched) Mathematics Made Simple. Example 2: A 30 m ladder on a fire engine has to reach a window 26 m from the ground which is horizontal and level Trigonometry for Secondary Schools. Angle $EOD$ is twice angle $EAD$, and angles $EAC$ and $DEC$ are equal. From which he derived the half angle and multiple angle formulae. [See Note 6 below] While many new aspects of trigonometry were being discovered, the chord, sine, versine and cosine were developed in the investigation of astronomical problems, and conceived of as properties of angles at the centre of the heavenly sphere pdf. You'll get personalized, one-to-one help during every session Trigonometry and double algebra [Paperback] [2010] (Author) Augustus De Morgan. He considered every triangle as being inscribed in a circle, so that each side became a chord. While chords were easy to calculate in some special cases with Euclidean knowledge, in order to complete his table Hipparchus would have needed to know many formulas of plane trigonometry that he either derived himself or borrowed from elsewhere Essentials of Precalculus, Algebra and Trigonometry. Sum to product trig worksheets, solving linear systems by linear combination cheat, algebra 1 worsheet. Printable practice GCF and LCM tests, hardest math equation ever, pre algebra quizzes free. Algebra structure and method book 1 chapter 5 booktest, find the root if its a real number, printable worksheets problem sums for primary 1, The Latest in math, solving a binomial expression College Algebra and Trigonometry: Plus MyMathLab Student Access Kit, 2nd Edition (Books a la Carte). You see, the infinite series for eθ looks like this: eθ = 1 + θ + θ2 + θ3 + θ4... 2! 3! 4! It is immediately obvious that the form of the equation is similar. To show us best just how similar it really is, we need to bring in imaginary number s. What we can do is multiply θ by i in the expansion ( i is - loosely speaking - the square root of minus one; so i × i = -1) College Algebra and Trigonometry, Ratti, Mcwaters, Annotated Teachers Edition 2008. | 677.169 | 1 |
Fall 2016
MATH061Basic Mathematics
An introduction the concepts needed for further study in Mathematics. Topics covered include operations with whole and rational numbers, decimals, percents, ratio and proportions, and their applications. (Nontransferable, nondegree applicable)
Code #
Division
Department
Meeting Days
Time
Location
Status
10072
M&S
MATH
Tuesday Thursday
Start Time: 11:20 AM End Time: 12:45 PM
Building: 400 Room: 404
Enrolled: 40 Available: -5
Class begins: 2016-08-15, Class ends: 2016-12-09
MATH081Beginning Algebra
Prerequisite: MATH 071 with a grade of "C" or better or appropriate placement.
An introduction to the concepts of Algebra. Topics covered include solving equations, polynomials, factoring, rational expressions, graphs and linear equations, systems of linear equations, and inequalities. (Nontransferable, nondegree applicable) | 677.169 | 1 |
Introduction to Geometry
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
2.64 MB | 124 pages
PRODUCT DESCRIPTION
This document contains geometry concepts introduced in elementary and middle school. It has 10 chapters of vocabulary terms, definitions, symbols, drawings, formulas, and examples, and would be a good resource for any student preparatory to a high school level geometry class. This booklet has been used in a summer school math enrichment class as well as a resource tool for other teachers.
By Math Fan | 677.169 | 1 |
Having students create their own problems and work can lead to powerful engagement and motivation.
For the Quadratic Quandaries Performance Task, I find that students love the flexibility and creative room that the performance task allows. Although it can be difficult, I try to reach a good balance of clear expectations and flexibility for this task.
Student work shows the variety of not only topics, from Football Functionsto Flappy Bird Rage, but also in terms of depth of understanding. Many projects had relatively straight forward quandaries (when does an object hit the ground, what is the maximum or minimum height that the object reaches, etc.), but some had complex questions, like the Volleyball Complex Questionwhere the group found an interesting application of quadratic functions.
An important note to the student work in this reflection is it is a mix of groups from my advanced honors section and groups from my fundamentals section. I am continuing to try and design and implement more engaging and open-ended tasks like the Quadratic Quandaries Performance Task for ALL students.
The reason I have students complete this exercise is twofold. First, I want students to review how to create equations to model real-life scenarios because part of today's lesson is a performance task that asks students to do just that. Second, I want to help frame the unit on quadratics to students by asking questions that not only have them think about what quadratics are, but how quadratic functions are different from linear and exponential functions.
It is important to remind/preview with students that they will be asked to create their own problem that can be modeled with quadratic functions later in the lesson to point out the relevance and importance of the class notes.
Resources
In this section I lay out the expectations and grading components for the Quadratic Quandaries Performance Task/Project. I hand out a copy of the Performance Task Checklist and Rubric: Quadratic Quandaries to each student and give students a few minutes to silently review the assignment before discussing it.
I then go through the overall intent of the project as well as the different components. I also review logistics (materials and where they are in the classroom, expectations around positive behavior during group work, etc..).
Time for questions is then provided and students dive into working on the project! It also is a good idea to explicitly remind students that they already have made excellent headway with the project through the work they did on their entry ticket in today's class.
This project is geared to give students an opportunity to further develop the Math Practice standards MP3 and MP4. Students have time to create arguments and critique the perspective of their peers in a setting that encourages students to make connections of how exponential functions can be used to model situations and better understand real-world phenomena.
During this time my role is that of a facilitator. I begin the time by making sure all of the groups have chosen a topic to model with a quadratic function. I ask reflective questions to help students monitor their progress and also to encourage the group to converse and collaborate with each other.
For homework, each group of students needs to complete the project outside of class. As an Exit Ticket, I have each group of students reflect on the following prompts:
What did your group do well on?
What can your group improve upon for next time? What concrete suggestions do you have for how your group can make those improvements a reality?
Before ending class, I ask each group to tell me what their plan is to complete the project. This closing activity helps students break down the steps needed to complete the project. I push them to articulate a plan for completing the project and monitoring their progress as they do so. My goal here is to promote a skill that is important as they continue with their secondary and post-secondary education.
Next class, when students turn in their projects, I will take the time to display all of the presentations and wonderful work that students have created. I also try to invite school administrators and other teachers to see the work and, again, start the year on a positive note of high expectations, challenge with just the right balance of support. | 677.169 | 1 |
Summary and Info
The second in a series of systematic studies by a celebrated mathematician I. M. Gelfand and colleagues, this volume presents students with a well-illustrated sequence of problems and exercises designed to illuminate the properties of functions and graphs. Since readers do not have the benefit of a blackboard on which a teacher constructs a graph, the authors abandoned the customary use of diagrams in which only the final form of the graph appears; instead, the book's margins feature step-by-step diagrams for the complete construction of each graph. The first part of the book employs simple functions to analyze the fundamental methods of constructing graphs. The second half deals with more complicated and refined questions concerning linear functions, quadratic trinomials, linear fractional functions, power functions, and rational functions. | 677.169 | 1 |
Math for the Frightened
45 View(s),
published at 2011, written by
Colin Pask, published by
Math for the frightened by Colin Pask Introduces the reader to the main ideas of mathematics, and painlessly demonstrates how they are expressed in symbols while helping to overcome the fear of math and begin to appreciate the science tha.... Published date on: 2011 with total page: 380 pages. Publisher of Math for the Frightened is .
Introduces the reader to the main ideas of mathematics, and painlessly demonstrates how they are expressed in symbols while helping to overcome the fear of math and begin to appreciate the science that Einstein called the poetry of logical ideas.
Are you want to read online this ebook math for the frightened? If you have read an ebook before don't be hasitate to leave review about the book math for the frightened investigation into the spatial politics of separation and division in South Africa, principally during the apartheid years, and the effects of these physical and conceptual barriers on the land. In contrast to the weight of literature focusing on ...
An exploration of why women were singled out as witches in 15th-century in Germany. Sigrid Brauner examines the connections between three central developments in early modern Germany: a shift in gender roles for women; the rise of a new urban ideal o...
No matter how simple it may be, solving mathematical equations can be a challenge for quite a number of persons. Through his book titled "Secret Of Mental Math Arithmetic: 70 Secrets To Super Speed Calculation & Amazing Math Tricks", Jason Scotts see | 677.169 | 1 |
Product Information
'A Level Mathematics for Edexcel' covers the latest 2008 curriculum changes and also takes a completely fresh look at presenting the challenges of A Level. It specifically targets average students, with tactics designed to offer real chance of success to more students, as well as providing more stretch and challenge material 225 | 677.169 | 1 |
You'll gain access to interventions, extensions, task implementation guides, and more for this lesson.
Big Ideas:
Rational functions are expressed as algebraic fractions.
In an inverse proportion, as the magnitude of the independent variable increases, the magnitude of the dependent variable decreases, and their product stays the same.
Rational functions can be used to model many different relationships. In this task, students develop a rational function to model the inverse relationship between the per-person cost of a trip and the number of attendees. Students are encouraged to use a table of values to create an explicit function rule, and then to consider what happens to the per-person cost of the trip as the number of attendees increases. In this way, students have the opportunity to develop intuitions about the asymptotic behavior of rational functions.
Vocabulary: function, input, output, domain, range, fraction, average
Special Materials:
None | 677.169 | 1 |
This well-respected text gives an introduction to the modern approximation techniques and explains how, why, and when the techniques can be expected to work. The authors focus on building students' intuition to help them understand why the techniques presented work in general, and why, in some situations, they fail. With a wealth of examples and exercises, the text demonstrates the relevance of numerical analysis to a variety of disciplines and provides ample practice for students. The applications chosen demonstrate concisely how numerical methods can be, and often must be, applied in real-life situations. In this edition, the presentation has been fine-tuned to make the book even more useful to the instructor and more interesting to the reader. Overall, students gain a theoretical understanding of, and a firm basis for future study of, numerical analysis and scientific computing. A more applied text with a different menu of topics is the authors' highly regarded NUMERICAL METHODS, Third Edition.
"synopsis" may belong to another edition of this title.
About the Author:
Richard L. Burden is a Professor of Mathematics at Youngstown State University. His research interests include numerical linear algebra and numerical solution of partial differential equations.
Book Description Brooks Cole. Hardcover. Book Condition: New. 053439200834392008 | 677.169 | 1 |
PACKAGED FREE WITH:
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BEGINNING ALGEBRA Seventh Edition
R. David Gustafson and Peter D. Frisk, both of Rock Valley College
BUILD SKILLS AND DEVELOP CONFIDENCE WITH THE BEST-SELLING TEXT FOR THE BEGINNING ALGEBRA COURSE!
Tried and true, Gustafson and Frisk's Beginning Algebra teaches solid mathematical skills while supporting students with careful pedagogy. Each book in this series maintains the authors' proven style through clear, no-nonsense explanations, as well as the mathematical accuracy and rigor that only Gustafson and Frisk can deliver. The text's clearly useful applications emphasize problem solving to effectively develop the skills students need for future mathematics courses, such as College Algebra, or on the job.
SUPPORTED BY ONE-OF-A-KIND RESOURCES FOR INSTRUCTORS AND STUDENTS
The Seventh Edition of Beginning Algebra also features a robust suite of online course management, testing, and tutorial resources for instructors and students. This includes BCA/iLrn Testing and Tutorial, vMentor™ live online tutoring, the Interactive Video Skillbuilder CD-ROM with MathCue, a Book Companion Web Site featuring online graphing calculator resources, and The Learning Equation (TLE) Labs. | 677.169 | 1 |
Numerical Methods for Engineers
The seventh edition of Chapra and Canale's Numerical Methods for Engineers retains the instructional techniques that have made the text so successful. Chapra and Canale's unique approach opens each part of the text with sections called "Motivation," "Mathematical Background," and "Orientation" Each part closes with an "Epilogue" containing "Trade-Offs," "Important Relationships and Formulas," and "Advanced Methods and Additional References." Much more than a summary, the Epilogue deepens understanding of what has been learned and provides a peek into more advanced methods. Helpful separate Appendices. "Getting Started with MATLAB" and "Getting Started with Mathcad" which make excellent references | 677.169 | 1 |
If you're seeking solutions to advanced and even esoteric problems, Advanced Analytical Models goes beyond theoretical discussions of modeling by facilitating a thorough understanding of concepts and their real-world applications—including the use of embedded functions and algorithms | 677.169 | 1 |
Algebra I EOC Study Guide
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This study guide is a must have reference when teaching and reviewing for the Texas STARR Algebra I End of Course Exam or any Algebra I EOC. I created this product after working with average students and special populations. I address the most frequent areas of confusion and difficulty. I added, changed then edited again to give you a most effective tool for students. Print it as a wall poster for easy reference or print it in black & white then have students use their own colored pencils to connect and remember! Your students will use it and love it! Also, use it as a quick reference when teaching Algebra II. Try it25. | 677.169 | 1 |
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$81 sixth edition of Elementary Algebra retains the same basic format and style as the fifth edition. The book is intended to be used in a lecture format class. Each section of the book can be discussed in a forty-five to fifty minute class session. Intended for schools that teach the course in a lecture environment, students are able to prepare for class by answering the Getting Ready for Class questions. This text is targeted at a one-semester or one-quarter first course in algebra, (elementary algebra, introductory algebra, beginning algebra, developmental mathematics). | 677.169 | 1 |
Description: Phylogenetics is the area of biology concerned with inferring ancestral relationships between collections of taxa (e.g. species). An ancestral history of species is usually summarized by a phylogenetic tree. Phylogenetics is a rich area for interactions between mathematics and biology, and mathematical tools in phylogenetics include: discrete math, probability, statistics, algebra and geometry.
This REU will focus on the mathematical study of tree space. Tree space is a geometric object which consists of the set of all metric trees and with a natural distance measure comparing those trees. In many situations, different data sets or different methods for analyzing the same data can yield different trees. Given a collection of inferred trees, how do we compare them? Is there a natural notion of a consensus tree or an average tree? Tree space provides useful ways to address these questions, by phrasing the questions as geometric operations.
After spending a few days developing background on the mathematical aspects of phylogenetics, we will work on reading papers and identifying open problems to tackle.
Basic Reading:
Semple and Steel, Phylogenetics (Excellent introductory text on mathematical phylogenetics. I will provide two copies of the book.)
Billera, Louis J., Holmes, Susan P., Vogtmann, Karen.
Geometry of the space of phylogenetic trees.
Adv. in Appl. Math.27 (2001), no. 4, 733–767. (This paper introduces the space of phylogenetic trees and proves some main properties about it: especially: it is a CAT(0) space so has unique geodesics.)
Federico Ardila, Megan Owen, Seth Sullivant. Geodesics in CAT(0) Cubical Complexes Advances in Applied Mathematics, 48: 142-163, 2012 (This paper gives a generalization of the Owen-Provan algorithm to arbitrary CAT(0) cubical complexes. Can the algorithm be improved? Is there a polynomial time algorithm?)
Moulton, Vincent; Steel, Mike Peeling phylogenetic `oranges'. Adv. in Appl. Math.33 (2004), no. 4, 710–727. (The "phylogenetic orange" space is a variation on tree space that takes into account probabilistic models of evolution. What do geodesics look like in this space? Can distances be computed quickly?) | 677.169 | 1 |
The past and present methods for teaching Calculus are now obsolete. There is a new way of learning Calculus. Professor Pine has discovered and invented a unique and wonderful way of learning this subject. Why not understand it, take pleasure in Calculus, instead of rote memorization? Tens of thousands of grateful students have learned Calculus and who never would have understood and passed their courses but for this exceptional teacher. Even non-students, professionals of every discipline, doctors, engineers, etc., or just plain curious people, have spoken to and written to the author expressing their joy, pleasure, even relief, at finally understanding this once mysterious subject. Students have call him "e pluribus unim" (one of a kind) compared to author Isaac Asimov, The Calculus Doctor, with the capacity to make simple what was complicated. Linus Pauling, the only double Nobel Prize winner, has commented on it, and the Inner London Educational Authority in typical British understatement says "this book is indispensable for all those who want to learn Calculus but who have missed the real meaning of Calculus as a result of the rigorous mathematics normally associated with this topic". The book How To Enjoy Calculus does it finally. Yes, it is very possible to understand and Enjoy Calculus. | 677.169 | 1 |
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Carefully developed for one-year courses that combine and integrate material from Precalculus through Calculus I, this text is ideal for instructors who wish to successfully bring students up to speed algebraically within precalculus and transition them into calculus. The Larson Calculus texts continue to offer instructors and students new and innovative teaching and learning resources. The Calculus series was the first to use computer-generated graphics, to include exercises involving the use of computers and graphing calculators, to be available in an interactive CD-ROM format, to be offered as a complete, online calculus course, and to offer this two-semester Calculus I with Precalculus text. Every edition of the series has made the mastery of traditional calculus skills a priority, while embracing the best features of new technology and, when appropriate, calculus reform ideas. Two primary objectives guided the authors in writing this book: to develop precise, readable materials for students that clearly define and demonstrate concepts and rules of calculus and to design comprehensive teaching resources for instructors that employ proven pedagogical techniques and saves the instructor time | 677.169 | 1 |
Calculus II (Integral Calculus)
Videos on a second course in calculus (Integral Calculus).
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About This Course
Published 1/2013
English
Course Description
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The course includes several techniques of integration, improper integrals, antiderivatives, application of the definite integral, differential equations, and approximations using Taylor polynomials and series. This course is required of engineering, physics, and mathematics majors.
This course contains a series of video tutorials that are broken up into various levels. Each video builds upon the previous one. Level I videos lay out the theoretical frame work to successfully tackle on problems covered in the next videos.
These videos can be used as a stand along course or as a supplement to your current calculus II course.
This course is consistently being populated with new videos.
This course is for anyone who wants to fortify their understanding of calculus II or anyone that wishes to learn calculus II can benefit from this course.
This course is consistently monitored ready to reply to any questions that may arise.
What are the requirements?
Calculus I (Differential Calculus)
What am I going to get from this course?
Fortify your understanding of Calculus II.
What is the target audience?
College Students goes over a second integration technique used to find indefinite integrals formed by a product of functions. This video goes over the derivation of the integration by parts formula by using the product rule as a starting point. In addition, the video goes over an example covering the application of this new integration technique.
This video goes over 3 examples, covering the proper way to use the integration by parts formula. This video includes an example covering the two forms of the integration by parts formula, an example where rewriting of the integrand is required, and a final example where an integral contains a single function.
This video goes over 2 examples, covering the proper way to find integrals that require the repeated application of the integration by parts formula. In addition, the tabular method for integration by parts is also introduced.
This video goes over an example, covering the proper way to find integrals that require the repeated application of the integration by parts formula specifically an integral that generates a constant multiple of the original integral. In addition, this integral will also be found by using the tabular method for integration by parts.
This video goes over three examples, covering the proper way to find definite integrals that require the application of the integration by parts formula. An example covering the tabular method is also presented cosine is odd sine is odd.
This video is an introduction to solving trigonometric integrals that contain combinations of trigonometric functions. Specifically, those that contain powers of sine and cosine. This video covers 1 basic example illustrating the case when the power of sine and cosine are odd.
This video is an introduction to solving trigonometric integrals that contain combinations of trigonometric functions. Specifically, those that contain powers of sine and cosine. This video covers 2 basic example illustrating the case when the power of sine and cosine are even.
This video continues illustrating methods in solving trigonometric integrals that contain combinations of trigonometric functions. Specifically, those that contain powers of sine and cosine. This video covers 2 challenging examples illustrating the case when the power of sine and cosine are even.
This video continues illustrating methods for solving trigonometric integrals that contain combinations of trigonometric functions. Specifically, those that contain products of sine and cosine with distinct arguments (angles). This video covers 3 examples illustrating the use of the product to sum identities for sine and cosine.
This video concludes the methods for solving trigonometric integrals that contain combinations of sine and cosine. This video covers 4 challenging examples that require the use of different trigonometric identities, multiplying by the conjugate, and factoring. | 677.169 | 1 |
ISBN 9789384089139
ISBN-10
9384089133
Binding
Paperback
Edition
3rd
Number of Pages
408 Pages
Language
(English)
Subject
Schools
NCERT Solutions Mathematics - Includes Chapter-wise Problems (Class 12) is a comprehensive book for class 12 students studying in schools affiliated to the Central Board of Secondary Education. The book covers the latest syllabus set by the National Council for Educational Research and Training, and helps students understand the problems in the textbook by providing detailed solutions for all the problems in the official NCERT textbook. The book includes step-by-step solutions to problems so that students understand the method of solving the problems. It helps students by presenting the questions in the exact same order as in the official textbook. The book is an essential resource for all the students studying in class-12.
About Disha Experts
Disha Publications is one of the leading publishers of India, publishing books and study materials for schools, medical entrance examinations, JEE, and other competitive exams conducted throughout the country. They have authored and published books such as 101 Speed Tests for IBPS-CWE Bank PO / MT Exam, Class 10 - Mental Ability for NTSE Stage I, DPP for AIIMS/NEET Biology with Solution Book, etc. | 677.169 | 1 |
Description of the book "Cambridge Mathematics Direct 6 Calculations Solutions":
Provides everything you need to plan, teach and assess the daily maths lesson from Reception to Key Stage 2. Cambridge Maths Direct 6 Calculations Solutions provides answers to all the questions posed in Cambridge Maths Direct 6 Calculations textbook and copymasters. They are listed under the title of the lesson in the teacher's handbook and follow the same order. Answers are given in several ways: complete solutions are listed wherever it is useful; facsimiles of completed copymasters are included where this is most helpful; for open-ended questions and investigations, the possibilities are indicated through examples. Solutions is a photocopiable resource.
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Mathematical Olympiad Challenges
Mathematical Olympiad Challenges is a rich collection of problems put together by two experienced and well-known professors and coaches of the U.S. International Mathematical Olympiad Team. Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory were selected from numerous mathematical competitions and journals. An important feature of the work is the comprehensive background material provided with each grouping of problems.
The problems are clustered by topic into self-contained sections with solutions provided separately. All sections start with an essay discussing basic facts and one or two representative examples. A list of carefully chosen problems follows and the reader is invited to take them on. Additionally, historical insights and asides are presented to stimulate further inquiry. The emphasis throughout is on encouraging readers to move away from routine exercises and memorized algorithms toward creative solutions to open-ended problems.
Aimed at motivated high school and beginning college students and instructors, this work can be used as a text for advanced problem- solving courses, for self-study, or as a resource for teachers and students training for mathematical competitions and for teacher professional development, seminars, and workshops.
À propos de l'auteur (2001)
Titu Andreescu received his Ph.D. from the West University of Timisoara, Romania. The topic of his dissertation was "Research on Diophantine Analysis and Applications." Professor Andreescu currently teaches at The University of Texas at Dallas. He is past chairman of the USA Mathematical Olympiad, served as director of the MAA American Mathematics Competitions (1998a "2003), coach of the USA International Mathematical Olympiad Team (IMO) for 10 years (1993a "2002), director of the Mathematical Olympiad Summer Program (1995a "2002), and leader of the USA IMO Team (1995a "2002). In 2002 Titu was elected member of the IMO Advisory Board, the governing body of the world's most prestigious mathematics competition. Titu co-founded in 2006 and continues as director of the AwesomeMath Summer Program (AMSP). He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1994 and a "Certificate of Appreciation" from the president of the MAA in 1995 for his outstanding service as coach of the Mathematical Olympiad Summer Program in preparing the US team for its perfect performance in Hong Kong at the 1994 IMO. Titua (TM)s contributions to numerous textbooks and problem books are recognized worldwide.
Dorin Andrica received his Ph.D. in 1992 from "Babes-Bolyaia University in Cluj-Napoca, Romania; his thesis treated critical points and applications to the geometry of differentiable submanifolds. Professor Andrica has been chairman of the Department of Geometry at "Babes-Bolyai" since 1995. He has written and contributed to numerous mathematics textbooks, problem books, articles and scientific papers at various levels. He is an invited lecturer atuniversity conferences around the world: Austria, Bulgaria, Czech Republic, Egypt, France, Germany, Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. Dorin is a member of the Romanian Committee for the Mathematics Olympiad and is a member on the editorial boards of several international journals. Also, he is well known for his conjecture about consecutive primes called "Andrica's Conjecture." He has been a regular faculty member at the Canadaa "USA Mathcamps between 2001a "2005 and at the AwesomeMath Summer Program (AMSP) since 2006.
Zuming Feng received his Ph.D. from Johns Hopkins University with emphasis on Algebraic Number Theory and Elliptic Curves. He teaches at Phillips Exeter Academy. Zuming also served as a coach of the USA IMO team (1997-2006), was the deputy leader of the USA IMO Team (2000-2002), and an assistant director of the USA Mathematical Olympiad Summer Program (1999-2002). He has been a member of the USA Mathematical Olympiad Committee since 1999, and has been the leader of the USA IMO team and the academic director of the USA Mathematical Olympiad Summer Program since 2003. Zuming is also co-founder and academic director of the AwesomeMath Summer Program (AMSP) since 2006. He received the Edyth May Sliffe Award for Distinguished High School Mathematics Teaching from the MAA in 1996 and 2002. | 677.169 | 1 |
This ebook is available for the following devices:
iPad
Windows
Mac
Sony Reader
Cool-er Reader
Nook
Kobo Reader
iRiver Story
The aim in this graduate level text is to outline the key mathematical concepts that underpin these important questions in applied mathematics. These concepts involve discrete mathematics (particularly graph theory), optimization, computer science, and several ideas in biology.
Two Classical Optimization Problems.- Gauss' Question.- What Does Solution Mean?.- Network Design Problems.- A New Challenge: The Phylogeny.- An Analysis of Steiner's Problem in Phylogenetic Spaces.- Tree Building Algorithms.
In the press
From the reviews of the first edition:
"The aim of this graduate-level text is to summarize mathematical concepts concerned with problems of shortest connectivity, and to demonstrate important applications of the theory, in particular in biology. … The book contains extensive references and gives rise to many problems for further research. … Examples are discussed in the history of evolution, taxonomy, historical linguistics and others." (Günther Karigl, Zentralblatt MATH, Vol. 1086, 2006) | 677.169 | 1 |
Maths@Bosworth
Revising For Maths : Revising For Maths
Maths @ Bosworth : Maths @ Bosworth Here at Bosworth we take the AQA Modular maths course. Students have taken module 1 – handling data and module 3 – number, which are both exams, and have also completed 2 pieces of coursework This gives a total of 50% towards the final grade.
Slide 3: The final 50% of the marks are made up of 2 papers. Paper 1 is non-calculator, and Paper 2 is calculator. This is made up mainly of Algebra, Shape, Space and Measures, with a small amount of number. Paper 1 – May 19th Paper 2 – June 2nd
Intranet : Intranet The Maths page on the intranet contains past papers, topics to cover (in the self supported study section) and links to some useful websites. To access the intranet from home type this link into the address bar
Slide 5: If there are any problems with that then: Google Bosworth college, and go to the college website. Click on the link called 'intranet' on the left hand side of the page. Then click on the link named 'Departments I-Z' and select Maths to enter the maths area.
Past Papers : Past Papers The best way to revise for maths is to practice questions. The best place to find questions is on past papers. The intranet contains past papers from June 2002 up until June 2006 for the old 3 tier system. To access them from the maths page on the intranet select GCSE, then module 5, and finally select the link 'Past Papers'.
Slide 7: To access more recent past papers you will need to visit the AQA website To go straight to the papers copy this link into your address bar Alternatively go to The past papers are all the old 3 tier system, as no module 5 exam has been sat for the new 2-tier system.
My Maths Website : My Maths Website This contains teaching pages, booster packs and games for all levels of maths. Website Username bosworth Password fraction
Slide 9: Every student has an individual login, which enables themselves and their maths teacher to track progress through the online worksheets. Each topic area on the website has an online homework attached, which is invaluable in testing understanding of that particular topic.
Slide 10: There are 4 appropriate booster sections on the website that target key topics with a revision section and then has an online worksheet to follow. A-A* Higher tier C's 2 B's Higher tier D's to C's Higher or Foundation tier 6 Boosters Foundation tier up to grade D Progress though these can also be tracked providing individual logins are used before starting any of the worksheets.
Other Useful Websites : Other Useful Websites
Slide 12: Revision Lists Booster pack tracking sheets Individual login passwords Revision CDs Revision guides
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Algebra - One On One manufacturer description
Algebra One on One is an educational game for those wanting a fun way to learn and practice Algebra. This program covers 21 functions which includes maximums, minimums, absolute values, averages, x/y, ax + b, axy + b, ax + by + c, squares, cubes, and so on. It has a practice and a game area.
It has a great help system that makes it easy for the beginner to do and understand algebra. It also has a "Einstein" level that even algebra experts will find fun and challenging. You can choose from a ten problem, a time trial, or a two-player game. High scores are saved and you are given a rank according to your score. The ranks are Novice, Learner, Veteran, Calculator, Math Pro, Math Whiz, Math Genius, and Einstein.
The practice menu lets you practice each function individually. The game menu lets you choose one function, two functions, and so on up to 21 functions. You can choose from calculate value (1 x and 1y value and the equation to solve), choose formula (you figure out the equation using the given x, y, and z values), or figure formula and calculate (you figure out the equation and solve for the missing z value).
If you get stuck trying to figure out what the function (equation) is a hint will be displayed. If you choose the wrong answer it will help you figure out the right one. The calculate option combined with the practice game enables students to practice solving the problems in the area they are having trouble with.
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Algebra One On One is NOT a public domain program. It is copyrighted by Sheppard Software. This software and accompanying documentation are protected by United States copyright law and also by international treaty provisions.
Sheppard Software grants you a license to use this software for evaluation. If you continue using this software after the evaluation period, and would like to receive the latest version and additional modules, make a registration payment to Sheppard Software
You All rights not expressly granted here are reserved to Sheppard Software. | 677.169 | 1 |
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modes of representation given
will be considered im-
provements upon the prevailing methods.
but they unquestionably furnish a very good antidote against 'the tendency of school algebra to degenerate into a mechanical application of
memorized
rules. such examples.
nobody would find the length Etna by such a method. and hence the student is more easily led to do the work by rote
than when the arrangement
braic aspect of the problem.
of the Mississippi or the height of Mt. " Graphical methods have not only a great practical value. Moreover. based upon statistical abstracts.
while in the usual course proportions
are studied a long time after their principal application.
viz.
to solve a
It is
undoubtedly more interesting for a student
problem that results in the height of Mt. the student will be able to utilize this knowledge where it is most
needed. But on the other hand very few of such applied examples are
genuine applications of algebra.PREFACE
vii
and graphical methods into the first year's work. elementary way.
and they usually involve difficult numerical calculations."
Applications taken from geometry. but the true study of algebra has not been sacrificed in order to make an impressive display of sham
life
applications. The entire work in graphical methods has been so arranged that teachers who wish
a shorter course
may omit
these chapters.
is such problems involves as a rule the teaching of physics by the
teacher of algebra.
ARTHUR SCHULTZE. 1910. Manguse for the careful reading of the proofs and
many
valuable suggestions.
William P.
genuine applications of elementary algebra work seems to have certain limi-
but within these limits the author has attempted to
give as
many
The author
for
simple applied examples as possible.
NEW
YORK. desires to acknowledge his indebtedness to Mr.
edge of physics.
.
pupil's knowlso small that an extensive use of
The average
Hence the
field of
suitable for secondary school
tations.
April. however.viii
PREFACE
problems relating to physics often
offer
It is true that
a field
for genuine applications of algebra.
Six
2
. 2-6 of the exercise.
. 6 = 1. if
:
a = 2. a
a=3.
27. = 3.
Twice a3 diminished by 5 times the square root of the quantity a minus 6 square.
22. 6 = 6. 6 = 3. 6. 38. a = 4. physics. 6 = 5. 25.
12 cr6
-f-
6 a6 2
6s.
Six times the square of a minus three times the cube of Eight x cube minus four x square plus y square. a =4.
a = 3. 6 = 2.
w
cube plus three times the quantity a minus
plus 6 multiplied
6.
Read the expressions
of Exs.c) (a .
34.
Express in algebraic symbols 31.
6.
6 = 4. 6 = 6.
30.
:
6. 6 = 5.
24. 30.
then
8 = \ V(a + 6 + c) (a 4.
and
If the three sides of a triangle contain respectively c feet (or other units of length). and other sciences.
29.
23.
The quantity a
6
2 by the quantity a
minus
36. of this exercise?
What kind of expressions are Exs. and the area of the
is
triangle
S
square feet (or squares of other units selected).
28. a = 2.
26. a = 4.
a.6 -f c) (6
a
+ c). 6 = 7. a = 3. 33.
37.
a =3. 10-14
The
representation of numbers by letters makes it posvery briefly and accurately some of the principles of arithmetic.
6=2.
35.6 . Six times a plus 4 times
32.12
17
&
*
ELEMENTS OF ALGEBRA
18
'
8
Find the numerical value of 8 a3
21. geometry.
sible to state
Ex.
By
using the formula
find the area of a triangle
whose
sides are respectively
(a) 3. and 5 feet.
. and 15 feet. A carrier pigeon in 10 minutes. 15 therefore feet.
(c) 4.) Assuming g
.16 centimeters per second. count the resistance of the atmosphere. if v
:
a.16
1
= 84.seconds. b.e.INTRODUCTION
E. then a 13. 14. if v = 30 miles per hour.
the area of the triangle equals
feet.
i.
S = | V(13-hl4-fl5)(13H-14-15)(T3-14-i-15)(14-13-f-15)
= V42-12-14. 13.
=
(a)
How
far does
a body fall from a state of rest in 2
seconds ?
(b)
*
stone dropped from the top of a tree reached the ground in 2-J. if v . b 14. A train in 4 hours. An electric car in 40 seconds. d. if v = 50 meters per second 5000 feet per minute. How far does a body fall from a state of rest in T ^7 of a (c)
A
second ?
3.
(b) 5. c.
2.
4. and c
13
and
15
=
=
=
. A body falling from a state of rest passes in t seconds 2 over a space S (This formula does not take into ac^gt 32 feet. Find the height of the tree.g.
84 square
EXERCISE
1.
9
distance s passed over by a body moving with the uniform velocity v in the time t is represented by the formula
The
Find the distance passed over by A snail in 100 seconds. the three sides of a triangle are respectively 13. 12. and 13 inches.
(c)
10
feet.
2 inches.
$ = 3.
If the diameter of a sphere equals d units of length.14d (square units).
denotes the number of degrees of temperature indi8.
meters.14
square meters.
(c)
5
F.
square units (square inches.
If the
(b) 1 inch. If cated on the Fahrenheit scale.
32 F.
~
7n
cubic feet.
diameter of a sphere equals d
feet.
to Centigrade readings:
(b)
Change the following readings
(a)
122 F.
5. and the value given above is only an
surface
$=
2
approximation. This number cannot be expressed exactly.
(c)
8000 miles. the equivalent reading C on the Centigrade scale may be found by the formula
F
C
y
= f(F-32).
fo
If
i
represents the simple interest of
i
p
dollars at r
in
n
years.
.) Find the surface of a sphere whose diameter equals
(a)
7.
:
8000 miles.14
4.). (The number 3. then
=p
n
*
r
%>
or
Find by means
(a)
(b)
6. the
3.
is
H
2
units of length (inches.
6
Find the volume of a sphere whose diameter equals:
(b)
3
feet.
on $ 500 for 2 years at 4 %.
the area
etc.
(c)
5 miles.14 is frequently denoted by the Greek letter TT.
then the
volume
V=
(a) 10 feet. Find the area of a circle whose radius is
It
(b)
(a) 10 meters.
of this formula
:
The The
interest on interest
$800
for 4 years at
ty%.).
ELEMENTS OF ALGEBRA
If
the radius of a circle
etc.
Thus a gain of $ 2 is considered the sum of a gain of $ 6 and a loss of $ 4. we define the sum of two numbers in such a way that these results become general.
of
$6 and a gain
$4
equals a
$2 may be
represented thus
In a corresponding manner we have for a loss of $6 and a
of
loss
$4
(.$6) + (-
$4) = (-
$10). but we cannot add a gain of $0 and a loss of $4. the fact that a loss of
loss of
+ $2. or positive and negative numbers. AND PARENTHESES
ADDITION OF MONOMIALS
31. in algebra this word includes also the results obtained by adding negative.
. we call the aggregate value of a gain of 6 and a loss of 4 the sum of the two. however. In arithmetic we add a gain of $ 6 and a gain of $ 4. or that
and
(+6) + (+4) = +
16
10. Or in the symbols of algebra
$4) =
Similarly.
Since similar operations with different units always produce analogous results.CHAPTER
II
ADDITION. SUBTRACTION. In algebra.
While
in arithmetic the
word sum
refers only to
the
result obtained
by adding positive numbers.
5.3.
(-17)
15
+ (-14).
of 2.16
32.
l-f(-2). 4.
The average
of two
numbers
is
average of three numbers average of n numbers is the
is one half their sum.
of:
20.
(_
In Exs.
23. find the numerical values of a + b
-f c-j-c?.
4
is
3 J.
-
0.
Thus.
33.
5.
is 0. c = = 5. 23-26. the average of 4 and 8
The average The average
of 2. is 2. + -12. c =
4.
d = 5. '.
. add their absolute values if they have opposite signs.
18.
21. and the sum of the numbers divided by n. 12.
ELEMENTS OF ALGEBRA
These considerations lead to the following principle
:
If two numbers have the same sign.
19. if :
a
a
= 2. subtract their absolute values and
. d = 0.
6
6
= 3. = 5. 10.
EXERCISE
Find the sum
of:
10
Find the values
17. 24.
22.
(always) prefix the sign of the greater. the one third their sum.
+ (-9).
43. . & = 15.
:
34. -11 (Centigrade).
.
and
-8
F.
-'
1?
a
26.
33. and 3 yards. 74. }/ Add 2 a. 55.
are similar terms.
3 and 25. SUBTRACTION. if his yearly gain or loss during 6 years was $ 5000 gain.
.
31. \\ Add 2 a.. 10.
27.
42. Find the average gain per year of a merchant. affected
by the same exponents.
4
F. and $4500 gain. = -13. 0. Find the average temperature of New York by taking the average of the following monthly averages 30. 32.4. or
and
.
40.
:
Find the average temperature of Irkutsk by taking the average of the following monthly temperatures 12. 7 a. $7000 gain. and 3 a. 5 and
12. 72. 34. 7 yards.
2. 6. and 3 F. . 66.ADDITION.
. 7 a. $3000 gain. 09. 60.
or 16
Va + b
and
2Vo"+~&. 32.
36.. 3. 10.5. 12. .
&
28. 10.
d=
3. and 3 a..
13.
35. $1000 loss.
'
Find the average of the following
34.
Similar or like terms are terms which have the same
literal factors. 41.
:
48.
sets of
numbers:
13.
What number must be added to 9 to give 12? What number must be added to 12 to give 9 ? What number must be added to 3 to give 6 ? C* What number must be added to 3 to give 6? **j Add 2 yards.
5 a2 &
6 ax^y and
7 ax'2 y. -4. 38. 1.
30.
25.
6.
.
= -23. and 4. c=14.7.
which are not
similar.
:
and
1.
2.13.
^
'
37. 39.3. 6.
Find the average of the following temperatures 27 F. c = 0. 37.
.
29.7.
Dissimilar or unlike terms are terms
4 a2 6c and o
4 a2 6c2 are dissimilar terms.
= 22.
AND PARENTHESES
d = l.5. $500 loss.
b wider sense than in arithmetic. either the difference of a and b or the sum of a and
The sum
of
a. in algebra it may be considered b.
The sum
x 2 and
f
x2 .
9(a-f-6).
.
10. While in arithmetic a denotes a difference only.ii.
Vm
-f.
-3a
.
5Vm + w.
5l
3(a-f-6).18
35.
14
.
11
-2 a +3a -4o
2.
2 a&.
In algebra the word sum is used in a 36.
:
2 a2.
b
a
-f (
6).
12Vm-f-n.13 rap
25 rap 2.
-f
4 a2.
1
\
-f-
7 a 2 frc
Find the sum of
9.
12
2 wp2 .
The sum The sum
of a of a
Dissimilar terms cannot be united into a single term.
12(a-f b)
12.
ELEMENTS OF ALGEBRA
The sum
of 3
of
two similar terms
x2
is
is
another similar term. Algebraic sum.
2
.
11.
ab
7
c
2
dn
6.
13. The indicated by connecting and a 2 and
a
is
is
-f-
a2
.
2(a-f &).
+ 6 af
.
EXERCISE
Add:
1.
5 a2
. 7 rap2.
sum of two such terms can only be them with the -f.
12
13
b sx
xY xY 7 #y
7. and 4 ac2
is
a
2 a&
-|-
4 ac2. or
a
6.sign.
and the
required number the difference. called the minvend.
(-
6)
-(-
= . In subtraction.
State the other practical examples which show that the number is equal to the addition of a
40. two numbers are given. SUBTRACTION.g. ing the sign of the subtrahend thus to subtract 6 a 2 6 and 8 a 2 6 and find the sum of change mentally the sign of
. from What 3.
Ex.
To
subtract. and their algebraic sum is required.
5
is
2.
1.ADDITION.
AND PARENTHESES
23
subtraction of a negative
positive number. the given number the subtrahend. In addition. Therefore any example in subtraction
different
. the other number is required.
From
5 subtract
+ 3. the algebraic sum and one of the two numbers is
The algebraic sum is given. Subtraction is the inverse of addition. may be stated number added to 3 will give 5? To subtract from a the number b means to find the number which added to b gives a.2.
41.
6
-(-3) = 8.
This gives by the same method.3.
3.
2.
may
be stated in a
:
5 take form e. a-b =
x.
3 gives 5
is
evidently 8.
From
5 subtract
to
The number which added
Hence. change the sign
of the subtrahend and
add.
a.
NOTE.
+b
3.
From
5 subtract
to
. The student should perform mentally the operation of chang8 2 6 from 6 a 2 fc.
. Or in symbols.
if
x
Ex.
The
results of the preceding examples could be obtained
by the following
Principle.
3 gives
3)
The number which added
Hence.
7.
Ex.
45.a
-f-
= 4a
sss
7a
12
06
6.
II.& c
additions
and sub-
+ d) = a + b
c
+ d.
tractions
By using the signs of aggregation. one occurring within the other.
(b
c)
a
=a
6 4-
c.
If there is no sign before the first term within a paren*
-f-
thesis.
46.
a+(b-c) = a +b . the sign
is
understood. If we wish to remove several signs of aggregation.a^6)]
-
}
. The beginner will find it most convenient
at every step to
remove only those parentheses which contain
(7 a
no others.a~^~6)]} = 4 a -{7 a 6 b -[.
4a-{(7a + 6&)-[-6&-f(-2&.
6
o+(
a
+ c) = a =a 6 c) ( 4-.
AND PARENTHESES
27
SIGNS OF AGGREGATION
43.6 b -f (. I.c. we may begin either at the innermost or outermost.2 b .
& -f
c.
A
moved
w
may be resign of aggregation preceded by the sign inserted provided the sign of evei'y term inclosed is
E.
changed. Ex.
66
2&-a + 6
4a
Answer.
Hence
the
it is
sign
may
obvious that parentheses preceded by the -f or be removed or inserted according to the fol:
lowing principles
44.b c = a a
&
-f-
-f.g. A sign of aggregation preceded by the sign -f may be removed or inserted without changing the sign of any term.ADDITION. may be written as follows:
a
-f ( 4.
. Simplify 4 a f
+ 5&)-[-6& +(-25.c. SUBTRACTION.
m
x
2
4.
12.7-fa.
8.
13. SUBTRACTION. )X
6.
The
difference of a
and
6.
'
NOTE.
The sum
of tKe squares of a
and
b.
6.
Three times the product of the squares of The cube of the product of m and n.
In each of the following expressions inclose the last three in a parenthesis preceded by the minus sign
:
-27i2 -3^ 2 + 4r/.
4 xy
7 x* 4-9 x + 2. The minuend is always the of the two numbers mentioned.
p + q + r-s. .
z
+ d.
of the cubes of
m and
n.
terms
5.
2. 9.
The sum^)f
m
and
n.
4.
7.
II.
2m-n + 2q-3t.
The square of the difference of a and b.
10.
The product The product
m
and
n.
The sum
of the fourth powers of a of
and
6.
3.
The The
difference of the cubes of
m
and
n.
6 diminished
.
7. The product of the sum and the difference
of
m and n.
y
-f-
8
.
a-\-l>
>
c
+
d.
first.ADDITION.
EXERCISE
AND PARENTHESES
16
29
In each of the following expressions inclose the last three terms in a parenthesis
:
1.
EXERCISES
IN"
ALGEBRAIC EXPRESSION
17
:
EXERCISE
Write the following expressions
I.
Nine times the square of the sum of a and by the product of a and b.1.
3.
5^2
_ r .
and the subtrahend the second.
5 a2
2.
difference of the cubes of n and m.2 tf .
5.4 y* .
m and n.
difference of the cubes of a
and
b divided
by the
difference of
a and
6.
a plus the prod-
uct of a and
s
plus the square of
-19. The difference of the squares of two numbers divided by the difference of the numbers is equal to the sum of the two numbers.
18.)
.
dif-
of the squares of
a and
b increased
by the
square root of
15. 6.
and
c
divided by the
ference of a and
Write algebraically the following statements:
V 17.
b.30
14. (Let a and b represent the numbers.
ELEMENTS OF ALGEBRA
The sum
x. 16.
6 is equal to the square of
b.
The sum
The
of a
and
b multiplied
b is equal to the difference of
by the difference of a and a 2 and b 2
.
d.
x cube minus quantity 2 x2 minus 6 x plus
The sum
of the cubes of a.
If the two loads
what
What. weight at A ? What is the sign of a 3 Ib.
If the
two loads balance. let us consider the and JB. weights.
two loads balance. applied at let us indicate a downward pull at by a positive sign.
force is produced
therefore. weight at B ?
If the
addition of five 3
plication example. what force
31
is
produced by tak(
ing away 5 weights from
B ? What therefore is
5)
x(
3) ?
.
3. If the two loads balance.
By what sign is an upward pull at A represented ? What is the sign of a 3 Ib.
5. therefore. and forces produced at by 3 Ib. 2. is 5 x ( 3) ?
7.CHAPTER
III
MULTIPLICATION
MULTIPLICATION OF ALGEBRAIC NUMBERS
EXERCISE
18
In the annexed diagram of a balance. weights at A ? Express this as a multibalance. is
by
taking away 5 weights from
A?
5
X 3?
6. what force is produced by the Ib. what force is produced by the addition of 5 weights at B ? What.
4.
A
A
A
1.
4)-(. such as given in the preceding exercise. becomes meaningless
if
definition.
(
(.4) x
braic laws for negative
~ 3> = -(. a result that would not be obtained by other assumptions. To take a number 7 times.
In multiplying integers we have therefore four cases
trated
illus-
by the following examples
:
4x3 = 4-12. 4 multi44-44-4 12.
. (-5)X4.
examples were generally
method of the preceding what would be the values of
(
5x4.9) x
11.
Multiplication
by a negative
integer is a repeated
sub-
traction. 5x(-4).
however. times is just as meaningless as to fire a gun
tion
7
Consequently we have to define the meaning of a multiplicaif the multiplier is negative.
the multiplier is a negative number.
(-
9)
x (-
11) ?
State a rule by which the sign of the product of
two
fac-
tors can be obtained.4)-(-4) = +
12. 4 multiplied by 3.
Thus. 9 x
(-
11).
4
x(-8) = ~(4)-(4)-(4)=:-12. or
4x3 =
=
(_4) X
The preceding
3=(-4)+(-4)+(-4)=-12.
x
11. or plied by 3. and we may choose any definition that does not lead to contradictions.32
8.
4x(-3)=-12. Practical examples^
it
however. Multiplication by a positive integer is a repeated addition.
48. thus. This definition has the additional advantage of leading to algenumbers which are identical with those for positive numbers.
NOTE.
9
9.
(.
ELEMENTS OF ALGEBRA
If
the signs obtained by the
true. make venient to accept the following definition
:
con-
49.
the work becomes simpler and more symmetrical by arranging these expressions according to either ascending or descending powers.
Ex. To multiply two polynomials.3
b
by a
5
b. Since errors. however. multiply each term of one by each term of the other and add the partial products thus formed.
2a-3b a-66 2 a .
The most convenient way of adding the partial products is to place similar terms in columns. If the polynomials to be multiplied contain several powers of the same letter.
If
Arranging according to ascending powers
2
a
. 1 being the most convenient value to be substituted for all letters.
Multiply 2
+ a -a. as illustrated in the following example
:
Ex.1. are far more likely to occur in the coefficients than anywhere else.3 a 2 + a8 .
.
59.
Check.4.a6 4 a 8 + 5 a* . the student should
apply this test to every example.3 a
3
2
by 2 a
:
a2 + l.
2. Since all powers of 1 are 1. Multiply 2 a .3 ab
2
2 a2
10 ab
-
13 ab
+ 15 6 2 + 15 6 2
Product. this method
tests only the values of the coefficients
and not the values of
the exponents.
a2
+ a8 + 3 .M UL TIP LICA TION
37
58.3 a 2 + a8
a
a = =-
I
1
=2
-f
2
a
4.a6
=2
by numerical
Examples
in multiplication can be checked
substitution.a
.2 a2 6 a8
2 a*
*
-
2"
a2
-7
60.
5. (3m + 2)(m-l).
The middle term
or
Wxy-12xy
Hence in general. plus the product of the
EXERCISE
Multiply by inspection
1.
:
25
2.
or
The student should note
minus
signs.
2 2 2 2 (2a 6 -7)(a & +
5).
6.
7.
.
The square
2
(a 4.
(x
i-
5
2
ft
x 2 -3 6 s). (100 + 3)(100 + 4).
(5a6-4)(5a&-3).
that the square of each term is while the product of the terms may have plus always positive.
2
(2x y
(6
2
2
+ z )(ary + 2z ).
2
2
+ 2) (10 4-3).
65.
2
10. (4s + y)(3-2y).
8. 9.42
ELEMENTS OF ALGEBRA
of the result is obtained
product of 5 x
follows:
by adding the These products are frequently called the cross products. 13. 14.-f
2 a&
-f
2 ac
+ 2 &c. plus the
last terms.
(2a-3)(a + 2).
2
(2m-3)(3m + 2).
11. ) (2
of a polynomial. and are represented as
2 y and 4y 3 x.
((5a?
(10
12.
7%e square of a polynomial is equal to the sum of the squares of each term increased by twice the product of each term with each that follows it.&
+ c) = a + tf + c
. the product of two binomials whose corresponding terms are similar is equal to the product of the first two
terms.
sum of the
cross products.
4.
(5a-4)(4a-l).
3.
is the process of finding one of two factors and the other factor are given. The dividend is the product of the two factors, the divisor the given factor, and the quotient is the required factor.
67.
Division
if
their product
is
Thus
by
-f
to divide
12.
12
by
+
3,
we must find
is
the
;
number which
3 gives
But
this
number
4
hence
_
multiplied
12 r +3
=4.
68.
Since
-f
a
-
-f b
-fa
_a
and
it
-f-
a
= -f ab = ab b = ab b = ab,
b
-f-
follows that
4-a
=+b
ab
a
ab
a
69.
Hence the law
:
of signs
is
the same in division as in
multiplication
70.
Like signs produce plus, unlike signs minus.
Law
of
,
a8 -5- a5
=a
3
for a 3
It follows from the definition that Exponents. X a5 a8
=
.
Or
in general, if
greater than
m n, a
-f-
and n are positive integers, and m ~ n an = a m a" = a'"-", for a
<
m
m
is
45
46
ELEMENTS OF ALGEBRA
71. TJie exponent of a quotient of two powers with equal bases equals the exponent of the dividend diminished by the exponent
of the divisor.
DIVISION OF MONOMIALS
7 3 72. To divide 10x y z by number which multiplied by number is evidently
2x y
6
2
,
we have
z
to
find
the
2x*y
gives 10 x^ifz.
This
Therefore,
the quotient
*
,
= - 5 a*yz.
is
Hence,
sign,
of two monomials of their
part
coefficients,
is the
a monomial whose
coefficient is the quotient
preceded by the proper
literal
and whose
literal
found
in accordance with the
quotient of their law of exponents.
parts
73. In dividing a product of several factors by a number, only one of these factors is divided by that number. Thus (8 12 20)-?-4 equals 2 12 20, or 8 3 20 or 8 12 5.
-
-
.
-
.
-
.
EXERCISE
Perform the divisions indicated
'
:
28
'
2
.
76-H-15.
-39-*- 3.
2
15
3"
7
7'
3.
-4*
'
4.
5.
-j-2
12
.
4
2
9
5 11
68
3 19 -j-3
5
10.
(3
38
-
-2 4 )^(3 4 .2 2).
56
'
11.
3
(2
.3*.5 7 )-f-(
2
'
12
'
2V
14
36 a
'
13
''
y-ffl-g
35
-5.25
-12 a
2abc
15
-42^
'
-56aW
'
UafiV
DIVISION
lg
47
-^1^. 16 w
7
20>
7i
9
_Z^L4L.
22.
10 iy.
132 a V* 14 1
*
01
-240m
120m-
40
6c
fl
/5i.
3J)
c
23.
2 (15- 25. a ) -=- 5.
25. 26.
(18
(
.
5
.
2a )-f-9a.
2
24.
(7- 26 a
2
)
-f-
13.
DIVISION OF POLYNOMIALS BY MONOMIALS
To divide ax-}- fr.e-f ex by x we must find an expression which multiplied by x gives the product ax + bx -J- ex.
74.
But
TT
x(a
aa?
Hence
+ b e) ax + bx + ex. + bx -f ex = a 4- b +
-\.
,
.
c.
a?
To divide a polynomial by a monomial, cfc'wde each term of the dividend by the monomial and add the partial quotients thus
formed.
3 xyz
EXERCISE
Perform the operations indicated
1.
:
29
2.
5.
fl
o.
(5*
_5* + 52)
-5.
52
.
3.
97
.
(2
(G^-G^-G^-i-G
(11- 2
4.
(8- 3
+
11 -3
+ 11
-5)-*- 11.
18 aft- 27 oc
Q y.
9a
4
-25 -2 )^-2
<?
2
.
+8- 5 + 8-
7) -*-8.
5a5 +4as -2a
2
-a
-14gV+21gy
Itf
15 a*b
-
12
aW + 9 a
2
2
3a
48
,
ELEMENTS OF ALGEBRA
22
4,
m n - 33 m n
4
s
2
-f
55
mV
- 39 afyV + 26 arVz 3
- 49 aW + 28 a -W - 14 g 6 c
4 4
15. 16.
2 (115 afy -f 161 afy
- 69
4
2
a;
4
?/
3
- 23 ofy
3
4
)
-5-
23 x2y.
(52
afyV - 39
4
?/
oryz
- 65 zyz - 26 tf#z)
-5-
13 xyz.
-f-
,
17.
(85 tf
- 68 x + 51 afy - 34 xy* -f 1 7
a;/)
- 17
as.
DIVISION OF A POLYNOMIAL BY A POLYNOMIAL
75.
Let
it
be required to divide 25 a
- 12 -f 6 a - 20 a
3
2
by
2 a 2 -f 3 a, divide
4
a, or, arranging according to
2
descending powers of
6a3 -20a
-f
25a-12
2 by 2a -
The term containing the highest power of a in the dividend (i.e. a 8 ) is evidently the product of the terms containing respectively the highest power of a in the divisor and in the quotient.
Hence the term containing the highest power
of a in the quotient is
If
the product of 3 a and 2
2
4 a
+
3, i.e.
6 a3
12 a 2
-f
9 a, be sub-
8 a 2 -f 16 a tracted from the dividend, the remainder is 12. This remainder obviously must be the product of the divisor and the rest of the quotient. To obtain the other terms of the quotient we have
therefore to divide the remainder,
8 a2
-f-
16 a
12,
2 by 2 a
4 a
+
3.
consequently repeat the process. By dividing the highest term in the new dividend 8 a 2 by the highest term in the divisor 2 a 2 we obtain
,
We
4,
the next highest term in the quotient. 4 by the divisor 2 a2 4 a Multiplying
-I-
+ 3, we
obtain the product
8 a2
16 a
12,
which subtracted from the preceding dividend leaves
the required quotient.
no remainder. Hence 3 a
4
is
DIVISION
The work
is
49
:
usually arranged as follows
- 20 * 2 + 3 0a-- 12 a 2 +
a3
25 a
{)
-
12
I
2 a2 8 a
-
4 a 4
a
_
12
+3
I
-
8 a? 4- 16
a-
76. The method which was applied in the preceding example may be stated as follows 1. Arrange dividend and divisor according to ascending or
:
descending powers of a common letter. 2. Divide the first term of the dividend by the first term of the divisor, and write the result for the first term of the quotient.
3.
Multiply this term of the quotient by the whole divisor, and
subtract the result
4.
from
it
the dividend.
the same order as the given new dividend, and proceed as before.
Arrange
the
remainder in
as a
expression, consider
5.
until the highest poiver
Continue the process until a remainder zero is obtained, or of the letter according to which the dividend
is less
was arranged
the divisor.
than the highest poiver of the same
letter in
77.
Checks.
Numerical substitution constitutes a very con-
venient, but not absolutely reliable check. An absolute check consists in multiplying quotient and divisor. The result must equal the dividend if the division
was
exact, or the dividend diminished by the remainder division was not exact.
The
first
member
or left side of an equation
is
that part
The secof the equation which precedes the sign of equality.
A set of numbers which when substituted for the letters an equation produce equal values of the two members.
83.CHAPTER V
LINEAR EQUATIONS AND PROBLEMS
79.
x
20.
Thus.
the
80. ond member or right side is that part which follows the sign of
equality.
which
is
true for all values
a2 6 2 no matter what values we assign to a Thus.
=11.
ber
equation is employed to discover an unknown num(frequently denoted by x.
.
. second member is x
+
4
x
9.
in
Thus x
12 satisfies the equation x
+
1
13. in the equation 2 x 0. (a + ft) (a b) and b.
81.r
-f9
= 20
is
true only
when
a.
(rt+6)(a-ft)
=
2
-
b'
2
. y y or z) from its relation to
63
An
known numbers.
y
=
7 satisfy
the equation x
y
=
13. An equation of condition is usually called an equation. An equation of condition is an equation which is true only for certain values of the letters involved. The sign of identity sometimes used is = thus we may write
.
.
82. is said to satisfy an equation.
hence
it
is
an equation
of
condition. An identity is an equation of the letters involved.
the
first
member
is
2 x
+
4.
a.b.
NOTE. the quotients are equal. A
2
a.
Like powers or
like roots
of equals are equal.
The process
of solving equations depends upon the
:
lowing principles.
Transposition of terms.g.
E.
89.
If equals be divided by equals. 87. x
I.
To
solve
an equation
to find its roots. the products are equal.
Axiom
4
is
not true
if
0x4
= 0x5.
.
2. the remainders are
equal.
expressed in arithmetical numbers
literal
is
as (7
equation
is
one in which at least one of the
known
quantities as x -f a letters
88.
4. Consider the equation b Subtracting a from both members.
one member to another by changing
x + a=.
fol-
A
linear equation is also called a simple equation.
5.
the divisor equals zero.
If equals be subtracted from equals.
9
is
a root of the equation 2 y
+2=
is
20.
86.
.
= bx
expressed by a letter or a combination of
c.
ELEMENTS OF ALGEBRA
If
value of the
an equation contains only one unknown quantity. an^ unknown quantity which satisfies the equation is
a root of the equation.
If equals be added
to equals. the
sums are
equal.
90.2.
A
term may be transposed from
its sign.
If equals be multiplied by equals.e. called axioms
1.
(Axiom
2)
the term a has been transposed from the left to thQ
right
member by changing
its
sign.
85.
2
= 6#-f7.
A numerical
equation is one in
which
all
.
3.
A
linear equation or
which when reduced
first
to its simplest
an equation of the first degree is one form contains only the
as
9ie
power of the unknown quantity.
but 4 does not equal
5.
the
known quan
x) (x -f 4)
tities are
=
.54
84.
greater one is g.
4.
1.
EXERCISE
1.
two numbers
and the and the
2
Find the greater one. The difference between two numbers Find the smaller one.
ELEMENTS OF ALGEBRA
What must
be added to a to produce a sum b ?
:
Consider the arithmetical question duce the sum of 12 ?
What must
be added to 7 to pro-
The answer is 5.
smaller one
16.
13.
7.
x
-f-
y yards cost $ 100
.
10.
14. so that
of c ?
is
p. 11.58
Ex. 15.
one part equals
is 10.
Ex.
$> 100 yards cost one hundred dollars.
6. is d.
33
2. Hence 6 a must be added
to a to give
5.
5.
By how much does a exceed 10 ? By how much does 9 exceed x ? What number exceeds a by 4 ? What number exceeds m by n ? What is the 5th part of n ? What is the nth part of x ? By how much does 10 exceed the third part of a? By how much does the fourth part of x exceed b ? By how much does the double of b exceed one half Two numbers differ by 7.
a.
Find the greater one. one yard will cost
100
-dollars.
is
a?
2
is
c?. 6.
17.
What number divided by 3 will give the quotient a? ? What is the dividend if the divisor is 7 and the quotient
?
. so that one part
The
difference between
is s. or 12 7.
Divide a into two parts. Divide 100 into two
12. so that one part Divide a into two parts.
If 7
2.
find the cost of one yard. is b.
3. one yard will cost -
Hence
if
x
-f
y yards cost $ 100.
9. and the smaller one
parts.
Find the sum of their ages
5 years ago.
A man
had a
dollars.
20. and B's age is y years.
numbers
is x.
is
A A
is
# years
old.
32. square feet are there in the area of the floor ?
How many
2 feet longer
29.
34. 28.
?/
31.
59
What must
The
be subtracted from 2 b to give a?
is a.
19.
If
B
gave
A
6
25.
feet wider than the one
mentioned in Ex.
A
dollars.
22. find the of their ages 6 years hence.
A
feet wide.
26. 33.
Find
35. A room is x feet long and y feet wide.
and
B
is
y years old.
What What What What
is
the cost of 10 apples at x cents each ?
is
is is
x apples cost 20 cents ? the price of 12 apples if x apples cost 20 cents ? the price of 3 apples if x apples cost n cents ?
the cost of 1 apple
if
. The greatest of three consecutive the other two.
smallest of three consecutive numbers
Find
the other two.
How many
cents had he left ?
28.
and 4
floor of a room that is 3 feet shorter wider than the one mentioned in Ex.
How many
cents are in d dollars ? in x dimes ?
A has
a
dollars.
24.
Find
21.
sum
If A's age is x years. amount each will then have. 28.
How many
cents
has he ?
27. and B has n dollars. b dimes. find the
has ra dollars. rectangular field is x feet long and the length of a fence surrounding the field.
and
c cents.LINEAR EQUATIONS AND PROBLEMS
18.
Find the area of the Find the area of the
feet
floor of
a room that
is
and 3
30.
y years
How
old was he 5 years ago ?
How
old will he be 10 years hence ?
23.
and spent
5 cents.
How many years
A
older than
is
B?
old.
50. The first pipe x minutes.
of 4. in how many hours he walk n miles ?
40.
how many
how many
miles will
he walk in n hours
38.
A
was 20 years
old.
.
miles does
will
If a man walks r miles per hour.
-46. and the second pipe alone fills it in
filled
y minutes.
c
a
b
=
-
9.
m is the
denominator.
How many
x years ago
miles does a train
move
in
t
hours at the
rate of x miles per hour ?
41.
% % %
of 100
of
x." we have to consider that in this by statement "exceeds" means minus ( ). What fraction of the cistern will be filled by one pipe in one minute ?
42. and "by as much as" Hence we have means equals (=)
95. If a man walks 3 miles per hour.
of m. If a man walks n miles in 4 hours.
The numerator
If
of a fraction exceeds the denominator
by
3.
Find
a. find the fraction. What fraction of the cistern will be second by the two pipes together ?
44.
If a
man walks
?
r miles per hour.
b
To express in algebraic symbols the sentence: " a exceeds much as b exceeds 9.
as
a exceeds
b
by as much
as c exceeds 9.
48.60
ELEMENTS OF ALGEBRA
wil\
36.
Find the
number. how many miles he walk in n hours ?
37.
-. Find a
47.
How
old
is
he
now ?
by a pipe in x minutes.
A
cistern can be filled
in
alone
fills it
by two pipes.
The two
digits of a
number
are x and
y.
Find x
% %
of 1000. he walk each hour ?
39.
per
Find 5 Find 6
45.
A
cistern
is
filled
43.
49.
a.
same
result as 7
subtracted from
. a is greater than b by b is smaller than a by
c.
3.
9.
c.
of a and 10 equals 2
c. 2.
third of x equals
difference of x
The
and y increased by 7 equals
a.
c.
of a increased
much
8.
The product
of the
is
diminished by 90 b divided by 7.
double of a
is
10.
5.
of x increased by 10 equals
x.
8
-b ) + 80 = a
. etc.
EXERCISE
The The double The sum
One
34
:
Express the following sentences as equations
1. Four times the difference of a and b exceeds c by as
d exceeds
9.
equal to the
sum and the difference of a and b sum of the squares of a and
gives the
Twenty subtracted from 2 a
a.
cases it is possible to translate a sentence word by in algebraic symbols in other cases the sentence has to be changed to obtain the symbols.
The double
as
7.LINEAR EQUATIONS AND PROBLEMS
Similarly. the difference of the squares of a
61
and
b increased
-}-
a2
i<5
-
b'
2
'
by 80 equals the excess of a over
80
Or.
-80.
=
2
2
a3
(a
-
80.
6.
80.
The
excess of a over b
is c.
a exceeds b by c.
In
many
word
There are usually several different ways of expressing a symbolical statement in words. thus:
a
b
= c may
be expressed as follows
difference between a
:
The
and
b is
c.
4.
by one third of b equals 100.
11. sum equals $20.
#is5%of450. B's age
20. 3 1200 dollars.
is
If A's age is 2 x. and
(a)
(6)
A
If
has $ 5 more than B. 14.
a. 16.
amounts. the first sum exceeds b % of the second sum by
first
(e)
%
of the first plus 5
%
of the second plus 6
%
of the
third
sum
equals $8000.
.
symbols
B. a third sum of 2 x + 1 dollars.
5x
A sum of money consists of x dollars. and C have respectively 2 a.
x
4-
If A. B's.
(c)
If each
man
gains $500.
m is x %
of n.
A
If
and
B
B together have $ 200 less than C.
x
is
100
x%
is
of 700.
18.
a. pays to C $100.
A
gains
$20 and B
loses
$40. Express as
:
equations of the (a) 5
(b)
(c)
% a%
of the second
(d)
x c of / a % of
4
sum equals $ 90.
(a)
(b)
(c)
A is twice as old as B. and C's ages will be 100. of 30 dollars. the first sum equals 6 % of the third sura.000.62
10. B.
->.
17.
ELEMENTS OF ALGEBRA
Nine
is
as
much below a
13. In 10 years the sum of A's. the
sum
and C's
money
(d)
(e)
will be $ 12.
50
is
x % of
15.
first
00
x % of the
equals one tenth of the third sum.
as 17
is
is
above
a.. A is 4 years older than
Five years ago A was x years old. express in algebraic symbols
:
-700. B's.
6
%
of m. a second sum. express in algebraic
3x
:
10. they have equal amounts. (d) In 10 years A will be n years old. In 3 years A will be twice as old as B.
and C's age
4
a.
12. they have equal
of A's.*(/)
(g)
(Ji)
Three years ago the sum of A's and B's ages was 50. (e) In 3 years A will be as old as B is now.
by 20
40 exceeds 20 by 20.
but
30
=3
x
years. The solution of the equation (jives the value of the unknown number.
Let x
The
(2)
= A's present age.
3 x or 60 exceeds 40
+ x = 40 + 40.
Transposing. 15.
Three times a certain number exceeds 40 by as Find the number. the
. x + 15 = 3 x
3x 16
15.
.
A
will
Check.
3
x
+
16
=
x
x
(x
-
p)
Or. 4 x = 80.
much
as 40 exceeds the number.
equation is the sentence written in alyebraic shorthand.
-23 =-30.
number by x (or another letter) and express the yiven sentence as an equation. Three times a certain no. = x x
3x
-40
3x
40-
Or. Find A's present age.
In 15 years
10.
6 years ago he was 10
.
be 30
. In order to solve them.
Ex. denote the unknown
96.
Let x = the number. exceeds 40 by as much as 40 exceeds the no. The equation can frequently be written by translating the sentence word by word into algebraic symbols in fact. verbal statement (1)
(1) In 15 years
A
will
may be expressed in symbols (2).
Uniting.
Uniting. x = 20.
1. the required
.
NOTE.
Ex.
Simplifying.LINEAR EQUATIONS AND PROBLEMS
63
PROBLEMS LEADING TO SIMPLE EQUATIONS
The simplest kind of problems contain only one unknown number. The student should note that x stands for the number of and similarly in other examples for number of dollars.
Check. In 15 years A will be three times as old as he was 5 years ago. Write the sentence in algebraic symbols. x= 15. number of
yards.
Transposing. etc. 2. be three times as old as he was 5 years ago.
x+16 = 3(3-5).
number.
Dividing.
3z-40:r:40-z.
Uldbe
66
| x x
5(5 is
=
-*-. A train moving at uniform rate runs in 5 hours 90 miles more than in 2 hours. Find the number.
35
What number added
to twice itself gives a
sum
of
39?
44.
Six years hence a
12 years ago. exceeds the width of the bridge.
ELEMENTS OF ALGEBRA
56
is
what per cent
of 120 ?
= number
of per cent.
.
14 50
is
is
4
what per cent of 500 ? % of what number?
is
12. then the
problem expressed in symbols
W
or.
A
number added
number.
300
56.
Find the number whose double exceeds 30 by as much
as 24 exceeds the number.
120.
A will
be three times as old as to-da3r
.
Hence
40
= 46f. 14.
4.
13. How long is the Suez
Canal?
10.
twice the number plus
7. How many miles per hour does it run ?
.
3.
How
old
is
man will be he now ?
twice as old as he was
9. 11.
to
42 gives a
sum
equal to 7 times the
original
6. by as much as 135 ft.
Let x
3.
Find the width of the Brooklyn Bridge. % of
120.
What number
7
%
of
350?
Ten times the width of the Brooklyn Bridge exceeds 800 ft.
Forty years hence
his present age.
Find the number.
Find the number whose double increased by 14 equals Find the number whose double exceeds 40 by 10.2.
Find
8.
EXERCISE
1.
Dividing.
5. Four times the length of the Suez Canal exceeds 180 miles by twice the length of the canal.
47 diminished by three times a certain number equals 2.64
Ex.
B
How
will loses $100.
How many
dol-
A has
A
to
$40.
65
A
and
B
$200. written in algebraic
symbols. The sum of the two numbers is 14. Vermont's population increased by 180. then dollars has each ? many
have equal amounts of money. One number exceeds the other one by II.
.
The problem consists of two statements I.
numbers (usually the smaller one) by
and use one of the
given verbal statements to express the other unknown number in terms of x.
1. If the first farm contained twice as many acres as
A man
number
of acres.
x. two verbal statements must be given.
14. If a problem contains two unknown quantities. B will have lars has A now?
17.
times as
much
as A. which gives the value of
8.
the second one.
make A's money
equal to 4 times B's
money
wishes to purchase a farm containing a certain He found one farm which contained 30 acres too many. Maine's population increased by 510.
Ex.
statements are given directly.
is
the equation. The other verbal statement. During the following 90 years. Find
the population of Maine in 1800.
F
8.
A
and
B
have equal amounts of money. In 1800 the population of Maine equaled that of Vermont. and
as
15.000. If A gains A have three times as much
16.
One number exceeds another by
:
and their sum
is
Find the numbers.
five
If
A
gives
B
$200. and Maine had then twice as many inhabitants as Vermont.
how many
acres did he wish to
buy
?
19. while in the more complex probWe denote one of the unknown
x.
97.000.LINEAR EQUATIONS AND PROBLEMS
15.
How many dollars
must
?
B
give to
18. Ill the simpler examples these two
lems they are only implied. and
B
has $00. and another which lacked 25 acres of the required number.
A has three times as many marbles as B.
/
.
terms of the other.
x
3x
4-
and
B
will gain. to
Use the simpler statement.
<
Transposing.
Statement
x
in
=
the larger number.66
ELEMENTS OF ALGEBRA
Either statement may be used to express one unknown number in terms of the other. although in general the simpler one should be selected. unknown quantity
in
Then.
expressed symbols is (14 x) course to the same answer as the first method.
+
a-
-f
-f
8
= 14.
Then.
To
express statement II in algebraic symbols.
Let
x
14
I
the smaller number. = 3.
. 8 the greater number.
If
we
select the first one.
= B's number of marbles. the sum of the two numbers is 14.
Dividing.
2x
a?
x
-j-
= 6.
A will lose. A gives B 25 marbles.
The two statements
I. the smaller number. 25 marbles to B.= The second statement written
the equation ^
smaller number.
x
= 8.
and
Let x
= the
Then x -+. 2.
o\
(o?-f 8)
Simplifying. B will have twice as many as A.
. 8 = 11.
26
= B's number of marbles after the exchange. B will have twice as
viz. 26 = A's number of marbles after the exchange.
in algebraic
-i
symbols produces
#4a.
. = 14.
Uniting.
which leads
ot
Ex. consider that by the
exchange
Hence.
If
A gives
are
:
A
If
II. the greater number. = A's number of marbles. I. Let
x
3x
express one
many as A.
x
x =14
8.
Another method for solving this problem is to express one unknown quantity in terms of the other by means of statement II viz.
has three times as many marbles as B.
240.
60. 15 + 25 = 40. consisting of half dollars and dimes.
Dividing.
Find
the numbers.
(Statement II)
Qx
.
Two numbers
the smaller.10.
*
'
. and the Find the numbers. The value of the half
:
is 11..
Uniting.
Selecting the cent as the denomination (in order to avoid fractions).
by 44. 6 half dollars = 260 cents.
is 70. the
price.
x x
+
= 2(3 x = 6x
25
25). cents.25 = 20.10.
The sum of two numbers is 42. 40 x . . x = 15.$3. have a value of $3.
50. etc.
50 x
Transposing..
2..5 x .
we
express the statement II in algebraic symbols.10.
Simplifying. Dividing. the number of dimes.LINEAR EQUATIONS AND PROBLEMS
Therefore.
. 45 .
x
from
I.
The numbers which appear
in the equation should
always
be expressed in the same denomination.
Never add the number number of yards to their
Ex.
Let
11
= the number of dimes.
Simplifying. x = 6. 3 x = 45. their sum
+ +
10 x 10 x
is
EXERCISE
36
is five
v
v.
6 dimes
= 60
= 310.
Check. Find the numbers. Check.
differ
differ
and the greater and their sum
times
Two numbers
by
60.
dollars
and dimes
is
$3. the number of half dollars.
greater
is
. Eleven coins. A's number of marbles.
1. 3.
*
98.
w'3. 11 x = 5.75. The number of coins II.
Uniting. x = the number of half dollars. of dollars to the number of cents. B's number of marbles.
50(11 660 50 x
-)+ 10 x = 310. then.
6 times the smaller. How many are there of each ?
The two statements are I.550 -f 310.
67
x
-f
25 25
Transposing. but 40 = 2 x 20.
one of which increased by
9.. Twice 14.
On December
21.
Find
Find two consecutive numbers whose sum equals 157. and the
greater increased by five times the smaller equals 22. Everest by 11. and B's age is as below 30 as A's age is above 40.
3 shall be equal to the other increased by
10.000
feet.
as the larger one.68
4. the number.
United
States.
tnree times the smaller by 65.
of volcanoes in
Mexico exceeds the number
of volcanoes in the United States by 2. What are their ages ?
is
A A
much
line 60 inches long is divided into two parts. the night in
Copenhagen
lasts 10 hours
longer than the day. and twice the altitude of Mt. McKinley exceeds the altitude of
Mt.
find the
weight of a cubic
Divide 20 into two parts. Mount Everest is 9000 feet higher than Mt. How many inches are in each part ?
15.
ELEMENTS OF ALGEBRA
One number
is
six
times another number.
2 cubic feet of iron weigh 1600 foot of each substance. McKinley.
it
If the smaller
one contained 11 pints more.
11. and in 5 years A's age will be three times B's.
7.
5.
and in Mexico
?
A
cubic foot of aluminum.
?
Two
vessels contain together 9 pints. How many volcanoes are
in the
8. the larger part exceeds five times the smaller part by 15 inches.
A's age is four times B's.
would contain three times as
pints does each contain ?
much
13.
6.
and twice the greater exceeds Find the numbers.
9. and four times the former equals five times the latter.
How many
14 years older than B.
Two numbers
The number
differ
by
39.
What
is
the altitude of each
mountain
12.
How many
hours does the day last ?
. Find their ages.
cubic foot of iron weighs three times as much as a If 4 cubic feet of aluminum and
Ibs.
number of dollars A had.
times as
much
as
A.
Ex.
1.
If
4x
= 24. III. B. then
three times the
money by
I. then three times the sum of A's and B's money would exceed C's money by as much as A had originally.
A
and B each gave $ 5
respectively.LINEAR EQUATIONS AND PROBLEMS
99. and C together have $80. If A and B each gave $5 to C. x = 8.
69
If a
verbal statements must be given.
5
5
Expressing in symbols Three times the sum of A's and B's money exceeds C's money by A's 3 x ( x _5 + 3z-5) (90-4z) = x. and 68.
has. and the other
of x
problem contains three unknown quantities.
Let
x
II. B has three times as much as A. = 48. bers is denoted by x.
8(8
+ 19)
to C.
number
had. 4 x = number of dollars C had after receiving $10.
. and C together have $80. and B has three as A.
the
the
number
of dollars of dollars of dollars
A
B
C
has. B. The solution gives
:
3x
80
Check. they would have 3.
are
:
C's
The three statements
A.
sum of A's and B's money would exceed much as A had originally. try to obtain
it
by a
series of successive steps. or 66 exceeds 58 by 8.
number
of dollars of dollars
B
C
had. three One of the unknown num-
two are expressed in terms by means of two of the verbal statements."
To
x
8x
90
= number of dollars A had after giving $5. let us consider the words ** if A and B each gave $ 5 to C. II. original amount. 19. The third
verbal statement produces the equation. = number of dollars B had after giving $5.
has. Tf it should be difficult to express the selected verbal state-
ment
directly in algebraical symbols. If A and B each gave $5 to C.
first
According to
3 x
number
number
and according
to
80
4
x
=
the
express statement III by algebraical symbols.
I.
28 x 15 or 450
5 horses. 9 cows. and Ex. number of cows. number of sheep.
and. according to III.
first. 90 x -f 35 x + GO x = 140 20 + 1185.
+ 35 (x +-4) -f 15(4z-f 8) = 1185. 2.
The
I. according to II. The number of sheep is equal to twice tho number of horses and
x 4
the
cows together. Let
then. = the number of dollars spent for sheep
Hence statement
90 x
Simplifying.
three statements are
:
IT.
x
-j-
= the
number of horses.70
ELEMENTS OF ALGEBRA
man spent $1185 in buying horses.
first
the third exceeds the second by and third is 20. 28 2 (9 5). and the difference between the third and the second is 15
2.140 + (50 x x 120 = 185.
1 1
Check. each horse costing $ 90. number of cows.
x
Transposing. number of horses.
The total cost equals $1185. = the number of dollars spent for horses. x -f 4 = 9. and each sheep $ 15.
x 35
-f
+
=
+
EXERCISE
1. cows. The number of cows exceeds the number of horses by 4. and 28 sheep would cost 6 x 90 -f 9 + 316 420 = 1185. each cow $ 35.
+
8
90 x
and. 90
may
be written.
85 (x 15 (4 x
I
+ 4)
+
8)
= the number of sheep.
37
Find three numbers such that the second is twice the first. sheep.
III. and the sum of the
.
A
and the number of sheep was twice as large as the number How many animals of each kind did he buy ?
of horses and cows together.
Find three numbers such that the second is twice the 2.
Dividing. 4 x -f 8 = 28. the third five times the first. x = 5. + 35 x 4. = the number of dollars spent for cows. 185 a = 925. 9 -5 = 4 . The number of cows exceeded the number of horses by
4. 2 (2 x -f 4) or 4 x
Therefore.
Uniting.
the second one is one inch longer than the first. the copper.000 more than Philadelphia (Census 1905).
twice the
6.LINEAR EQUATIONS AND PROBLEMS
3.
9.
New York
delphia.000. what is the length of each?
has 3.
7. and 2 more men than women.
13.
the third
2. and is 5 years younger than sum of B's and C's ages was 25 years.
and
of the three sides of a triangle is 28 inches.
"Find three
is 4. women. and the third exceeds the
is
second by
5.
v
.
The
three angles of any triangle are together equal to
180. and the sum of the first and third is 36.
Find three consecutive numbers whose sum equals
63. increased by three times the second side. men.
A
12. If twice
The sum
the third side. In a room there were three times as many children as If the number of women.000 more inhabitants than Philaand Berlin has 1. and the third part exceeds the second by 10.
If the second angle of a triangle is 20 larger than the and the third is 20 more than the sum of the second and
first.
v
-
Divide 25 into three parts such that the second part first. what is the population of each city ?
8. The gold.
the
first
Find three consecutive numbers such that the sum of and twice the last equals 22. how many children
were present ?
x
11.
71
the
Find three numbers such that the second is 4 less than the third is three times the second.
first.
what are the three angles ?
10. and the pig iron produced in one year (1906) in the United States represented together a value
. equals 49 inches.
first.000. and children together was 37.
twice as old as B.
-
4.
A
is
Five years ago the What are their ages ?
C.
is five
numbers such that the sum of the first two times the first. If the population of New York is twice that of Berlin.
such as length. First fill in all the numbers given directly.000. it is frequently advantageous to arrange the quantities in a systematic manner.
.
Find the value of each.
3z + 4a:-8 = 27. then x 2 = number of hours B walks.72
of
ELEMENTS OF ALGEBRA
$ 750.
how many
100.e.
has each state
?
If the example contains Arrangement of Problems. 3 and 4. and quantities
area. together. number of miles A
x
x
walks.000. and 4 (x But the 2) for the last column. of 3 or 4 different kinds. 8 x = 15.
California has twice as
many
electoral votes as Colorado.
of
arid the value of the iron
was $300. we obtain 3 a.
= 35. number of hours. statement "A and B walk from two towns 27 miles apart until they meet " means the sum of the distances walked by A and B equals 27 miles.
The copper had twice
the value of the gold.
and Massachusetts has one more than California and Colorado If the three states together have 31 electoral votes. and distance.
Dividing. width. Let x = number of hours A walks. but stops 2 hours on the way.000.
B
many
miles does
A
walk
?
Explanation. speed.
Hence
Simplifying.000 more than that
the copper.
14. Since in uniform motion the distance is always the product of
rate
and time. and A walks at the rate of 3 miles per hour without stopping.
A and B
apart.g. After how many hours will they meet and how
E. or time. = 5. i.
7
Uniting.
start at the same hour from two towns 27 miles walks at the rate of 4 miles per hour.
3x
+
4 (x
2)
=
27.
55. were increased by 3 yards. and its width decreased
by 2 yards.
A
If its length
rectangular field is 2 yards longer than it is wide. how much did each cost per yard ?
6. each of the others had to pay
$ 100 more.
2. and how far will each then have traveled ?
9.
as a
4. and a second sum.
Twenty men subscribed equal amounts
of
to raise a certain
money. and the sum Find the length of their areas is equal to 390 square yards.
A
of each. and in order to raise the required sum each of the remaining men had to pay one dollar more.
invested at 5 %. and follows on horseback traveling at the rate of 5 miles per hour. and the cost
of silk
of the auto-
and 30 yards of cloth cost together much per yard as the cloth.
Ten yards
$
42.
Six persons bought an automobile. How many pounds of each kind did he buy ?
8. A man bought 6 Ibs.
mobile. the area would remain the same. The second is 5 yards longer than the first. but four men failed to pay their shares.
sions of the field. After how many hours will B overtake A. What are the
two sums
5. paid 24 ^ per pound and for the rest he paid 35 ^ per pound.74
ELEMENTS OF ALGEBRA
EXERCISE
38
rectangular field is 10 yards and another 12 yards wide. but as two of them were unable to pay their share.
sum $ 50
larger invested at 4
brings the same interest Find the first sum. twice as large.
A sum
?
invested at 4 %. of coffee for $ 1.
Find the share of each. How much did each man subscribe ?
sum
walking at the rate of 3 miles per hour.
Find the dimen-
A
certain
sum
invested at 5
%
%. together bring $ 78 interest.
1.
3.
A
sets out later
two hours
B
.
If the silk cost three times as
For a part he 7.
walking at the same time in the same If A walks at the rate
of 2
far
miles per hour. but
A has
a start of 2 miles. and another train starts at the same time from New York traveling at the rate of 41 miles an hour.LINEAR EQUATIONS AND PROBLEMS
v
75
10. how many miles from New York will they meet?
X
12. Albany and travels toward New York at the rate of 30 miles per hour without stopping. and B at the rate of 3 miles per hour.
A
sets out
two hours
later
B
starts
New York to Albany is 142 miles. After how many hours.will they be 36 miles apart ?
11.
A
and
B
set out
direction. traveling by coach in the opposite direction at the rate of 6 miles per hour.
The
distance from
If a train starts at
. how must B walk before he overtakes A ?
walking at the rate of 3 miles per hour. and from the same point.
An expression is integral with respect to a letter. if this letter does not occur in any denominator.
An
after simplifying.
The
factors of
an algebraic expression are the quantities
will give the expression.
An
expression
is integral
and rational with respect
and rational.
stage of the work.
a factor of a 2
A
factor is said to be prime. if it is integral to all letters contained in it. at this
6
2
.
-f-
db
6
to b. it contains no indicated root of this letter
.
104.
76
.
it is
composite.
expression is rational with respect to a letter.
a2
to 6.
vV
.CHAPTER
VI
FACTORING
101. 5.
this letter. if it does contain
some indicated root of
.
irrational. if.
The prime
factors of 10 a*b are 2.
which multiplied together
are considered factors.
\-
V&
is
a
rational with respect to
and
irrational with respect
102.
a.
+ 62
is
integral with respect to a. as. 6. but fractional with respect
103.
J Although Va'
In the present chapter only integral and rational expressions
b~
X
V
<2
Ir
a2
b'
2
2
?>
.
a-
+
2 ab
+ 4 c2
. if it contains
no other
factors (except itself
and unity)
otherwise
. consider
105.
we
shall not. a.
107.
it fol-
lows that every method of multiplication will produce a method
of factoring.62 + &)(a 2
.3 6a + 1).
2.
?/.
Divide
6
a% . .g.
factors of 12
&V
is
are 3. 2.
in the
form
4)
+3. 2.
Since factoring
the inverse of multiplication. for this result is a sum.62
can be
&).
Factor G ofy 2
.
TYPE
I.
An
the process of separating an expression expression is factored if written in the
form of a product.
110.
It (a.
55. 8) (s-1).
y.FACTORING
106.
it
follows
that a 2
.
x.
E. dividend
is
2 x2
4
2
1/
.
Ex.
01. since (a + 6) (a 2 IP factored.
Hence
6 aty 2
= divisor x quotient.3 sy + 4 y8).
1.
The factors
of a
monomial can be obtained by inspection
2
The prime
108.
2 4 x + 3) is factored if written (x' would not be factored if written x(x and not a product. x.9 x2^ + 12 sy* = 3 Z2/2 (2 #2 .
POLYNOMIALS ALL OF WHOSE TERMS CONTAIN A COMMON FACTOR
(
mx + my+ mz~m(x+y + z).
Factor
14 a*
W-
21 a 2 6 4 c2
+ 7 a2 6
2
c2
7
a2 6 2 c 2 (2 a 2
.9 x2 y 8 + 12
3 xy
-f
by
3
xy\
and the quotient
But.
.
109.
77
Factoring
is
into its factors.
or
Factoring examples may be checked by multiplication by numerical substitution.)
Ex.9 x if + 12 xy\
2
The
greatest factor
common
2
to all terms
flcy*
is
8
2
xy'
. or that a
=
6)
(a
= a .
1 afy 8 The two numbers whose product is equal to 12 yp and whose sum equals 3 8 7 y are -4 y* and -3 y*. Hence fc -f 10 ax
is
10 a are 11 a
-
12 /. or 11 and 7 have a sum equal to 4.
If
30 and whose
sum
is
11 are
5
a2
11
a = 1.
.6 = 20.
and the greater one has the same sign
Not every trinomial
Ex.11 a + 30.
Factor a2
.
Ex.
Therefore
Check.30 = (a .FACTORING
Ex.
. + 30 = 20. the two numbers
have opposite
signs.
3. a 2 .
2. If q is positive.
77 as the product of 1 77.11.
4. If q is negative.
.
We may consider
1.
EXERCISE
Besolve into prime factors :
40
4.
or
77
l.
79
Factor a2
-4 x .11 a
2
.1 1 a
tf
a 4.a). the student should first all terms contain a common monomial factor.5) (a 6). however.
determine whether
In solving any factoring example. but only in a limited number
of ways as a product of two numbers.
as p.4 . can be factored.
is
The two numbers whose product and -6.
Ex. and (a . 2 11 a?=(x + 11 a) (a.4 x .
tfa2 -
3.11) (a
+
7).
m -5m + 6. Factor x? .
11 a2 and whose sum The numbers whose product is and a.
11
7.77 =
(a.
+
112.
but of these only
a:
Hence
2
.
Factor
+ 10 ax .G) = .
Since a number can be represented in an infinite number of ways as the sum of two numbers.
5. Hence z6 -? oty+12 if= (x -3 y)(x*-4 y ).
2
6. the two numbers have both the same sign as p. or 7 11..5) (a . it is advisable to consider the factors of q first. of this type.
FACTORING
If
81
we consider that the
factors of -f 5
as
must have
is
:
like signs.
3. X x 18.
a. If py? -\-qx-\-r does not contain any monomial factor. and after a little practice the student possible should be able to find the proper factors of simple trinomials
In actual work
at the first trial. Hence only 1 x 54 and 2 x 27 need
be considered.
all
it is not always necessary to write down combinations. exchange the
signs of the second terms of the factors.e-5
V A
x-1
3xl \/ /\
is
3
a.17 x
2o?-l
V A
5
-
13 a
combination
the correct one.13 x + 5 = (3 x . and r is negative. Since the first term of the first factor (3 x) contains a 3.
The work may be shortened by the
:
follow-
ing considerations
1.
If
the factors
a combination should give a sum of cross products. 27 x 2.83 x
-f-
54. If p is poxiliw. none of the binomial factors can contain a monomial factor.
Factor 3 x 2
.5 . but the opposite sign.
. 18 x 3.31 x
Evidently the
last
2
V A
6.
and that they must be negative.
The
and
factors of the first term consist of one pair only.
sible
13 x
negative. all pos-
combinations are contained in the following
6x-l
x-5 .
Ex. 64 may be considered the
:
product of the following combinations of numbers 1 x 54.5) (2 x . or
G
114.1). then the second terms of
have opposite signs. 9 x 6. 2. which has the same absolute value as the term qx. 6 x 9. we have to reject every combination of factors of 54 whose first factor contains a 3. 54 x 1. viz. 2 x 27. the second terms of the factors have same sign as q.
11 x
2x.
. the signs of the second terms are minus.
the
If p and r are positive. 3 x and x.
C. If the expressions have numerical coefficients. F.) of
two or more
.
89
. C.
are prime can be found by inspection. C.
The
highest
is
common
factor (IT. 12 tfifz.
+
8
ft)
and
cfiW is
2
a 2 /) 2
ft)
.CHAPTER
VII
HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE
HIGHEST COMMON FACTOR
120. of the algebraic expressions.
II
2
.
24
s
.
13 aty
39 afyV. F. of a 4 and a 2 b is a2
The H.
expressions which have no are prime to one another. F.
5
2
3
.
the algebraic factor of highest degree common expressions to these expressions thus a 6 is the II.
Two
common
factor except unity
The H.
33
2
7
3
22 3 2
.
6. F. F. C.
5.
F.
5
7
34 2s
. and prefix it as a coefficient to H.
121. The H.
5
s
7
2
5. F. of (a
and (a
+
fc)
(a
4
is
(a
+ 6)
2
.
54
-
32
. of 6 sfyz.
122.
8
.
C.
C. of
aW.
3
. F. is the lowest
that the power of each factor in the power in which that factor occurs in any
of the given expressions.
EXERCISE
Find the H.
C. of a 7 and a e b 7
.
The student should note
H. aW.
25
W.
2
2
. F.
3.
15
aW. The H. of
two or more monomials whose factors
.
. Thus the H. and GO aty 8 is 6 aty.
2.
-
23 3
. C. F. C. find by arithmetic the greatest common factor of the coefficients. C. of
:
48
4.
2 multiples of 3 x
and 6 y are 30 xz y. 60
x^y'
2
. M.
NOTE.
. C.
Ex.(a + &) 2 (a
have the same absolute value. To find the L.LOWEST COMMON MULTIPLE
91
LOWEST COMMON MULTIPLE
multiple of two or more expressions is an which can be divided by each of them without a expression
124. C.
300 z 2 y. thus.
M.6)2. If the expressions have a numerical coefficient.
a^c8
3
.
of 4 a 2 6 2 and 4 a 4
-4 a 68
2
.C.
Common
125.
Find the L.
2.
&)
2
M. C.
of 12(a
+
ft)
and (a
+ &)*( -
is
12(a
+ &)( . find by arithmetic their least common multiple and prefix it as a coefficient to the L. C. C. of the
general.C.
4 a 2 &2
_
Hence.
M.
C.
2
The The
L. C. two lowest common multiples.
.
The
L.) of
two or more
expressions is the common multiple of lowest degree.
= (a -f
last
2
&)'
is
(a
-
6) .
The
lowest
common
multiple (L.
6
c6 is
C a*b*c*. of several expressions which are not completely factored. M.
of 3
aW.
128. C.M. is equal to the highest power in which it occurs in any of the
given expressions. of tfy and xy*. L. resolve each expression into prime factors and apply the method for monomials. C.
=4 a2 62 (a2 .
etc. M.
1.
M of the algebraic expressions.M.
127. of as -&2 a2 + 2a&-f b\ and 6-a.
A
common
remainder.
126.
Find the L. M. Obviously the power of each factor in the L. M. ory is the L. L.
M. but opposite
.
Hence the L. C. each set of expressions has
In example ft).6 3 ). which
also
signs.
Ex.
a?.
131.
common
6
2
divisors of
numerator and denomina-
and z 8
(or divide the terms
.
The dividend a is called the numerator and the The numerator and the denominator
are the terms of the fraction.
successively all
2
j/' . F.
TT
Hence
24
2 z = --
3x
. only positive integral numerators shall assume that the
all
arithmetic principles are generally true for
algebraic numbers.
an indicated quotient. the value of a fraction is not altered by multiplying or dividing both its numerator and its denominator by the same number.
130.
and denominators are considered. etc.
a b
= ma
mb
. C. as 8. Thus.
All operations with fractions in algebra are identical
with the corresponding operations in arithmetic. If both terms of a fraction are multiplied or divided by the same number) the value of the fraction is not altered.
fraction
is
in its lowest
when
its
numerator
and
its
denominator have no
common
factors.
Ex. however.ry ^
by
their H.CHAPTER
VIII
FRACTIONS
REDUCTION OF FRACTIONS
129.
rni
Thus
132.
Reduce
~-
to its lowest terms.
and
i
x mx = my y
terms
A
1.
the product of two fractions is the product of their numerators divided by the product of their denominators. but we
In arithmetic.
Remove
tor. thus -
is identical
with a
divisor b the denominator.
A
-f-
fraction is
b.
mon
T denominator.
1).
multiply each quotient by the corresponding numerator. and 6rar 3 a? kalr
.
Multiplying these quotients by the corresponding numerators and writing the results over the common denominator.
1. we have
(a
+ 3) (a -8) (-!)'
NOTE. by the denominator of each fraction.~16
(a
+ 3) (x. we may use the same process as in arithmetic for reducing fractions to the lowest
common
denominator.C.
we have
the quotients (x
1).
we have
-M^.
by any quantity without altering the value of the fraction.C.
Reduce -^-.
Ex
-
Reduce
to their lowest
common
denominator.
Since a
(z
-6 + 3)(s-3)O-l)'
6a. Divide the L.
and
Tb reduce fractions to their lowest common denominator. of the denominators for the common denominator.
Ex.r
2
2
.-1^22
' .
M.
3 a\ and 4
aW
is
12 afo 2 x2 .
we may extend this method
to integral expressions.
ELEMENTS OF 'ALGEBRA
Reduction of fractions to equal fractions of lowest common Since the terms of a fraction may be multiplied
denominator.
. take the L.3)O -
Dividing this by each denominator. =(z
(x
+ 3)(z. multiplying the terms of
22
.D.M.
^
to their lowest
com-
The
L.3) (-!)'
=
.
and
(a-
8).M. C.
+
3).by 3 ^
A
2
' .
-
of
//-*
2
.
To reduce
to a fraction with the
denominator 12 a3 6 2 x2 numerator
^lA^L O r 2 a 3
'
and denominator must be multiplied by
Similarly.
-
by 4
6' .96
134.
and the terms of
***.
. C. and
135.
TheL.
.
2>
.
(In
order to cancel
common
factors.)
Ex.
or.
fractions to integral numbers.
Simplify 1 J
The
expreeaion
=8
6
.
-x
b
c
=
numerator by
To multiply a fraction by an
that integer.
we may extend any
e.102
ELEMENTS OF ALGEBRA
MULTIPLICATION OF FRACTIONS
140. each
numerator and denomi-
nator has to be factored.
2. Common factors in the numerators and the denominators should be canceled before performing the multiplication. multiply the
142.
F J Simplify
. Fractions are multiplied by taking the product of tht numerators for the numerator.
2
a
Ex.
!. and the product of the denominators for the
denominator.
Since -
= a.
integer.g.
expressed in symbols:
c
a
_ac b'd~bd'
principle proved for
b
141.
x a + b
obtained by inverting
reciprocal of a fraction
is
the fraction. and the principle of division follows
may
be expressed as
145. :
a 4-1
a-b
* See page 272.
.
The The
reciprocal of a
is
a
1
-f-
reciprocal of J
is
|
|.
1. expression by the reciprocal of the fraction.
Divide X-n?/
.
The reciprocal of ?
Hence the
:
+*
x
is
1
+ + * = _*_.
144.
* x* -f xy 2
by
x*y
+y
x'
2
3
s^jf\ =
x'
2
x*
. Integral or mixed divisors should be expressed in fractional form before dividing.y3
+
xy*
x*y~ -f y
8
y
-f
3
2/
x3
EXERCISE 56*
Simplify the following expressions
2
x*
'""*'-*'
:
om
2 a2 6 2
r -
3
i_L#_-i-17
ar
J
13 a& 2
5
ft2
'
u2
+a
. invert the divisor and multiply it by the dividend.
8
multiply
the
Ex.
The
reciprocal of a
number
is
the quotient obtained by
dividing 1
by that number. To divide an expression by a fraction. To divide an expression by a fraction.104
ELEMENTS OF ALGEBRA
DIVISION OF FRACTIONS
143.
100
C. 2 3
.
. 1.
A would do
each day ^ and
B
j. Ex. A can do a piece of work in 3 days and B in 2 days. 2.
~^ = 15
11 x
'
!i^=15. When between 3 and 4 o'clock are the hands of
a clock together
?
is
At
3 o'clock the hour hand
15 minute spaces ahead of the minute
:
hand.
C
is
the circumference of a circle whose radius
R. = the number of minute spaces the minute hand moves
over..
.
and
12
= the number
over.
Find
R in terms of C and
TT.
Ex.minutes after x=
^
of
3 o'clock.
of minute spaces the
hour hand moves
Therefore x
~ = the number of minute spaces the minute hand moves
more than the hour hand.
then
= 2 TT#.
12.180.
Multiplying by
Dividing.
x
Or
Uniting.
PROBLEMS LEADING TO FRACTIONAL AND LITERAL EQUATIONS 152.20
C.
ELEMENTS OF ALGEBRA
(a) Find a formula expressing degrees of Fahrenheit terms of degrees of centigrade (<7) by solving the equation
(F)
in
(ft)
Express in degrees Fahrenheit 40
If
C. = 16^.. In how many days can both do it working together ?
If
we denote
then
/-
the required
number
by
1.
is
36. hence the question would be formulated After how many minutes has the minute hand moved 15 spaces more than the hour hand ?
Let then
x x
= the required number of minutes after 3 o'clock.114
35.
days by x and the piece of work while in x days they would do
respectively
ff
~ and and hence the sentence written in algebraic symbols ^.
Explanation
:
If
x
is
the rate of the accommodation train.
the rate of the express train. u The accommodation train needs 4 hours more than the express train.
32
x
= |."
:
Let
x -
= the
required
number
of days. what is
the rate of the express train
?
180
Therefore.
Ex.
180
Transposing.
= 100 + 4 x. and the statement.
Clearing.
in
Then
Therefore. The speed of an express train is $ of the speed of an If the accommodation train needs 4 accommodation train. the required
number
of days." gives the equation /I). hours more than the express train to travel 180 miles. 3.
Solving. then
Ox
j
5
a
Rate Hence the rates can be expressed.FRACTIONAL AND LITERAL EQUATIONS
A
in symbols the following sentence
115
more symmetrical but very similar equation is obtained by writing ** The work done by A in one day plus the work done by B in one day equals the work done by both in one day.
= the
x
part of the
work both do
one day. 4x = 80.
or 1J.
But
in
uniform motion Time
=
Distance
.
fx
xx*
=
152
+4
(1)
Hence
=
36
= rate
of express train.
A man left ^ of his property to his wife. and one half the greater Find the numbers.
length in the ground. How
did the
much money
man
leave ?
11. and J of the greater Find the numbers. How much money had he at first?
12
left
After spending ^ of his
^ of his money and $15. which was $4000.116
ELEMENTS OF ALGEBRA
EXERCISE
60
1.
J-
of the greater
increased by ^ of the smaller equals
6.
Find a number whose third and fourth parts added
together
2. and of the father's age.
are the
The sum of two numbers numbers ?
and one
is
^ of the other.
is
equal
7.
is oO. A man lost f of his fortune and $500.
make
21.
3.
Find two consecutive numbers such that
9.
by 3.
Twenty years ago A's age was |
age. a man had How much money had he
at first?
.
its
Find the number whose fourth part exceeds part by 3.
ceeds the smaller by
4.
of his present age.
9
its
A
post
is
a fifth of
its
length in water.
Two numbers
differ
l to s of the smaller. to his daughand the remainder. one half of What is the length
of the post ?
10
ter. and found that he had \ of his original fortune left. to his son.
Find A's
8.
fifth
Two numbers
differ
2.
ex-
What
5.
money and $10.
The sum
10 years hence the son's age will be
of the ages of a father and his son is 50.
and 9
feet above water. -|
Find their present ages.
by 6.
If the rate of the express train is -f of the rate of the accommodation train.
investments.
At what time between 4 and
(
5 o'clock are the hands of
a clock together?
16. and
it
B in 6 days. what is the
14.FRACTIONAL AND LITERAL EQUATIONS
13.
A has invested capital
at
more
4%. In how many days can both do it working together ? ( 152. and an ounce of silver -fa of an ounce.
air. ^ at 5%.
.
A man
has invested
J-
of his
money
at
the remainder at
6%.
152. and B in 4 days.) (
An express train starts from a certain station two hours an accommodation train.
at 4J % and P> has invested $ 5000 They both derive the same income from their How much money has each invested ?
20. Ex. Ex. If the accommodation train needs 1 hour more than the express train to travel 120 miles. and has he invested if
his animal interest therefrom is
19. 1. A can do a piece of work in 4 clays.
At what time between 7 and
8 o'clock are the hands of
?
a clock in a straight line and opposite
18.)
22.
117
The speed
of an accommodation train
is
f of the speed
of an express train. ounces of gold and silver are there in a mixed mass weighing
20 ounces in
21.
?
In
how many days can both do
working together
23. Ex. and after traveling 150 miles overtakes the accommodation train.)
At what time between 7 and 8
o'clock are the
hands of
a clock together ?
17. 3. 2.
and losing
1-*-
ounces when weighed in water?
do a piece of work in 3 days. and B In how many days can both do it working together
in
?
12 days. what is the rate of the express train? 152.
A can
A
can do a piece of work in 2 days.
after
rate of the latter ?
15. An ounce of gold when weighed in water loses -fa of an How many ounce.
How much money
$500?
4%.
Answers to numerical questions of this kind may then be found by numerical substitution.
they can both do
in 2 days. Find the numbers if m = 24 30. 2. Ex. A in 4.e.
To
and
find the numerical answer.
is 42. B in 12. A in 6. n x
Solving. therefore.
ELEMENTS OF ALGEBRA
The
last three questions
and their solutions differ only two given numbers. and apply the
method of
170.
:
In
how many days
if
can
A
and
it
B
working together do a
piece of
work
each alone can do
(a)
(6)
(c)
in the following
number
ofdavs:
(d)
A in 5. is 57.
6
I
3
Solve the following problems
24. it is possible to solve all examples of this type by one example. B in 30.
. Hence. .
.
25. is A can do a piece of work in m days and B in n days. A in 6.g.
26. Find three consecutive numbers whose sum equals m. by taking for these numerical values two general algebraic numbers. Then
ft
i. In how
in the numerical values of the
:
many days
If
can both do
we
let
x
= the
it working together ? required number of days.= -.118
153. The problem to be solved. if
B
in 3 days.414.
make
it
m
6
A can do this work in 6 days Q = 2. m and n.
B in 5. B in 16.
we
obtain the equation
m m
-. 3.009 918.
Find three consecutive numbers whose sum
Find three consecutive numbers whose sum
last
:
The
two examples are
special cases of the following
problem 27. e. and n = 3.=
m
-f-
n
it
Therefore both working together can do
in
mn
-f-
n
days.
and how many miles does each travel ?
32.FRACTIONAL AND LITERAL EQUATIONS
28. the
Two men start at the same time from two towns. If each side of a square were increased by 1 foot.
squares
30.
:
(c)
64 miles. respectively (a) 60 miles. two pipes together ? Find the numerical answer.
A cistern can
be
filled
(c)
6 and 3 hours.
33.
. (d) 1. if m and n are.000.
Two men
start at the
first
miles
apart. the second at the apart.
and
the rate of the second are. (b) 8 and 56 minutes. 88 one traveling 3 miles per hour. (c) 16. After how many hours do they meet.001.721. the rate of the
first.
is (a)
51. Find the side of the square. respectively. (b) 35 miles.
4J-
miles per hour.
meet.
same hour from two towns.
squares
29.
Find two consecutive numbers -the difference of whose
is 21. the area would be increased by 19 square feet. 3 miles per hour. d miles the first traveling at the rate of m. After how many hours do they rate of n miles per hour.
The
one:
31. solve the following ones Find two consecutive numbers the difference of whose squares
:
find the smaller number.
34. 2 miles per hour.
is ?n
.
by two pipes in m and n minutes In how many minutes can it be filled by the respectively. 5 miles per hour.
last three
examples are special cases of the following
The
difference of the squares of
two consecutive numbers
By using the result of this problem. and how many miles does each travel ? Solve the problem if the distance. (b) 149. 2 miles per hour. 3J miles per hour. (a) 20 and 5 minutes.
119
Find two consecutive numbers the difference of whose
is 11. and the second 5 miles per hour.
antecedent.
terms are multiplied or divided by the same number. the antecedent.
term of a ratio
a
the
is
is
the antecedent.
:
A somewhat shorter way
would be to multiply each term by
120
6.g. etc."
we may
write
a
:
b
= 6.CHAPTER X
RATIO AND PROPORTION
11ATTO
154.
1.
Thus the
written a
:
ratio of a
b
is
.
The
ratio -
is
the inverse of the ratio -.
A
ratio
is
used to compare the magnitude of two
is
numbers.) The ratio of 12 3 equals 4.
:
:
155. instead of writing
6 times as large as
?>.
Ex.
is
numerator of any fraction
consequent.
. b is the consequent. the denominator
The
the
157.
Simplify the ratio 21 3|. a ratio
is
not changed
etc. the second
term the consequent. all principles
relating
to
fractions
if its
may
be af)plied to ratios.
b.
the symbol
being a sign of division.
The
ratio of
first
dividing the
two numbers number by the
and
:
is
the quotient obtained by
second.
b
is
a
Since a ratio
a
fraction.
The
first
156.
In the ratio a
:
ft.or a *
b
The
ratio is also frequently
(In most European countries this symbol is employed as the usual sign of division.
E.
b. 158.5.
" a Thus. 6 12 = .
:
is
If the means of a proportion are equal. the second
and fourth terms of a proportion are the and third terms are the means.
proportional between a
and
c.
J:l.
8^-
hours.
AND PROPORTION
ratio 5
5
:
121
first
Transform the
3J so that the
term will
33
:
*~5
~
3
'4*
5
EXERCISE
Find the value of the following
1.
equal
2.
3:4.
= |or:6=c:(Z are
The
first
160.
10.
16.
18.
11.
16 x*y
64 x*y
:
24 48
xif.
A
proportion
is
a statement expressing the equality of
proportions. and the last term the third proportional to the first and second
161.
$24: $8.
61
:
ratios
72:18.
159.
Simplify the following ratios
7. The last
first three.
5 f hours
:
2.
62:16.
:
ratios so that the antecedents equal
16:64.
7|:4 T T
4
.
17. a and d are the extremes.
two
|
ratios.
4.RATIO
Ex.
12. b and c the means.
4|-:5f
:
5.
27 06: 18 a6.
9.
6. b.
:
a-y
.
extremes.
terms.
:
1.
16a2 :24a&. either mean the mean proportional between the first and the last terms.
1.
Transform the following
unity
15.
and
c
is
the third proportional to a and
.
In the proportion a b
:
=
b
:
c.
3:1}.
term
is
the fourth proportional to the
:
In the proportion a b = c c?.
3
8.
3.
7f:6J. The last term d is the fourth proportional to a.
b is the
mean
b. and c.
and the
other pair the extremes. or 8 equals the inverse ratio of 4 3. then G ccm. Instead of u
If 4
or 4 ccm.
ccm.
If 6 men can do a piece of work in 4 days. then 8 men can do it in 3 days.
2
165.
:
c.
:
:
directly proportional
may say.) b = Vac. 6 ccm.
briefly.
q~~ n
.30 grams. Hence the weight of a mass of iron is proportional to its volume. if the ratio of any two of the first kind is equal \o the inverse ratio of the corresponding two of
the other kind.
163. of a proportion.__(163.
!-. " we " NOTE. In any proportion product of the extremes. and we
divide both
members by
we have
?^~ E. Hence the number of men required to do some work. if the ratio of any two of the first kind. of iron weigh 45 grams.
t/ie
product of the means
b
is
equal
to the
Let
a
:
=c
:
d.
Clearing of fractions.
If
(Converse of
nq.122
162. is equal to the ratio of the corresponding two
of the other kind.
The mean proportional
of their product. 3 4.
163. are
: : :
inversely proportional.
164. and the time necessary to do it. a b
:
bettveen two
numbers
is
equal to
the square root
Let the proportion be
Then Hence
6
=b = ac.
pro-
portional.)
mn = pq. = 30 grams 45 grams. If the product of two numbers is equal to the product of two other numbers^ either pair may be made the means. of iron weigh .e. i.
ad =
be.
ELEMENTS OF ALGEBRA
Quantities of one kind are said to be directly proper
tional to quantities of another kind.'*
Quantities of one kind are said to be inversely proportional to quantities of another kind.
What
will be the
volume
if
the pressure
is
12 pounds per square inch ?
.
and the speed
of the train.
56. State whether the quantities mentioned below are directly or inversely proportional (a) The number of yards of a certain kind of silk. and the time. and
the time necessary for it.
(b)
The time a
The length
train needs to travel 10 miles.
A
line 11 inches long
on a certain
22 miles.
A
line 7^. The number of men (m) is inversely proportional to the number of days (d) required to do a certain piece of work. the volume of a
The temperature remaining
body of gas inversely proportional to the pressure.
ELEMENTS OF ALGEBEA
State the following propositions as proportions : T (7 and T) of equal altitudes are to each. and the
:
total cost.
(d)
The sum
of
money producing $60
interest at
5%.
what
58. (c) The volume of a body of gas (V) is
circles are to
each
inversely propor-
tional to the pressure (P). under a pressure of 15 pounds per square inch has a volume of
gas
is
A
16 cubic
feet.
areas of circles are proportional to the squares of If the radii of two circles are to each other as
circle is
4
:
7.
(d)
The
areas
(A and
A') of two circles are to each other as
(R and R').inches long represents
map corresponds to how many miles ?
The
their radii. and the area
of the rectangle.126
54.
1
(6) The circumferences (C and C ) of two other as their radii (R and A").
57.
the squares of their radii
(e)
55. (e) The distance traveled by a train moving at a uniform rate.
(c)
of a rectangle of constant width.
and the area of the smaller
is
8 square inches.
the area of the larger? the same. othei
(a) Triangles
as their basis (b
and
b').
11
x
x
7
Ex. 7 x = 42 is the second number. What is the greatest distance a person can see from an elevation of 5 miles ? From h miles
the
Metropolitan
Tower (700
feet high) ?
feet
high) ?
From Mount
McKinley (20. 2. produced to a point C. AB = 2 x. = the second number. 11 x = 66 is the first number.
Hence
or
Therefore
Hence
and
= the first number.000
168. x = 6.
Divide 108 into two parts which are to each other
7.
2 x
Or
=
4. When a problem requires the finding of two numbers which are to each other as m n.
Therefore
7
=
14
= AC.RATIO AND PROPORTION
69. 11 x -f 7 x = 108. 4 inches long.
:
Ex.
is
A line AB.
127
The number
is
of miles one can see from an elevation of
very nearly the mean proportional between h and the diameter of the earth (8000 miles).
as 11
Let
then
:
1.
4
'
r
i
1
(AC): (BO) =7: 5.
Let
A
B
AC=1x. 18 x = 108. so that
Find^K7and BO. it is advisable to represent these unknown numbers by mx and nx.
.
x=2.
Then
Hence
BG = 5 x.
:
4.000.
consists of 9 parts of copper and one part of ounces of each are there in 22 ounces of gun-
metal ?
Air is a mixture composed mainly of oxygen and nitrowhose volumes are to each other as 21 79. Water consists of one part of hydrogen and 8 parts of
If the total surface of the earth
oxygen.
11. Brass is an alloy consisting of two parts of copper and one part of zinc.
A line 24 inches
long
is
divided in the ratio 3
5.
m
in the ratio x:
y
%
three sides of a triangle are 11. and c inches. 9.
12.
3. and the longest is divided in the ratio of the other two.
Divide 20 in the ratio 1 m. 12. How
The
long are the parts ? 15.)
. How many gen.
14.
How many
7.
:
Divide a in the ratio 3
Divide
:
7.
Divide 44 in the ratio 2
Divide 45 in the ratio 3
:
9. If c is divided in the ratio of the other two. cubic feet of oxygen are there in a room whose volume is 4500
:
cubic feet?
8.
What
are the parts ?
5.
Gunmetal
tin. The three sides of a triangle are respectively a.
of water?
Divide 10 in the ratio a
b. How many ounces of copper and zinc are in 10 ounces of brass ?
6.
:
197.
How many
grams of hydrogen are contained in 100
:
grams
10.
:
Divide 39 in the ratio 1
:
5.
7. 6.000 square miles. 2. find the number of square miles of land and of water.
The
total area of land is to the total area of
is
water as
7 18. and 15 inches. what are
its
parts ?
(For additional examples see page 279.128
ELEMENTS OF ALGEBRA
EXERCISE
63
1.
13.
the equation is satisfied by an infinite number of sets Such an equation is called indeterminate.
a?
(1)
then
I.
the equations have the two values of
y must be equal. is x = 7. =.-.y=--|. there is only one solution. if
. such as
+ = 10.-L
x
If
If
= 0.
The
root of (4)
if
K
129
.
However.
If
satisfied
degree containing two or more by any number of values of
2oj-3y =
6. From (3) it follows y 10 x and since
by the same values of x and
to be satisfied
y. etc.CHAPTER XI
SIMULTANEOUS LINEAR EQUATIONS
169.
Hence.
2 y = . expressing a y.
An
equation of the
first
unknown numbers can be the unknown quantities. x = 1.e. values of x and y.
if
there
is
different relation
between x and
*
given another equation.
y
(3)
these
unknown numbers can be found. y = 1. y =
5
/0 \ (2)
of values.
Hence
2s -5
o
= 10 _ ^
(4)
= 3. which substituted in (2) gives y both equations are to be satisfied by the same Therefore.
viz.
The process of combining several equations so as make one unknown quantity disappear is called elimination.
unknown
quantity. 3.
21 y
. y
I
171. y = 2.
are simultaneous equations.
174.
4y
.3 y = 80.24.X. 30 can be reduced to the same form -f 5 y Hence they are not independent.
Substitution.
(3)
(4)
Multiply (2) by
-
Subtract (4) from (3).130
170.
= . 6 and 4 x y not simultaneous. The first set of equations is also called consistent.
6x
.
172.
to
The two methods
I.26.
Solve
-y=6x
6x
-f
Multiply (1) by
2.
E. 26 y = 60. Any set of values satisfying 5 x + 6 y = 60 will also satisfy the equation 3 x -f.
of elimination
most frequently used
II. for they cannot be satisfied by any value of x and y. Independent equations are equations representing different relations between the unknown quantities such equations
.
Therefore. for they express the x -f y 10.
cannot be reduced to the same form.
ELEMENTS OF ALGEBRA
A
system
of simultaneous equations is
tions that can be satisfied
a group of equa by the same values of the unknown
numbers. same relation. the last set inconsistent. for they are 2 y = 6 are But 2 x 2.
ELIMINATION BY ADDITION OR SUBTRACTION
175.
~ 50.
A
system of two simultaneous equations containing two
quantities is solved by combining them so as to obtain
unknown
one equation containing only one
173.
By By
Addition or Subtraction. and 3 x + 3 y =.
x
-H
2y
satisfied
6 and 7 x 3y = by the values x = I.
symbols:
x
+
y
+z-
8. Problems involving several unknown quantities must contain. the number. either directly or implied.
.
(1)
100s
+ lOy + z + 396 = 100* + 10y + x. 1.
Check.
and Then
100
+
10 y
+z-
the digit in the units' place.
2
= 6. 1 digit in the tens place.
2
= 1(1+6).
unknown quantity by
every verbal statement as an equation.
x
:
z
=1
:
2.
The
three statements of the problem can
now be
readily expressed in
. The sum of three digits of a number is 8. and to express
In complex examples.
+
396
= 521.y
125
(3)
The solution of these equations gives x Hence the required number is 125. + z = 2p.
= 2 m.
1
=
2. and if 396 be added to the number.
+2+
6
= 8.
Find the number. y
31.
Obviously
of the other
. The digit in the tens' place is | of the sum of the other two digits.
to express
it is difficult
two of the required
digits in
terms
hence we employ 3
letters for the three
unknown
quantities.
# 4. z + x = 2 n. the first and the last digits
will be interchanged.2/
2/
PROBLEMS LEADING TO SIMULTANEOUS EQUATIONS
183.
y
*
z
30.
(
99. however.SIMULTANEOUS LINEAR EQUATIONS
143
x
29.)
it is advisable to represent a different letter.
M=i. Simple examples of this
kind can usually be solved by equations involving only one
unknown
every
quantity.
Ex.
Let
x
y z
= the
the digit in the hundreds' place. as many verbal statements as there are unknown quantities.
.
=
l.
. y = 3.
Since the three
men
traveled the
same
distance.
4
x
= 24.
+
I
2
(1)
and
These equations give x
Check.
=
Hence the
fraction
is
f. 5_
_4_
A.
3
xand y
I
1
(2)
5. C.144
Ex.
(1) (2)
12. and C travel from the same place in the same B starts 2 hours after A and travels one mile per hour faster than A.
By
expressing the two statements in symbols.
x
y
= the = the
x
denominator
.
x 3
= 24. B. starts 2 hours after B and overtakes A at the same How many miles has A then traveled? instant as B.
Ex.
3.
2.
2. = 8.
= the
fraction.
increased by one.
6
x 4
= 24.
Or
(4)-2x(3). the fraction is reduced to | and if both numerator and denominator of the reciprocal of the fraction be dimin-
ished by one. who travels 2 miles an hour faster than B. the fraction
Let and
then y
is
reduced to
nurn orator.
xy
a:
2y 4y
2.
3+1 5+1
4_2.
direction. From (3)
Hence xy
Check.
8
= xy + x xy = xy -f 3 x 2 y = 2.
ELEMENTS OF ALGE13KA
If both numerator and denominator of a fraction be
.
(3) C4)
=
24 miles. the distance traveled by A.
Find the
fraction. x 3x-4y = 12.
we
obtain.
the fraction equals . the value of the fraction is fa.
7.
1. to the number the digits will be interchanged. Find the number.
part of their difference equals
4.
added to the numerator of a fraction. Find the numbers. A fraction is reduced to J.
Half the sum of two numbers equals 4.
If 4 be
Tf 3 be
is J. If the denominator be doubled.
If 27 is
10. Four times a certain number increased by three times another number equals 33. the fraction is reduced
fraction. both terms. fraction is reduced to \-. Find the number.
If the
numerator of a fraction be trebled.
Five times a certain number exceeds three times another 11. If
9 be added to the number. and the second increased by 2 equals three times the first. the Find the fraction.)
added to a number of two digits.
tion ?
8.
to
L
<>
Find the
If the
numerator and the denominator of a fraction be If 1 be subtracted from increased by 3.
.
6.
The sum
of the first
sum
of the three digits of a number is 9. Find the fraction. the last two digits are interchanged.
5. and twice the numerator What is the fracincreased by the denominator equals 15. Find the numbers.
number by
the
first
3. and the second one increased by 5 equals twice
number.
the
number
(See Ex.
?
What
9. the digits will be interchanged. and the numerator increased by 4. and the two digits exceeds the third digit by 3. its value added to the denominator.
2. if its numerator and its denominator are increased by 1. and four times the first digit exceeds the second digit by 3. and the fourth 3.
Find the numbers.
183.}.SIMULTANEOUS LINEAR EQUATIONS
EXERCISE
70
145
1.
The sum
18
is is
and
if
added
of the digits of a number of two figures is 6. and
its
denomi-
nator diminished by one. it is reduced to J.
Twice A's age exceeds the sum of B's and C's ages by 30. respectively ?
16. bringing a total yearly interest of $530. and The 6 investment brings $ 70 more interest than the 5
%
%
4%
investments together.
14. What was the amount of each investment ?
A man
%
5%. 12. Find
the rates of interest. Two cubic centimeters of gold and three cubic centimeters of silver weigh
together 69 J.
partly at
5 %.
Ten years ago A was B was as
as old as
B
is
old as
will be 5 years hence .
13.
in 8 years to $8500.
and 5 years ago
their ages is 55. and partly at 4 %. now. and the 5% investment brings $15 more interest than the 4 % investment.grams. a part at 6 and the remainder bringing a total yearly interest of $260. A man invested $750.
A
sum
of $10. and in 5 years to $1125. If the sum of
how
old
is
each
now ?
at
invested $ 5000. How 6 %. and 4 %. Ten years ago the sum of their ages was 90.
19. partly at 5% and partly at 4%.
What was
the
sum and
rates
est
The sums of $1500 and $2000 are invested at different and their annual interest is $ 190.
and
money and
17. Three cubic centimeters of gold and two cubic centimeters of silver weigh together 78 grains.
.000
is
partly invested at
6%.146
ELEMENTS OF ALGEBRA
11. Find the weight of one cubic centimeter of gold and one cubic centimeter of silver. the rate of interest?
18. Find their present ages. and B's age is \ the sum of A's and C's ages. the rate of interest ?
What was
the
sum
of
A sum
of
money
at simple interest
amounted
in 2 years
to $090. 5 %.
much money
is
invested at
A sum
of
money
at simple interest
amounted
in 6 years
to $8000. What was the amount of each investment ?
15. If the rates of interwere exchanged. the annual interest would be $ 195.
andCL4 = 8. receiving $ 100 for each horse.
Find their
rates of walking.
is
the center of the circum-
scribed circle. and F. and GE = CF. and F '(see diagram).
the three sides of a triangle E.
points.
25. ED = BE. If one angle exceeds the sum of the other two by 20. three
AD = AF. A farmer sold a number of horses. and AC = 5 inches. and sheep.SIMULTANEOUS LINEAR EQUATIONS
147
20.
An C touch ing the sides in D. and F. and CF?
is
a circle
inscribed in the
7<7. The number of sheep was twice the number of horses and cows together. what is
that
=
OF.
23.
. then AD = AF. If angle ABC = GO angle BAG = 50. c. what are the angles of the triangle ?
22. are taken so
ABC.
the length of
NOTE. and their difference by GO . BD = HE. and angle BCA = 70.
1
NOTE. BC = 7 inches.
E.
In the annexed diagram angle a = angle b. and e.
Find the parts of the
ABC touching the three sides if AB = 9. cows. B find angles a. angle c = angle d. but if A would double his pace. for $ 740.
respectively. and angle e angle/. and CE If AB = G inches.
triangle
Tf
AD. and $15 for each sheep.
24. The sum of the 3 angles of a triangle is 180. BE. $ 50 for each cow.
On
/). he would walk it in two hours less than
than
to travel
B
B.
BC=7.
A
r
^
A
circle is inscribed in triangle
sides in D. How many did he sell
of each if the total
number
of animals
was 24?
21.
It takes
A two hours
longer
24 miles.
.
CHAPTER
XII*
GRAPHIC REPRESENTATION OF FUNCTIONS AND
EQUATIONS
184. jr.
PN. and PJ/_L XX'. B. and point the origin.
two fixed straight lines XX' and YY' meet in at right angles.
.
2).
?/. and r or its equal OA is
. Thus the points A.
first
3). then
the position of point is determined if the lengths
of
P
P3f and
185. hence
The
coordinates lying in opposite directions are negative. and PN _L YY'. and
respectively represented
Dare
and
by
(3 7 4).
The
abscissa
is
usually denoted by
line XX' is called the jr-axis.. and ordinates abore the x-axis are considered positive .
It'
Location of a point.
is
The point whose abscissa is a. Abscissas measured to the riyht of the origin. (7.
186. YY' they-axis.
-3).
(3.
* This chapter
may
be omitted on a
148
reading.
(2.
The
of
Coordinates. the ordinate by ?/.
PN are given.
lines
PM
the
and P^V are
coordinates
called
point P.
the ordinate of point P. and whose ordinate is usually denoted by (X ?/).
or its equal
OM.
is the abscissa. PM.
(2.
1). 0). 1).
.
4.(!. Graphic constructions are greatly facilitated by the use of cross-section paper.
What
Draw
is
the distance of the point
(3. -!).
=3?
is
If a point lies in the avaxis.
What
are the coordinates of the origin ? If
187.
Plot the points: (-4. 3). (4.
8.
(-1.
3. -2).e.
(-3.
Plot the points:
(4.
Where do Where do
Where do
all
points
lie
whose ordinates
tfqual
4?
9.
4)
and
(4.
2J-).
Plot the points
:
(0.1).
(4.2).
Draw
the triangle whose vertices are respectively
(-l.
4)
from the
origin ?
7.
4).
and measure
their
distance.4). -3).
2. -4).
12.
71
2).
(-5.
the quadrilateral whose vertices are respectively
(4.
Plot the points
(6. (-2.
What
is
the locus of
(a?. which of its coordinates
known ?
13.GRAPHIC REPRESENTATION OF FUNCTIONS
The
is
149
process of locating a point called plotting the point. (-4.
0). (See diagram on page 151.
-2). (4.
6.3).
all all
points
points
lie
lie
whose abscissas equal zero ?
whose ordinates equal zero?
y) if y
10. (4.
whose coordinates are given
NOTE.
Graphs. i. 3).)
EXERCISE
1. paper ruled with two sets of equidistant and parallel linos intersecting at right angles.and(l. (0. the mutual dependence of the two quantities may be represented
either by a table or
by a diagram. 11.
. 0).
6. (0.
two variable quantities are so related that
changes of the one bring about definite changes of the other. 0). (-4.
.
ABCN
y
the so-called graph of
To
15
find
from the diagram the temperature on June
to be 15
. we obtain an uninterrupted sequence
etc.
By representing
of points.
1.
Thus the average temperature on May
on April 20. but it indicates in a given space a great many more
facts than a table.
may be found
on Jan.
188.
representation does not allow the same accuracy of results as a numerical table.. C.
ically
each representing a temperature at a certain date. in like manner the average temperatures for every value of the time. 10
. may be represented graphby making each number in one column the abscissa.
from January 1 to December 1.150
ELEMENTS OF ALGEBRA
tables represent the average temperature
Thus the following
of
New
volumes
1
Y'ork City of a certain
to 8 pounds.
we meas1
.
15. or the curved line the temperature.
ure the ordinate of F. however.
A graphic
and
it
impresses upon the eye
all
the peculiarities of
the changes better and quicker than any numerical compilations. B. and the corresponding number in the adjacent column the ordinate of a point. Thus the first table produces 12 points. A. D. and the amount of gas subjected to pressures from
pound
The same data.
The engineer.GRAPHIC REPRESENTATION OF FUNCTIONS
151
i55$5St5SS 3{utt|s33<0za3
Graphs are possibly the most widely used devices of applied matheThe scientist uses them to compile the data found from experiments. the merchant. (b) July 15. uses them. (c) January 15.
.
physician. Daily papers represent ecpnoniical facts graphically. (d) November 20. and to deduce general laws therefrom. the rise and fall of wages. etc.
EXERCISE
From the diagram
questions
1.
:
72
find approximate answers to the following
Determine the average temperature of New York City on (a) May 1. the graph
is
applied. as the prices and production of commodities. Whenever a clear. the
matics. concise representation of a
number
of numerical data
is
required.
During what month does the temperature decrease most
rapidly ?
13. from what date to what date would it extend ?
If
. (freezing
point) ?
7. 1 to Oct.152
2.
June
July
During what month does the temperature increase most
?
rapidly
12.
When
the average temperature below
C.
on
1 to
the
average.
ELEMENTS OF ALGEKRA
At what date
(a) G
or dates
is
New York
is
C..
1 ?
does
the
temperature
increase from
11..
is
ture
we would denote the time during which the temperaabove the yearly average of 11 as the warm season.
How
much.
From what
date to what date does the temperature
increase (on the average)?
8.
is
10.
During what month does the temperature change least?
14.
Which month
is
is
the coldest of the year?
Which month
the hottest of the year?
16. 1?
11
0. (c)
the average temperature oi 1 C. At what date is the average temperature lowest? the lowest average temperature ?
5.
When
What
is
the temperature equal to the yearly average of
the average temperature from Sept. (1)
10
C.
How much warmer
1 ?
on the average
is it
on July 1 than
on
May
17. ?
-
3.
15.?
is
is
the average temperature of
New York
6. At what date is the average temperature highest the highest average temperature?
?
What What
is
4.
During what months
above 18 C.. (d) 9 0. ?
9.
20.
19.
Hour
Temperature
.
Represent graphically the populations
:
(in
hundred thou-
sands) of the following states
22. transformation of meters into yards.GRAPHIC REPRESENTATION OF FUNCTIONS
18. in a similar manner as the temperature graph was applied in examples 1-18.
Draw
a
graph for the
23. Construct a diagram containing the graphs of the mean temperatures of the following three cities (in degrees Fahren-
heit)
:
21.
Draw
.09 yards. One meter equals 1.
NOTE.
a temperature chart of a patient.
153
1?
When is the average temperature the same as on April
Use the graphs of the following examples for the solution of concrete numerical examples. From the table on page 150 draw a graph representing the volumes of a certain body of gas under varying pressures.
x
7 to 9.154
24.
190. if 1 cubic centimeter of iron weighs 7.
29. 9.
2.
.
+7
If
will
respec-
assume the values 7. amount to $8. 2 .
Show
graphically the cost of the
REPRESENTATION OF FUNCTIONS OF ONE VARIABLE
189.
26. e.
(Assume ir~
all circles
>2
2
.. 3.50.
4.
x*
x
19.
ELEMENTS OF ALGEBRA
If
C
2
is
the circumference of a circle whose radius
is J2.5
grams. 1 to 1200 copies.
3.
to 20 Represent graphically the weight of iron from cubic centimeters. the daily average expenses for rent. if he sells 0. binding.)
T
circumferences of
25. Represent graphically the cost of butter from 5 pounds if 1 pound cost $. Represent graphically the distances traveled by a train in 3 hours at a rate of 20 miles per hour. represent his daily gain (or loss). 2 x -f 7 gradually from 1 to 2.50. if each copy sells for $1.
x increases will change gradually from
13.
then
C
irJl.inch.) On the same diagram represent the selling price of the books. An expression involving one or several letters a function of these letters.
If
dealer in bicycles gains $2 on every wheel he sells.g.
28. gas. and $.
A
10 wheels a day.
books from
for printing..
to
27. etc.
from
R
Represent graphically the = to R = 8 inches.
function
If the value of a quantity changes. etc. 2 8 y' + 3 y is a function of x and
y. if x assumes
successively the
tively
values
1..
2
is
called
x
2 xy
+ 7 is a function of x. the value of a of this quantity will change.50 per copy
(Let 100 copies = about \.
The
initial cost of
cost of manufacturing a certain book consists of the $800 for making the plates.
2. The values of func192. for x=l.
9).
Q-.
Draw the graph of x2 -f. and (3.
values of x2
nates are the corresponding i.
may. 4).1).
it is
In the example of the preceding article.
155
-A
variable is a quantity
whose value changes in the
same
discussion.
etc.
2).1). be also represented by a graph. however. while 7 is a constant.
and join
the
points in order.
is
supposed to change.
(-
2.
9). (1.g.
Ex. plot points which
lie
between those constructed above. E.
3
(0.
To
obtain the values of the functions for the various values of
the
following arrangement
be found convenient
:
.
(1^.
1
the points (-3. x a variable.
. Thus the table on page 1G4 gives the values of the functions x 2 x3 and Vsr. hence
various values of x
The values of a function for the be given in the form of a numerical table.
-J).
may
. to con struct the graph x of x 2 construct a series of -3 points whose abscissas rep2 resent X) and whose ordi1
tions
.
a*.
2
(-1. (2.e. If a more exact diagram
is
required.
is
A
constant
a quantity whose value does not change in the
same discussion. 3 50. 4). as
1.0).
Graph
of a function.GRAPHIC REPRESENTATION OF FUNCTIONS
191. to
x = 4.2 x
may
4 from x
=
4. construct
'.
(-3..
Draw
y
z x
the graph of
= 2x-3. (4.
2.
r
*/
+*
01
.-.
Ex. and join(0.
.
(To avoid
very large ordinatcs.. the scale unit of the ordinatcs is taken smaller than that of the x. the function
is
frequently represented
by a single letter. (-2.4).. Thus in the above example. or ax + b -f c are funclirst
tions of the first degree.)
For brevity.
A
Y'
function of the
first
degree is an integral
rational function
involving only
the
power of the variable. = 4. 5). and joining in order produces
the graph
ABC. Thus 4x + 7.
4J.
rf
71
. y = 6. 4).
if
/*
4
>
1i >
>
?/
=
193.2 x
.
= 0.
It can be
proved that the
graph is a straight
of a function of the first degree
line.156
ELEMENTS OF ALGEBRA
Locating the points(
4. -1). straight line produces the required graph.
7
.
194. etc. 2 4 and if y = x -f. as y.. hence two points are sufficient for the construction
of these graphs. j/=-3.20).
If
If
Locating
ing
by a
3) and (4.
what values of x make the function x2 + 2x 4 = (see 192). i.
A body
moving with a uniform
t
velocity of 3 yards per
second moves in
this
seconds a distance d
=3
1. If two variables x and y are directly proportional..
.) scale are
expressed in
degrees of the Centigrade (C.
to Fahrenheit readings
:
Change
10
C. then
y = . 14 F.158
24..
y=
formula graphically..
that the graph of two variables that are directly proportional is a straight line passing through the origin (assume
for c
27.
25. then
cXj
where
c is a constant.
If
two variables x and y are inversely proportional.) scale by the formula
(a)
Draw
the graph of
C = f (F-32)
from
to
(b)
4 F F=l.
1
C.where x
c is
a constant.
that
graph with the o>axis.. we have to measure the abscissas of the intersection of the
195.24.24 or x =
P and
Q.
ELEMENTS OF ALGEBRA
Degrees of the Fahrenheit
(F. it is evidently possible Thus to find to find graphically the real roots of an equation. 9 F..
C. Therefore x = 1. Represent 26.
GRAPHIC SOLUTION OF EQUATIONS INVOLVING ONE
UNKNOWN QUANTITY
Since we can graphically determine the values of x make a function of x equal to zero. the abscissas of 3. 32 F.
Show
any convenient number).
From
grade equal to
(c)
the diagram find the number of degrees of centi-1 F.
if c
Draw
the locus of this equation
= 12.e.
3x
_
4
.
i. because their graphs are straight lines.
and joining by a straight
line.
produces the
7*
required locus.
unknown
quantities. that can be reduced
Thus
to represent
x
-
-
-L^-
\
x
=2
-
graphically.
fc
= 3.
?/
=4
AB.
4) and
them by
straight line
AB
(3.
if
y
=
is
0. 2).e.
If the given equation is of the we can usually locate two
y.
. locate points
(0.
Hence.
Ex.
Thus
If
in
points without solving the equation for the preceding example:
3x
s
.
4)
and
(2. y = -l.160
ELEMENTS OF ALGEBRA
GRAPHIC SOLUTION OF EQUATIONS INVOLVING TWO UNKNOWN QUANTITIES
198.1.
(f
.
y=
A
and construct
x
(
-
graphically.
1)
and
0).
Hence we may
join (0.
X'-2
Locating the points
(2.
first
degree.
If
x
=
0.
199. 0). and join the required graph.
Hence
if
if
x
x
-
2.
y y
2.
T
.
Graph
of
equations involving two
unknown
quantities.2 y ~ 2. solve for
?/.
represent graphically equations of the form y function of x ( 1D2). we can construct the graph or locus of any
Since
we can
=
equation involving two
to the above form. = 0.2.
?/. Represent graphically
Solving for
y ='-"JJ y.
Ex. Equations of the first degree are called linear equations. Draw the locus of 4 x + 3 y = 12.
== 2.
NOTE.
linear equations have only one pair of roots.
(2)
. P.
Since two straight lines which are not coincident nor simultaneous
Ex.
equation
x=
By measuring
3.1=0.
Graphical solution of a linear system.
AB
but only one point
in
AB
also satisfies
(2).
The
every
coordinates
of
point in satisfy the equation
(1).
To
find the roots of
the system. and every set of real values of x and y satisfying the given equation is represented by a point in
the locus.
203.
Solve graphically the equations
:
(1)
\x-y-\.
and CD. The roots of two simultaneous equations are represented by the coordinates of the point (or points) at which their
graphs intersect.
202.
3.
201.
parallel have only one point of intersection.
By
the
method
of
the preceding article construct the graphs
AB
and
and
CD
of
(1)
(2) respectively.GRAPHIC REPRESENTATION OF FUNCTIONS
161
200.57. we obtain the roots. The coordinates of every point of the graph satisfy the given equation. viz.15.
AB
y
= . the point of intersection of the coordinate of P.
.
obtain the graph (a circle)
AB C
joining.
4. 0) and (0.
3x
2 y = -6.
intersection. AB the locus of (1). 5.
There can be no point of
and hence no
roots.
-
4.
3. 1.y~ Therefore. 0.
4. e. the graph
of
points
roots.
y equals
3. parallel graphs indicate inconsistent equations.9. 0. (4.
. 3.
Using the method of the preceding para. there are two pairs of By measuring the coordinates of
:
P and Q we find
204. 2.
4.0).
Measuring the coordinates
of P.
P
graphs meet in two and $.
V25
5.
In general.
3.
Inconsistent equations.
Solve graphically the
:
fol-
lowing system
= =
25. etc. i. we of the
+
y*
= 25.e.
4.162
ELEMENTS OF ALGEBRA
graph. construct CD the locus of (2)
of intersection.
2 equation x
3).0. the point
we
obtain
Ex.
1. and
.
x2
.5. and joining by a
straight line. 4.
(1)
(2)
-C. (-2.g.
4. they are inconsistent.
5.5.
Locating the
points
(5. Since the two
-
we obtain DE.
4.
and
+ 3).
Solving (1) for y.
Locating two points of equation (2). 3). This is clearly shown by the graphs of (1) arid (2).
2. which consist of a
pair of parallel lines.
(1)
(2)
cannot be satisfied by the same values of x and y.
The equations
2
4
= 0. (-4.
4. = 0. if x equals
respectively
0.
215.
Every odd root of a quantity has
same sign as
and
2
the
quantity.
109
.
for (-f 3) 2
(
3)
equal
0. for (+ a) = a \/32 = 2.
27
=y
means
r'
=
27.
tity
.
Thus
V^I is an imaginary number.
2. Since even powers can never be negative.
a)
4
= a4
.CHAPTER XIV
EVOLUTION
213. for distinction. or y
~
3.
Evolution
it is
is the operation of finding a root of a quan the inverse of involution. called real
numbers.
numbers. and ( v/o* = a.
or x
&4 .
= x means
= 6-.
4
4
. and
all
other numbers are.
It follows
from the law of signs
in evolution that
:
Any
even root of a positive.
V9 = +
3. it is evidently impossible to express an even root of a negative quantity by Such roots are called imaginary the usual system of numbers.
or
-3
for
(usually written
3)
.
\/"^27=-3. (_3) = -27. etc.
1. which can be simplified no further. quantity
may
the
be either 2wsitive
or negative.
\/a
=
x means x n
=
y
?>
a.
V
\/P
214.
12.
2 2
218. a -f. however.172
7.b 2 2 to its square.2 ab + b
.
ELEMENTS OF ALGEBEA
4a2 -44a?> + 121V2 4a
s
.2 &c.
second term 2ab by the double of
by dividing the the so-called trial divisor.
+ 6 + 4a&. i. 11.
2
49a 8 16 a 4
9. it is not known whether the given
expression is a perfect square.
The work may be arranged
2
:
a 2 + 2 ab
+ W \a + b
.
2
.
. In order to find a general method for extracting the square root of a polynomial.e.
10.
14.
a-\-b
is
the root
if
In most cases.
and
b.
the given expression is a perfect square.
mV-14m??2)-f 49.2 ac . the that 2 ab -f b 2
=
we have then to consider sum of trial divisor 2 a.
multiplied by b must give the last two terms of the
as follows
square.
a2
+ & + c + 2 a& .
15.
#2
a2
-
16.
term a of the root
is
the square root of the
first
The second term
of the root can be obtained
a. let us consider the relation of a -f.
2ab
.
8
. and b (2 a -f b).72 aW + 81 &
4
.
The
term
a'
first
2
.>
13.
\
24 a 3
4-f
a2
10 a 2
Second remainder.
Extract the square root of
16 a 4
.
.
The square
.
is
As
there
is
no remainder.
1.
First complete divisor.
. 8 /-. 1.
-
24 a
3
+
25 a 2
-
12 a
+4
Square of 4 a First remainder.
. the first term of the answer. 2 Subtracting the square of 4x' from the trinomial gives the remainder '24 x'2 + y.
.
Ex. 6 a. we obtain the next term of the root 3 y 3 which has to be added to 2 the trial divisor.
219.
.
double of this term
find the next
is
the
new
trial divisor.
and so
forth. Second trial divisor.24 afy* -f 9 tf.
First trial divisor.
Arranging according to descending powers of
10 a
4
a.24 a + 4 -12 a + 25 a8
s
.EVOLUTION
Ex.
*/''
. 8 a 2 .
Explanation. and consider Hence the their sum one term. the required root
(4
a'2
8a
+
2}.
by division we
term of the
root. 8 a 2
2. As there is no remainder. By doubling 4x'2 we obtain 8x2 the trial divisor. Multiply the complete divisor Sx' 3y 3 by Sy 8 and subtract the product from the remainder.
10 a 4
8
a. We find the first two terms of the root by the method used in Ex.
Arrange the expression according to descending powers root of 10 x 4 is 4 # 2 the lirst term of the root.
4 x2
3
?/
8 is
the required square foot. The process of the preceding article can be extended to polynomials of more than three terms.
of x. 24# 2 y 3 by the trial divisor Dividing the first term of the remainder.
2. 8 a 2
-
12 a
+4
a
-f 2.
173
x*
Extract the square root of 1G
16x4
10 x*
__
. 8 a 2 Second complete divisor.
etc. Hence if we divide the digits of the number into groups. the integral part of the square root of a number less than 100 has one figure.
Find the square root of 7744.
the
consists of
group is the first digit in the root. the first of which is 8.. two figures. of 1. and we may apply the method used in algebraic process. the square root of 7744 equals 88. etc.1344. which may contain one or two).EVOLUTION
220.176.
As
8
x 168
=
1344. of a number between 100 and 10. the first of which is 4.
first
.000.
square root of arithmetical numbers can be found to the one used for algebraic
Since the square root of 100 is 10.
the preceding explanation it follows that the root has two digits. Therefore 6 = 8. and the complete divisor
168.000 is 1000.
175
The
by a method very similar
expressions.
2.
Ex.
a
f>2'41 '70
6
c
[700
+ 20 + 4 = 724
2 a
a2 = +6=
41)
00 00
1400
+ 20 = 1420
4
341 76
28400
=
1444
57 76
6776
.
Ex. Hence the root is 80 plus an unknown number. then the number of groups is equal to the number of digits in the square root. Thus the square root of 96'04' two digits.
From
A
will
show the
comparison of the algebraical and arithmetical method given below identity of the methods.
7744 80 6400
1
+8
160
+ 8 = 168
1344
1344
Since a
2 a
Explanation.
The
is
trial divisor
=
160.000.
a 2 = 6400. beginning at the
and each group contains two digits (except the last. and the first remainder is. the first of which is 9 the square root of 21'06'81 has three digits.000 is 100.
1.
= 80. and the square root of the greatest square in
units. of 10.
Find the square root of 524.
70
6.1T6
221.
The groups
of 16724.7 to three decimal places.
places.
we must
Thus the groups
1'67'24.10.688
4
45 2 70
2 25
508
4064
6168 41)600
41344
2256
222.
3.
Find the square root of
6/.
in . and if the righthand group contains only one digit.
12.
Roots of common fractions are extracted either by divid-
ing the root of the numerator by the root of the denominator.GO'61. annex a cipher.
EXERCISE
Extract the square roots of
:
82
.1 are
Ex. or by transforming the common fraction into a decimal.
ELEMENTS OF ALGEKRA
In marking
off groups in a number which has decimal begin at the decimal point.0961
are
'.
then
Since such a triangle
tangle.
29.
and their product
:
150.
r. 25.
'
4.
2
:
3.
=
a
2
2
(' 2
solve for solve for
= Trr
.
28.
If 22
= ~^-. Find the side of each field.
27.
If a 2 4.
is
one of
_____
b
The side right angle.
solve for d. 24.
A
right triangle is a triangle. opposite the right angle is called the hypotenuse (c in the diagram).
. and the two other sides respectively
c
2
contains
c
a and b units.
Find the numbers.)
of their squares
5.
Three numbers are to each other as 1 Find the numbers.
EXERCISE
1.b 2 If s
If
=c
.
2a
-f-
1
23. and the first exceeds the second by 405 square yards.
4.
If 2
-f 2 b*
= 4w
2
-f c
sol ve for
m.
A
number multiplied by
ratio of
its fifth
part equals 45.
2
.
solve for
r. is 5(5.
The
two numbers
(See
is
2
:
3.
:
6.
and the sum
The
sides of
two square
fields are as
3
:
5.
If s
= 4 Trr
'
2
.
solve for v.
108.
and they con-
tain together 30G square feet.
If
G=m m
g
.
Find the side
of each field.
.
84
is
Find a positive number which
equal to
its
reciprocal
(
144).
find a in terms of 6
.
22
a.
Find
is
the number.
2
.
9
&
-{-
c#
a
x
+a
and
c.
may
be considered one half of a
rec-
square units.
3.
26. If the hypotenuse
whose angles
a
units of length.
228. its area contains
=a
2
-f-
b2
.180
on
__!_:L
ELEMENTS OF ALGEBRA
a. The sides of two square fields are as 7 2.
2.
)
COMPLETE QUADRATIC EQUATIONS
229.2
7
. in how many seconds will a body fall (a) G4 feet. its surface
(Assume
ir
=
2 .
Find the
sides.
Two
circles together contain
:
3850 square
feet.
member can be made a complete square by adding 7 x with another term.
Method
of completing the
square. passes in t seconds 2 over a space s yt Assuming g 32 feet.QUADRATIC EQUATIONS
7. (b) 100 feet?
=
.
sides.
7r
(Assume
and their
=
2 7
2
.
add
(|)
Hence
2
.
.
radii are as 3
14.
make x2
Evidently 7 takes the place 7x a complete square
to
to
which corresponds
m
2
.
-J-
=
12.
.
24.
.
of a right triangle Find these sides.
Find the
radii.
Solve
Transposing.
2m. let us compare x 2
The
left
the perfect square x2
2
mx -f m
to
2
.
The following
ex-
ample
illustrates the
method
or
of solving a complete quadratic
equation by completing the square.
The area
:
sides are as 3
4.
the formula
= Trr
whose radius equals r is found by Find the radius of circle whose area S
equals (a) 154 square inches. To find this term.
The area $
/S
of a circle
2
.
The hypotenuse
of a right triangle is 2.7 x -f 10 = 0. and the
other two sides are as 3
4.
4.
is
and the other
two
sides are equal.
8. Find the unknown sides and the area.
and the two smaller
11. x* 7 x=
10. we have
of
or
m = |.
Find these
10.
8 = 4 wr2 Find 440 square yards. the radius of a sphere whose surface equals
If the radius of a sphere is r. The hypotenuse of a right triangle is to one side as 13:12. A body falling from a state of rest.
181
The hypotenuse
of a right triangle
:
is
35 inches. and the third side is 15 inches.)
13. (b) 44 square feet.
9.
o^
or
-}-
3 ax == 4 a9
7 wr
.
ao.
x
la
48.
Solving this equation we obtain
by the method of the preceding
2a
The
roots of
substituting the values of a. -\-bx-\.
Solution
by formula.
article.
231. and c in the general answer.
2x
3
4.
=0.
=8
r/io?.
49.
2
Every quadratic equation can be
reduced to the general form.
.184
ELEMENTS OF ALGEBRA
45
46.
= 12.
any quadratic equation may be obtained by 6.c
= 0.
Find the number.
The sum
of the squares of
two consecutive numbers
85.
Find a number which exceeds
its
square by
is
-|.
Find
the numbers.
56.
2.
-2.0.
5. and consequently many prob-
235.
-2. and the difference Find the numbers.1.
.
55.
57.
Problems involving quadratics have
lems of this type have only one solution. Find the number. but frequently the conditions of the problem exclude negative or fractional answers.
is
Find two numbers whose product
288. -5. Find the sides.3. -2.9. feet.
Divide CO into two parts whose product
is 875.
-2.
Find two numbers whose difference
is 40.
PROBLEMS INVOLVING QUADRATICS
in general two answers.3.
What
are the
numbers
of
?
is
The product
two consecutive numbers
210.
and whose sum
is
is 36.
88
its reciprocal
A
number increased by three times
equals
6J.0.
its
sides of a rectangle differ by 9 inches.
number by 10.
1.
The
difference of
|.
58.
two numbers is 4.
Twenty-nine times a number exceeds the square of the 190.
6.
189
the equations whose roots are
53.
8.
and whose
product
9.2.
-4.
52.
3.3.
of their reciprocals is
4.QUADRATIC EQUATIONS
Form
51.
0.
EXERCISE
1.
area
A
a perimeter of 380
rectangular field has an area of 8400 square feet and Find the dimensions of the field.
1.
:
3.
3. and equals 190 square inches. The
11.
7.
G.
2.
54.
he would have received 12 apples less for the same money.10. it would have needed two hours less to travel 120 miles.
14. At what rates do
the steamers travel ?
18. Find the rate
of the train. What did he pay for each
apple ?
A man bought a certain number of horses for $1200.
watch for $ 24. he would have received two horses more for the same money.190
12.
c equals 221
Find
AB and AD. a distance One steamer travels half a mile faster than the two hours less on the journey.
If he
each horse ?
. Two vessels.
as the
16. sold a horse for $144. had paid $ 20 less for each horse.
other. and lost as many per cent Find the cost of the watch.
A man
cent as the horse cost dollars. What did he pay for
21. If a train had traveled 10 miles an hour faster. exceeds its widtK AD by 119 feet. dollars.
ELEMENTS OF ALGEBRA
The length
1
B
AB of a rectangle.
. he had paid 2 ^ more for each apple.
and gained as many per Find the cost of the horse.
watch cost
sold a watch for $ 21. start together on voyages of 1152 and 720 miles respectively.
A man
A man
sold a
as the watch cost dollars. ABCD.
13.
15.
ply between the same two ports.
Two steamers
and
is
of 420 miles. and the slower reaches its destination one day
before the other.
19.
The diagonal
:
tangle as 5 4. and lost as many per cent Find the cost of the watch. and the line BD joining
two opposite
vertices (called "diagonal")
feet. one of which sails two miles per hour faster than the other.
vessel sail ?
How many
miles per hour did the faster
If 20. and Find the sides of the rectangle. A man bought a certain number of apples for $ 2.
of a rectangle is to the length of the recthe area of the figure is 96 square inches.
17.
and the area of the path
the radius of the basin. the two men can do it in 3 days.
24.
Ex.I) -4(aj*-l)
2
= 9.
. so that the rectangle. as
0.
1.
of the area of the basin.
^-3^ = 7.
Find the side of an equilateral triangle whose altitude
equals 3 inches. The number of eggs which can be bought for $ 1 is equal to the number of cents which 4 eggs cost. If the area of the walk is equal to the area of the plot. and the unknown factor of one of these terms is the square of the unknown factor of the
other.
By formula.
is
On the prolongation of a line AC. Equations in the quadratic form can be solved by the methods used for quadratics. how wide is the walk ?
23.
237. a point taken.
or x
= \/l = 1. is surrounded by a walk of uniform width. constructed with and CB as sides. and working together.
27.QUADRATIC EQUATIONS
22.
Find
TT r (Area of a circle .
Solve
^-9^ + 8 =
**
0.) 25.
EQUATIONS IN THE QUADRATIC FORM An equation is said to be in the quadratic form
if it
contains only two unknown terms. 30 feet long and 20 feet wide.
B
AB
AB
-2
191
grass plot.
A rectangular
A
circular basin is surrounded
is
-
by a path 5
feet wide. A needs 8 days more than B to do a certain piece of work. In how many days can B do the work ?
=
26. Find and CB. How many eggs can be bought for $ 1 ?
236. contains B 78 square inches.
=9
Therefore
x
=
\/8
= 2.
(tf. 23 inches long.
Then the law
of involution. such as 2*."
means "is greater than"
195
similarly
means "is
.
for all values
1
of
m and n. ~ a m -f.
we may choose
for such
symbols any definition that
is
con-
venient for other work. (a ) s=a m = aw bm
a
.
We assume.
= a""
<
.
(ab)
. (a m ) w
. very important that all exponents should be governed by the same laws. 4~ 3 have meaning according to the original definition of power. hence.
m
IV. and
.
a m a" = a m+t1 . The following four fundamental laws for positive integral exponents have been developed in preceding chapters
:
I.
244. we let these quantities be what they must be if the exponent law of multiplication is generally true. instead of giving a formal definition of fractional and negative exponents.
the direct consequence of the defiand third are consequences
FRACTIONAL AND NEGATIVE EXPONENTS
243.*
III.
II.
no
Fractional and negative exponents. that a
an
= a m+n
.
>
m therefore.CHAPTER XVI
THE THEORY OF EXPONENTS
242.
It is.a" = a m n
mn . while the second of the first.
provided
w > n. however.
must be
*The symbol
smaller than.
The
first
of these laws
is
nition of power.
0?=-^.
28.
(bed)*. ml.
= a.
'&M
A
27.
a\
26.
Hence
Or
Therefore
Similarly.
30.
a*.
24.
.
To
find the
meaning
of
a fractional exponent.g.196
ELEMENTS OF ALGEBRA
true for positive integral values of n.
31. 3*.
e.
a?*.
-
we
find
a?
Hence we
define a* to be the qth root of of. at.
Write the following expressions as radicals :
22. (xy$.
laws. since the raising to a positive integral power is only a repeated multiplication.
Let
x
is
The operation which makes the fractional exponent disappear evidently the raising of both members to the third power.
disappear. 25. fractional. or zero exponent
equal
x. a .
n 2 a.
Assuming these two
8*.
29.
m$.
etc. 4~ .
23.
^=(a^)
3*
3
. as.
245.
we
try to discover the
let the
meaning of
In every case we
unknown quantity
and apply to both members of the equation that operation which makes the negative.
2
=
a2
.
ELEMENTS OF ALGEBRA
To
find the
meaning
of a negative exponent.
etc.
a
a
a
= =
a a a
a1
1
a.
an x = a.
Multiplying both members by
a". e.
a8 a
2
=
1
1
.
Let
x=
or". consider the following equations.
248.
.
each
is
The
fact that a
if
=
we
It loses its singularity
1 sometimes appears peculiar to beginners.198
247.
Or
a"#
= l. in which
obtained from the preceding one by dividing both
members by
a.
vice versa.g.
by changing the sign of
NOTE. Factors
may
be transferred
from
the
numerator
to
the
denominator of a fraction. or
the exponent.
cr n.
6
35.202
ELEMENTS OF ALGEBRA
32.
lix
=
2x-l
=+1
Ex.
powers of x arranged are
:
Ex.
1.
34.
1 Multiply 3 or
+x
5 by 2 x
x.2 d
.
Arrange in descending powers of
Check.
Divide
by
^
2a
3 qfo
4.
V ra
4/
3
-\/m
33. the term which does not contain x may be considered as a term containing #.
we wish to arrange terms according to descending we have to remember that.
If
powers of
a?.
40.
1. The
252.
2.
(3V3-2Vo)(2V3+V5).
V3 .
-v/a
-
DIVISION OF RADICALS
267.
Ex.
is
1
2.
(V50-f 3Vl2)-4-V2==
however.
Va
-v/a.
it
more convenient to multiply dividend and divisor by a factor which makes the divisor rational.
a fraction.
(5V2+V10)(2V5-1).
60.
.
E.
Ex. Monomial surdn of the same order may be divided by multiplying the quotient of the coefficients by the quotient of the
surd factors.
53. the quotient of the surds
is
If.
268.214
42.
44.
51.
46.y.
43.
ELEMENTS OF ALGEHRA
(3V5-5V3)
S
.
47.
(5V7-2V2)(2VT-7V2).
48.
49.
all
monomial surds may be divided by
method.
(3V5-2V3)(2V3-V3). a
VS
-f-
a?Vy
= -\/ -
x*y
this
Since surds of different orders can be reduced to surds of
the same order.
52.
(2
45.V5) ( V3 + 2 VS).
Divide 4 v^a by
is
rationalizing factor
evidently \/Tb
hence.by the usual arithmetical method. e.57735.
we have
to multiply
In order to make the divisor (V?) rational. Evidently.
. is illustrated
by
Ex.
1.73205. the rationalizing factor
x
' g
\/2.g.
4\/3~a'
36
Ex. arithTo find. we have
V3
But
if
1.
by V7.
Divide
VII by v7.
is
Since \/8
12 Vil
=
2 V*2.RADICALS
This method.
3. To show that expressions with rational denominators are simpler than those with irrational denominators.
The
2.
+ 4\/5 _ 12v 3 + 4\/5 V8 V8
V2 V2
269.
Divide 12 V5
+ 4V5 by V.
/~
}
Ex.
Hence
in arithmetical
work
it
is
always best to
rationalize the denominators before dividing. the by 3 is much easier to perform than the division by
1.
VTL_Vll '
~~"
\/7_V77
.
. called rationalizing the
the following examples
:
215
divisor.
..73205
we
simplify
JL-V^l
V3
*>
^>
division
Either quotient equals . metical problems afford the best illustrations. however.
The
difference of
two even powers should always be
considered as a difference of two squares.
ELEMENTS OF ALGEBRA
positive integer.
and have
for
any positive integral value of
If n
is
odd.
It
y is
not divisible by
287.
Factor 27 a* -f
27 a 6
8. if n is even.g.
is
odd.
We may
6
n 6 either a difference of two squares or a
dif-
* The symbol
means " and so forth to.xy +/).
-
y
5
=
(x
-
can readily be seen that #n -f either x + y or x y.
:
importance.
2. xn y n y n y n = 0.
it
follows from the Factoi
xn y n is always divisible by x y."
.230
285. if n For ( y) n -f y n = 0.
Ex. If n is a Theorem that
1.
actual division
n.
Factor
consider
m
m
6
n9
.
By
we obtain the other
factors.
ar
+p=
z6
e. For substituting y for x.
xn -f.
286.
Two
special cases of the preceding propositions are of
viz.
2.y n is divisible by x -f ?/. if w is odd.
1.
x* -f-/
= (x +/)O .
2
Ex.
2 8 (3 a )
+8=
+
288.
creases.
306.
(1).
ToU"
^-100 a.
oo is
= QQ.
is satisfied
by any number. however
x approaches the value
be-
comes
infinitely large.
(1)
is
an
identity.
and becomes infinitely small.
customary to represent this result
by the equation ~
The symbol
304.i
solving
a problem
the result
or oo indicates that the
all
problem has no solution.
I.
= 10.increases
if
x
de-
x
creases.000
a.
the
If in an equation
terms containing
unknown quantity
cancel.
Or.
x
-f 2.
and
.e.can be
If
It is
made
larger than
number.x'2 2 x =
1. while the
remaining terms do not
cancelj the root is infinity.
Hence such an equation
identity. the answer is indeterminate.
1. or infinitesimal) This result is usually written
:
305.
The
~~f
fraction .
be the numbers.
TO^UU"
sufficiently small. equation.242
303. or that x may equal any finite number. without exception.
of the second exceeds the product of the first
Find three consecutive numbers such that the square and third by 1.
.
i.
(a:
Then
Simplifying.
ELEMENTS OF ALGEBRA
Interpretation of ?
e.e.
By making x
any * assigned
zero.
(1)
= 0.
as
+ l.
. it
is
an
Ex.
Interpretation of
QO
The
fraction
if
x
x
inis
infinitely large.decreases
X
if
called infinity.
1.
The
solution
x
=-
indicates that the problem
is indeter-
If all terms of an minate.g.
+
I)
2
x2
'
-f
2x
+
1
-x(x + 2)= . cancel.
Hence any number will satisfy equation the given problem is indeterminate.
great.
i.
Let
2.
Find the dimensions of the
field.
and the sum of
(
228. The volumes of two cubes differ by 98 cubic centimeters.
6.
the
The mean proportional between two numbers sum of their squares is 328. increased by the edge of the other.
rectangle is 360 square Find the lengths of the sides.
is 6. Find the edge of each cube.
255 and the sum of
5.
146 yards. Find the side of each square.
The hypotenuse
is
the other two sides
7.
Find the other two
sides.
103. and the edge of one.quals 20 feet.
9.
10. 12.
Find the
sides. the area becomes
-f%
of
the original area.)
The area
of a right triangle is 210 square feet. equals 4 inches. Find the numbers.
The area of a
nal 41 feet.
and
is
The area of a rectangle remains unaltered if its length increased by 20 inches while its breadth is diminished by 10 inches.
.)
53 yards.
and the
hypotenuse
is 37. Find these sides.
of a rectangular field
feet.
Find the
sides of the rectangle. and the edge of one exceeds the edge of the other by 2 centimeters. Find the edges.
ELEMENTS OF ALGEBRA
The
difference between
is
of their squares
325.
and the diago(Ex. To inclose a rectangular field 1225 square feet in area. But if the length is increased by 10 inches and
12. 148 feet of fence are required.244
3.
is
the breadth
diminished by 20 inches.
of a right triangle is 73. and
its
The diagonal
is
is
perimeter
11. The sum of the areas of two squares is 208 square feet.
14. 190.
two numbers Find the numbers. Find two numbers whose product whose squares is 514. p. Two cubes together contain 30| cubic inches.
8. and the side of one increased by the side of the other e.
13.
is
is
17 and the
sum
4.
Find the number.SIMULTANEOUS QUADRATIC EQUATIONS
15. irR *.) (Area of circle
and
=
1
16.
is
20 inches.
the quotient
is 2.)
17. (Surface of sphere
If a
number
of
two
digits be divided
its digits.
differ by 8 inches.
245
The sum of the radii of two circles is equal to 47 inches. Find the radii.
The
radii of
two spheres
is
difference of their surfaces
whose radius = 47T#2. their areas are together equal to the area of a circle whose radius is 37 inches.
and
if
the digits will be interchanged.
by the product of 27 be added to the number.
. and the equal to the surface of a sphere Find the radii.
CHAPTER XX
PROGRESSIONS
307. 3.. 19.) is a series. P..
17. (n 1) d must be added to a.
a. 2 d must be added to a.
+
2 d. P.11 246
(I)
Thus the 12th term of the
3
or 42. is derived from the preceding by the addition of a constant number.
The terms
ARITHMETIC PROGRESSION
308. to produce the nth term.
. each term of which. The progression is a.. 15 is 9 -f.7.
series 9.. .
-f
.. a + 2 d.
of the following series is
3. a
11. to produce the 4th term.
309. The first is an ascending. 16.1) d.
The common differences are respectively 4. the first
term a and
the
common difference d being given.
-4.
to each
term produces the next term. progression. the second a descending.
To
find the
nth term
/
of an A.
Hence
/
= a + (n .. and d. P. except the first.
10. a -f d.
a
+
d.
..
:
7.
of a series are its successive numbers. 12. to produce the 3d term..
The common
Thus each
difference is the
number which added
an A.. 3 d must be added to a..
Since d
is
a
-f
3
d. An arithmetic progression (A. a
3d.
11.
.
added to each term to obtain the next one.
to
A series
is
a succession of numbers formed according
some
fixed law.
2
sum
of the first 60
I
(II)
to find the
' '
odd numbers. -4^.
8.
the last term
and the common difference d being given. -7. 3.
. d
.
-|. 3.PROGRESSIONS
310.. P.. 5.
= a + (a
Reversing the order. 4. 6.
247
first
To
find the
sum s
19
of the first
n terms of an A. 8.4..
.
Or
Hence
Thus
from
(I)
= (+/). 5.
Which
(6)
(c)
of the following series are in A..
2
EXERCISE
1.
5.
Find the 5th term of the
4. 2J.
= -2.
6
we have
Hence
..
first
2
Write down the
(a)
(6)
(c)
6 terms of an A.
.-... 99) = 2600.' cZ == . 3.
if
a = 5. P..
7. -24. the
term
a..
of the series 10.
series 2. ?
(a) 1.
Find the 12th term of the
-4.
5..
= I + 49 = *({ +
. 7..
series
.
.3 a = -l..8.
9.
1.
Find the 10th term of the
series 17.
2.
1.
Adding.
= 99. -10. 21. .. 6.-
(a
+ + (a +
l)
l).
1-J..
.
3.
Find the 7th term of the Find the 21st term
series
. 19. 5.
-3..
(d) 1J. 2.
9.
Find the 101th term of the
series 1.
2*=(a + Z) + (a + l) + (a + l)
2s = n
*
. d = 3. 8.16..
Find the nth term of the
series 2.-.
6.. a = 2. P.
115.
-.
33.
3.
7.1 -f 3.
20.
22.
21.
Q^) How many times
in 12 hours ?
(&fi)
does a clock.
. 2J.
1.5
H + i-f
-f-
to 10 terms. 15.
to 8 terms.
rf.
12.
to 16 terms.
.
23.
'.
17. 1|. striking hours only.
:
3. and for each than for the preceding one.
to 15 terms.
2.
ELEMENTS OF ALGEBRA
last
term and the sum of the following series :
.
1.248
Find the
10.
Sum
the following series
14. and a yearly increase of $ 120. 11.
to 7 terms.
11. 7. 11.
4.
to 20 terms. 11.
. How much does he receive (a) in the 21st year (6) during the first 21 years ?
j
311.
+ 2-f-3 + 4 H
hlOO.
1+2+3+4H
Find the sum of the
first
n odd numbers. P.
(i)
(ii)
.
16. 7. hence if any three of them are given. the other two may be found by the solution of the simultaneous equations
.7 -f
to 12 terms.
6.
13. strike
for the first yard.
15.
8..
(x +"l) 4.
.
. 31.
15.
\-n.
19.
to 20 terms.
to 20 terms.
. 12.
to 10 terms. 29. In most problems relating to A.
$1
For boring a well 60 yards deep a contractor receives yard thereafter 10^ more How much does he receive all
together ?
^S5 A bookkeeper accepts a position at a yearly salary of $ 1000.
+ 3.
. Jive quantities are involved.
>
2-f
2.
18.(#
1
2) -f (x -f 3) H
to
a terms. 16.
1J.
17. Given a = |. = 45. = 83. = ^ 3 = 1.
produced. Find n.
man saved each month $2 more than in the pre 18. Find a Given a = 7. Find a and Given s = 44. How much did he save the first month?
19.
f
J 1 1
/
.
11. s == 440. s = 70.
14. Find?.3.
How much
.
6?
9. Given a = 1.
a x
-f-
b
and a
b. n = 4. n
has the series
^
j
. of 5 terms
6. 78.
10.
f?
.
T?
^. Find w.
y and #-f-5y.250
ELEMENTS OF ALGEBRA
EXERCISE
116
:
Find the arithmetic means between
1. = 17.
16.
has the series 82.
7. = 52.
12. ceding one. Find d and Given a = 1700.
15. Given a = .
Between 4 and 8
insert 3 terms (arithmetic
is
means)
so
that an A.
and
s. n = 17. Find d. n = 16.
3. n = 13.
I
Find
I
in terms of a. = 1870. 74.
a+
and
b
a
b
5. and all his savings in 5 years amounted to $ 6540.
. P. Given a = 4.
How many terms How many terms
Given d = 3.
Between 10 and 6
insert 7 arithmetic
means
.
= 16. n. Find d. n = 20.
13.
8. d = 5.
I.
A
$300
is
divided
among 6 persons
in such a
way
that each
person receives $ 10 did each receive ?
more than the preceding
one.
4.
m
and
n
2.
and
To
find the
nth term
/ of
a G.
...
2 a. 36.
Therefore
Thus the sum
= ^ZlD. P..
|... 108.
NOTE.
A geometric progression
first. except the
multiplying
derived from the preceding one by by a constant number. 24.
The progression is a. 36.
(I)
of the series 16. P.)
is
a series each term of
which.
.
.
ratios are respectively 3..
Hence
Thus the 6th term
l
= ar
n~l
.
-2.
is
16(f)
4
. called the ratio. is
it
(G.. rs =
s
2
-.arn ~ l .
If
n
is less
:
than unity.
g==
it is
convenient to write formula' (II) in
*.
E. the first
= a + ar -for ar -f ar Multiplying by r.
2
arn
(2)
Subtracting (1) from
(2).
r
n~ l
.
.PROGRESSIONS
251
GEOMETRIC PROGRESSION
313.
ar8
r.
or 81
315...
of a G. 36. 12.
fl
lg[(i)
-l]
==
32(W -
1)
= 332 J.
4-
(1)
. the first term a and
the ratios r being given.g.
4..
The
314. ar. To find the sum s of the first n terms term a and the ratio r being given. -I. P.
the following form 8
nf +
q(l-r")
1
r
. a?*2 To obtain the nth term a must evidently be multiplied by
. or. 24. +1.
s(r
1)
8
= ar"
7*
JL
a.
4.
. <zr .
(II)
of the
8 =s
first
6 terms of the series 16.
r^2.
.
is 16.
.
7.252
ELEMENTS OF ALGEBRA
316.
8..
576. f.18.
. ?
(c)
2.
6. hence.
series 5.
Evidently the total
number
of terms is 5
+ 2..288. In most problems relating to G.
Write down the first 5 terms of a G. 3.
or
7.
-fa. 4. I
= 670.
And the
required
means are
18.
144. 36.
0.
To
insert 5 geometric
means between 9 and 576. 36.
|.
series
.
72.
.
series
Find the llth term of the Find the 7th term of the
ratio is
^. + 5. whose and whose common ratio is 4.
\
t
series
.
+-f%9 %
.
Ex.5. 144.
i 288. 676. the other two be found by the solution of the simultaneous equations :
may
(I)
/=<!/-'. 9.
Hence the
or
series is
0.4.
Find the 6th term of the
series J.
288.
.
volved ..
.
Write down the first 6 terms of a G.-. 72..5.
.
36.
. 25.54.
.
. 18..
(it.*.
4.
Hence n
=
7.6. Jive quantities are in.
.
first
term
4. 1.
.
l. P.
676
t
Substituting in
= r6 = 64.
first
5.._!=!>.
f... 144.
first
term
is
125 and
whose common
.
-fa. 20. P.72.. P.
2
term
3.18.
(d) 5.
Find the 7th term of the Find the 6th term of the
Find the 9th term of the
^. 80.
a
=
I..
EXERCISE
1. if any three of them are given.
..
9. 9.
is 3. P. P.
Find the 5th term of a G. whose
.
series 6..
117
Which
(a)
of the following series are in G. whose and whose second term is 8.
10.l..
(b) 1.
181.
respectively. x*
185.266
173. the ana of the floor will be increased 48 square feet.
3 gives the
174.
What
is
the distance?
if
square grass plot would contain 73 square feet more Find the side of the plot. 12 m.
7/
191.
An
The two
express train runs 7 miles an hour faster than an ordinary trains run a certain distance in 4 h.
180. was three times that of the younger. 10x 2 192.
178.
The age
of the elder of
it
three years ago of each.
dimension
182.
188. Find the age
5 years older than his sister
183. 15 m.
.-36. How many are there in each window ?
.
A
each
177.
younger than his Find the age of
the father. A house has 3 rows of windows. 189.
number divided by
3.
. z 2 + x . aW + llab-2&. and the father's present age is twice what the son will be 8 years
hence.56.
179. + 11 ~ 6.
+
a.
father.
Find the number. and 5 h. 4 a 2
y-y
-42.
and | as old as his Find the age of the
Resolve into prime factors
:
184.
-ll?/-102.
187. 6 in each row the lowest row has 2 panes of glass in each window more than the middle row. z 2
-92.
The length
is
of a floor exceeds its width
by 2
feet.
2
2
+
a
_ no. Find the dimensions of the floor.
power one of the two Find the power of each.
+x-
2.
186. Four years ago a father was three times as old as his son is now. side were one foot longer.
176.
A
boy
is
father.
ELEMENTS OF ALGEBRA
A A
number increased by
3.
13 a + 3. and the middle row has 4 panes in each window more than the upper row there are in all 168 panes of glass.
if
each
increased 2 feet.
is
What are their ages ? Two engines are together
more than the
of 80 horse
16 horse power
other. the
sum
of the ages of all three is 51.
190.
.
sister
.
A
the
boy
is
as old as his father
and
3 years
sum
of the ages of the three is 57 years.
3 gives the
same
result as the
numbet
multiplied by
Find the number.
two boys is twice that of the younger.
train.
same
result as the
number
diminished by
175.
c) .
(x
.
A man
drives to a certain place at the rate of 8 miles an
Returning by a road 3 miles longer at the rate of 9 miles an hour.
4x
a
a
2 c
6
Qx
3 x
c
419.
-
a)
-2
6 2a.
418 ~j-o.278
410. he takes 7 minutes longer than in going. and was out 5 hours.(c rt
a)(x
-
b)
=
0. 18 be subtracted from the number.
a
x
)
~
a
2 b
2
ar
a
IJ a. Tn 6 hours
. 411. How long is each road ?
423. Find the number.
421.
420. far did he walk all together ?
A
. the order of the digits will be inverted.
(x
-f
ELEMENTS OF ALGEBRA
a)(z
-
b)
=
a
2 alb
=
a
(x
-f
b)(x
2
. x
1
a
x
x1
ab
1
1
a
x
a
c
+
b
c
x
a
b
b
~
c
x
b
416
417.
mx ~
nx
(a
~
mx
nx
c
d
d
c)(:r
lfi:r
a
b)(x
.a)(x b
b)
(x
b
~
)
412.
down again
How
person walks up a hill at the rate of 2 miles an hour.
A
in 9 hours
B walks
11 miles
number of two digits the first digit is twice the second.
a
x
a
x
b
b
x
c
b
_a
b
-f
x
414. In a
if
and
422. and at the rate of 3^ miles an hour. Find the number of miles an hour that A and B each walk.
hour.
-f
a
x
-f
x
-f c
1
1
a-b
b
x
415.
2 a
x
c
x
6
-f c
a
+
a
+
a
+
6
-f
walks 2 miles more than B walks in 7 hours more than A walks in 5 hours.(5 I2x
~r
l
a)
.
had each at first?
B
B
then has
J
as
much
spends } of his money and as A. There are two numbers the half of the greater of which exceeds the less by 2. Find their ages.
487. Find the
fraction.
485. and in 20 months to $275. Find the numbers. and the other number least. and in 18 months to $2180. A sum of money at simple interest amounts in 8 months to $260. and a fifth part of one brother's age that of the other. Find the numbers.
. age. fraction becomes equal to |. In a certain proper fraction the difference between the nu merator and the denominator is 12. also a third of the greater exceeds half the less by 2. Of the ages of two brothers one exceeds half the other by 4 is equal to an eighth of 482.
477.
by 4.
If 1
be added to the numerator of a fraction
it
if 1
be added to the denominator
it becomes equal becomes equal to ^.
latter
would then be twice the
son's
A
and B together have $6000.
years.
least
The sum
of three
numbers
is
is
21.
479. If 31 years were added to the age of a father it would be also if one year were taken from the son's age
. the Find their ages. What is that fraction which becomes f when its numerator is doubled and its denominator is increased by 1. Find two numbers such that twice the greater exceeds the by 30. Find the sum and the rate of
interest.
483.
Find
the number.282
ELEMENTS OF ALGEBRA
476.
if
the
sum of
the digits be multiplied by
the digits will be inverted.
to
.
481. A sum of money at simple interest amounted in 10 months to $2100. and becomes when its denominator is doubled and its numerator increased by 4 ?
j|
478. and if each be increased by 5 the Find the fraction. Find the principal and the rate of
interest.
A
number
consists of
two
digits
4.
thrice that of his son
and added to the father's.
whose difference
is
4. How much money
less
484.
half the
The greatest exceeds the sum of the greatest and
480.
A
spends \ of his.
486. and 5 times the less exceeds the greater by 3.
if and L. B and C and C and A in 4 days. 37 pounds of tin lose 5 pounds.
and third equals \\ the sum third equals \. and BE.
530. and 23 pounds of lead lose 2 pounds. if L and Af in 20 minutes. Tf and run together. In
circle
A ABC.
sum of the reciprocals of of the reciprocals of the first of the reciprocals of the second and
the
sum
528. and losing 14 pounds when weighed in water? (b) How many pounds of tin and lead are in an alloy weighing 220 pounds in air and 201 pounds in water ?
in 3 days. AB=6. (a) How many pounds of tin and lead are in a mixture weighing 120 pounds in air. Find the numbers. M. An (escribed) and the prolongations of BA and BC in Find AD. Tu what time will it be filled if all run
M
N
N
t
together?
529.
touches
and
F respectively. N. A vessel can be filled by three pipes. the first and second digits will change places. his father is half as old again as his mother was c years ago. What are their rates of
travel?
.
and B together can do a piece of work in 2 days. and one overtakes the other in 6 hours. A boy is a years old his mother was I years old when he was born.
532.
E
533. BC = 5.REVIEW EXERCISE
285
525.
AC
in /). If they had walked toward each other. When weighed in water. A number of three digits whose first and last digits are the same has 7 for the sum of its digits. CD. it is filled in 35 minutes.
. Find the present ages of his father and mother. 90. A can do a piece of work in 12 days B and C together can do the same piece of work in 4 days A and C can do it in half the time in which B alone can do it.
it
separately
?
531.
527.
and CA=7. In how many days can each alone do the same work?
526. in 28 minutes. Throe numbers are such that the
A
the
first
and second equals
. L. How long will B and C take to do
.
. they would have met in 2 hours. Two persons start to travel from two stations 24 miles apart. if the number be increased by Find the number.
formation of dollars into marks. GERMANY.
542.10 marks. 2|. i.
2. Represent the following table graphically
TABLE OF POPULATION (IN MILLIONS) OF UNITED STATES.
547.286
ELEMENTS OF ALGEBRA
:
534.
-
3 x.
550. If
to
feet is the length of a
seconds. x*
-
2
x. 2 x
+
5. The roots of the equation 2 + 2 x x z = 1.
c. x *-x
+
x
+
1.
545.
b. 2
-
x
-
x2
.
e.
from x
=
2 to x
= 4. then / = 3 and write
=
3. if x = f 1. the time of whose swing a graph for the formula from / =0
537.
-
7. FRANCE.
The values of y. The values of x if y = 2.3
Draw
down
the time of swing for a
pendulum
of length
8 feet.e.
d. z 2
-
x x
-
5.
How
is
t /
long will
I
take 11
men
2
t' . The greatest value of the function.
543. the function.
-
3 x.
546.
to
do the work? pendulum. x 2 544.
.
x*.
of
Draw
a graph for the trans-
The number
in
of
workmen Draw
required to finish a certain piece
the graph
work
D
days
it
is
from
D
1 to
D=
12. AND BRITISH ISLES
535.
Draw
the graphs of the following functions
:
538.
+
3.
540. x 8
549. The value of x that produces the greatest value of y. 2 541.
a. x
2
+
x. 3 x
539.
548. One dollar equals 4.
. Draw the graph of y 2 and from the diagram determine
:
+
2 x
x*. 536.
25 might have bought five more for the same money.
___ _ 2* -5 3*2-7
715.
722. if 1 more for 30/ would diminish
720. Find two numbers whose 719. 725. Find two consecutive numbers whose product equals 600.44#2 + 121 = 0. he
many
312?
he had waited a few days until each share had fallen $6. and working together they can build it in 18 days.
716. 717.
What two numbers
are those whose
sum
is
47 and product
A man
bought a certain number of pounds of tea and
10 pounds more of coffee. 2n n 2 2 -f-2aar + a -5 = 0. A man bought a certain number of shares in a company for
$375.
in value. Find four consecutive integers whose product is 7920.
217
.
sum is a and whose product equals J. Find the price of an apple. 16 x* . 724.40 a 2* 2 + 9 a 4 = 0.
ELEMENTS OF ALGEBRA
+36 = 0. In how many days can A build the wall?
718.
729.
723.
The area
the price of 100 apples by $1.
of a rectangle is 221 square feet and its perimeter Find the dimensions of the rectangle. 727. 721. How shares did he buy ?
if
726.292
709. **-13a: 2
710. what is the
price of the coffee per
pound ?
:
Find the numerical value of
728.l
+
8
-8
+
ft)'
(J)-*
(3|)*
+
(a
+
64-
+ i. 12
-4*+
-
8. If a pound of tea cost 30 J* more than a pound of coffee. Find the altitude of an equilateral triangle whose side equals a.
a:
713.
The
difference of the cubes of
two consecutive numbers
is
find them.
A
equals CO feet. paying $ 12 for the tea and $9 for the coffee. needs 15 days longer to build a wall than B.
. What number exceeds its reciprocal by {$.
714
2
*2
'
+
25
4
16
|
25 a2
711. 3or
i
-16 .
feet. The diagonal of a rectangle equals 17 feet.
ELEMENTS OF ALGEBRA
(*+s)(* + y)=10.
much and A then
Find at what
increases his speed 2 miles per hour. = ar(a? -f y + 2) + a)(* + y
933. In the first heat B reaches the winning post 2 minutes before A.
A
is
938.
935. y(x + y + 2) = 133. Find the sides of the rectangle.
rate each
man
ran in the
first
heat. a second rec8 feet shorter.
.
is
3
. *(* + #) =24. z(* + y + 2) = 76. (y + *) = . the area of the new rectangle would equal 170 square feet.
Assuming
= -y. The
sum
of the circumferences of
44 inches.
feet.
two numbers Find the numbers.
is 3.
944.
two squares is 23 feet. A plantation in rows consists of 10.
and the sum of their cubes
is
tangle
certain rectangle contains 300 square feet. A and B run a race round a two-mile course.
and the sum of
their areas 78$.
two squares equals 140 feet.
The sum
of the perimeters of
sum
of the areas of the squares is 16^f feet.
and the
difference of
936. Find the side of each two
circles is
IT
square. s(y
932.
931. (3 + *)(ar + y + z) = 96. Find the numbers. How many rows are there?
941.
+ z)=18.
is 20.square inches.
943. there would have been 25 more trees in a row. Find the length and breadth of the first rectangle. and the Find the sides of the and
its
is
squares.
the
The sum
of the perimeters of
sum
of their areas equals 617 square feet.300
930.
34
939. The perimeter of a rectangle is 92 Find the area of the rectangle. + z) =108. and B diminishes his as arrives at the winning post 2 minutes before B.
2240. If each side was increased by 2 feet.
937.
y(
934.000 trees. 152. and 10 feet broader. (y
(* + y)(y +*)= 50. Tf there had been 20 less rows.
The sum of two numbers Find the numbers.
942. The difference of two numbers cubes is 513. and also contains 300 square feet.
find
the radii of the two circles. In the second
heat
A
.
the difference of their
The
is
difference of
their cubes
270.102.
diagonal
940.
Find the number. The area of a certain rectangle is equal to the area of a square side is 3 inches longer than one of the sides of the rectangle. Find the number. Find its length and breadth. the area lengths of the sides of the rectangle.
overtook
miles. and
its
perim-
948. each block.
952. and that B.REVIEW EXERCISE
301
945. set out from two places. Find the width of the path if its area is 216 square yards. that
B
A
955. When
from
P
A
was found that they had together traveled 80 had passed through Q 4 hours before. its area will be increased 100 square feet.
Two
starts
travelers.
951. the difference in the lengths of the legs of the Find the legs of the triangle. the square of the middle digit is equal to the product of the extreme digits.
whose
946. A certain number exceeds the product of its two digits by 52 and exceeds twice the sum of its digits by 53.
P and
Q.
. triangle is 6. A rectangular lawn whose length is 30 yards and breadth 20 yards is surrounded by a path of uniform width.
. was 9 hours' journey distant from P. and the other 9 days longer to perform the work than if both worked together. The sum of the contents of two cubic blocks
the
of the heights of the blocks is 11 feet. The square described on the hypotenuse of a right triangle is 180 square inches.
Find the
eter
947.
sum
Find an edge of
954.
950. and if 594 be added to the number. Two men can perform a piece of work in a certain time one takes 4 days longer. Find in what time both will do it. If the breadth of the rectangle be decreased by 1 inch and its
is
length increased by 2 inches. The area of a certain rectangle is 2400 square feet. A number consists of three digits whose sum is 14. Find two numbers each of which
is
the square of the other. The diagonal of a rectangular is 476 yards.
. the digits are reversed. distance between P and Q.
unaltered. What is its area?
field is
182 yards. at Find the his rate of traveling.
949. and travels in the same direction as A.
A and
B.
is
407 cubic feet. at
the same time
A
it
starts
and
B
from
Q
with the design to pass through Q. if its length is decreased 10 feet and its breadth increased 10 feet.
953.
then this
sum multiplied by
(Euclid. P.
985. Insert 8 arithmetic means between
1
and
-. P. Find the value of the infinite product 4
v'i
v7-!
v^5
..
987.)
the last term
the series
a perfect number.
1.2
.
990. Find the sum of the series
988.
to 105?
981.-.
303
979.
Find
n.-. The sum
982.
of n terms of 7
+
9
+ 11+
is
is
40.04
+
.01
3..
to
n terms. Insert 22 arithmetic means between 8 and 54. named Sheran.3 '
Find the 8th
983.
How many
sum
terms of 18
+
17
+
10
+
amount
.
v/2
1
+ +
+
1
4
+
+
3>/2
to oo
+
+
.
989.
to infinity
may
be 8?
..
all
A
perfect number
is
a number which equals the sum
divisible. The Arabian Araphad reports that chess was invented by amusement of an Indian rajah.
"(.
980.
992. and the
common
difference.
of n terms of an A. doubling the number for each successive square on the board.REVIEW EXERCISE
978.
Find the number of grains which Sessa should have received.1
+
2.+ lY L V. Find four numbers in A.
If
of
2
of
integers + 2 1 + 2'2
by which
is
it
is
the
sum
of
the series
2 n is prime. is 225.
to oo. such that the product of the and fourth may be 55. 2 grains on the 2d. The
term. Find the
first
term. P. and the sum of the first nine terms is equal to the square of the sum of the first two.001
4. 5
11. and so on. and of the second and third 03. The 21st term of an A. who rewarded the inventor by promising to place 1 grain of wheat on
Sessa for the
the 1st square of a chess-board..
first
984.
986.001
+
.
Find four perfect
numbers.---
:
+
9
-
-
V2
+
... 4 grains on the 3d.
What
2 a
value must a have so that the
sum
of
+
av/2
+
a
+
V2
+
.
0.
.
1003. P. In an equilateral triangle second circle touches the first circle and the sides AB and AC. and so forth to Find (a) the sum of all perimeters. after how strokes would the density of the air be xJn ^ ^ ne original
density ?
a circle is inscribed. and G. prove that they cannot be in A.
997.
1001.
1000.
994.
998. In a circle whose radius is 1 a square is inscribed.304
ELEMENTS OF ALGEBRA
993. are
45 and 765
find the
numbers.
512
996. (I) the sum of the perimeters of
all
squares.
. P.
third circle touches the second circle and the
to infinity. P. and so forth to infinity. P. are unequal.
999. (a) after 5 strokes. Two travelers start on the same road. Find (a) the sum of all circumferences.
pump removes
J
of the
of air
is
fractions of the original amount contained in the receiver. in this square a circle. The other travels 8 miles the first day and After how increases this pace by \ mile a day each succeeding day.
and the
fifth
term
is
8 times
the second . (6) after n
What
strokes?
many
1002. Insert 4 geometric means between 243 and 32.
AB =
1004. at the same time. are 28 and find the numbers. 995. ft. Insert 3 geometric means between 2 and 162. Each stroke of the piston of an air
air contained in the receiver. find the series. in this circle a square.
is 4. The sum and product of three numbers in G.
ABC
A A
n
same
sides.
The sum and sum
. If a.
of squares of four
numbers
in
G. c. Under the conditions of the preceding example.
and
if
so forth
What
is
the
sum
of the areas of all circles. P. the sides
of a third triangle equal the altitudes of the second. One of them travels uniformly 10 miles a day. The side of an equilateral triangle equals 2. (6) the sum of the infinity. inches. The
fifth
term of a G. The sides of a second equilateral triangle equal the altitudes of the first. areas of all triangles. many days will the latter overtake the former?
.
which have been omitted from the body of the work Indeterminate Equahave been relegated to the Appendix. Ph.
A
examples are taken from geometry. The author
has emphasized Graphical Methods more than is usual in text-books of this grade. given.
THE MACMILLAN COMPANY
PUBLISHERS.ELEMENTARY ALGEBRA
By ARTHUR SCHULTZE.
$1. etc.
HEW TOSS
. 64-66 FIFTH AVBNTC. To meet the requirements of the College Entrance Examination Board. and the Summation of Series is here presented in a novel form. The more important subjects
tions. book is a thoroughly practical and comprehensive text-book. than by the
.
Half
leather.
$1. All subjects now required for admission by the College Entrance Examination Board
have been omitted from the present volume. save Inequalities. but the work in the latter subject
has been so arranged that teachers
who
wish a shorter course
may omit
it
ADVANCED ALGEBRA
By ARTHUR SCHULTZE.
xiv+563
pages. physics. The introsimpler and more natural than the
methods given
In Factoring.25
lamo. without the sacrifice of scientific accuracy and thoroughness. which has been retained to serve as a basis for higher work. very numerous and well graded there is a sufficient number of easy examples of each kind to enable the weakest students to do some work. but these few are treated so thoroughly and are illustrated by so many varied examples that the student will be much better prepared for further
The Exercises are superficial study of a great many cases.
xi 4-
373 pages. great many
work.D. but none of the introduced illustrations is so complex as to require the expenditure of
time for the teaching of physics or geometry.
Half leather. comparatively few methods are heretofore. Particular care has been bestowed upon those chapters which in the customary courses offer the greatest difficulties to the beginner. and commercial life. so that the Logarithms.10
The treatment of elementary algebra here is simple and practical.
not
The Advanced Algebra is an amplification of the Elementary. especially
duction into Problem
Work
is
very
much
Problems and Factoring. proportions and graphical methods are introduced into the first year's course.
i2mo.
$1.
12010. great many
A
examples are taken from geometry.10
The treatment
of elementary algebra here
is
simple and practical.
$1. book is a thoroughly practical and comprehensive text-book. All subjects now required for admission by the College Entrance Examination Board
have been omitted from the present volume. The more important subjects
which have been omitted from the body of the work Indeterminate Equahave been relegated to the Appendix.
Half leather.
THE MACMILLAN COMPANY
PUBLISHBSS. but the work in the latter subject
has been so arranged that teachers
who
wish a shorter course
may omit
it
ADVANCED ALGEBRA
By ARTHUR SCHULTZE.
In Factoring. To meet the requirements of the College Entrance Examination Board. proportions and graphical methods are introduced into the first year's course. physics. and commercial life. but these few are treated so thoroughly and are illustrated by so many varied examples that the student will be much better prepared for further
work. bestowed upon those chapters which in the customary courses offer the greatest difficulties to the beginner. which has been retained to serve as a basis for higher work.
has emphasized Graphical Methods more than is usual in text-books of this and the Summation of Series is here presented in a novel form. The introsimpler and more natural than the
methods given heretofore.D. The Exercises are very numerous and well graded. but none of the introduced illustrations is so complex as to require the expenditure of
time for the teaching of physics or geometry. The author
grade. without
Particular care has been the sacrifice of scientific accuracy and thoroughness.
HEW YOKE
.25
i2mo. Logarithms.
not
The Advanced Algebra is an amplification of the Elementary. especially
duction into Problem
Work
is
very
much
Problems and Factoring. Ph.
xiv+56a
pages.
xi
-f-
373 pages. etc. there is a sufficient number of easy examples of each kind to enable the weakest students to do some work.ELEMENTARY ALGEBRA
By ARTHUR Sen ULTZE. comparatively few methods are
given. save Inequalities. than by the superficial study of a great many cases.
HatF leather. 64-66
7HTH
AVENUE. so that the tions.
$1.
wor.
NEW YORK
. State: .
i2mo.D. These are introduced from the beginning 3. 10..
and no attempt has been made
to present these solutions in such form
that they can be used as models for class-room work. guides him in putting forth his efforts to the best
advantage. iamo.
. more than 1200 in number in 2.
of Propositions has a
Propositions easily understood are given first and more difficult ones follow . The Analysis of Problems and of Theorems is more concrete and practical than in any other
distinct pedagogical value.10
L.
xtt-t
PLANE GEOMETRY
Separate. Ph.r and. Many proofs are presented in a simpler and manner than in most text-books in Geometry 8. 64-66 FIFTH AVENUE.
PLANE AND SOLID GEOMETRY
F.
80 cents
This Geometry introduces the student systematically to the solution of geometrical exercises. Preliminary Propositions are presented in a simple manner .10
By ARTHUR
This key will be helpful to teachers who cannot give sufficient time to the Most solutions are merely outsolution of the exercises in the text-book. at the
It
same
provides a course which stimulates him to do original time. 4. under the heading Remarks". The numerous and well-graded Exercises the complete book.
Algebraic Solution of Geometrical Exercises is treated in the Appendix to the Plane Geometry .
Cloth.
lines.
izmo.
Half
leather. Difficult Propare made somewhat? easier by applying simple Notation .
By ARTHUR SCHULTZE and
370 pages. 6. aoo pages. $1. Cloth.
THE MACMILLAN COMPANY
PUBLISHERS.
ments from which General Principles may be obtained are inserted in the " Exercises. SCHULTZE.
text-book in Geometry
more
direct
ositions
7. 9.
KEY TO THE EXERCISES
in
Schultze and Sevenoak's Plane and Solid Geometry. Hints as to the manner of completing the work are inserted The Order 5. 7 he
. Pains have been taken to give Excellent Figures
throughout the book. Proofs that are special cases of general principles obtained from the Exercises are not given in detail. Attention is invited to the following important features I.
xii
+ 233 pages. The Schultze and Sevenoak Geometry is in use in a large number of the leading schools of the country.
SEVENOAK.
Typical topics the value and the aims of mathematical teach-
ing
. $1.
methods of teaching mathematics the first propositions in geometry the original exercise parallel lines methods of the circle attacking problems impossible constructions applied problems typical parts of algebra.
.
New York
DALLAS
CHICAGO
BOSTON
SAN FRANCISCO
ATLANTA
. 370 pages. a great deal of mathematical spite
teaching
is
still
informational.
12mo. enable him to
" The chief object of the speak with unusual authority.
THE MACMILLAN COMPANY
64-66 Fifth Avenue. of these theoretical views. making mathematical teaching less informational and more disciplinary.
.
causes of the inefficiency of mathematical teaching.
.
.The Teaching
of
Mathematics
in
Secondary Schools
ARTHUR SCHULTZE
Formerly Head of the Department of Mathematics in the High School Commerce."
The treatment
treated are
:
is concrete and practical. New York City. Most teachers admit that mathematical instruction derives its importance from the mental training that it But in affords. and Assistant Professor of Mathematics in New York University
of
Cloth.
.
. and not from the information that it imparts.
.25
The
author's long
and successful experience as a teacher
of mathematics in secondary schools and his careful study of the subject from the pedagogical point of view.
.
Students
to
still
learn
demon-
strations instead of learning
how
demonstrate. " is to contribute towards book/ he says in the preface.
Studies and Questions at the end of each chapter take the place of the individual teacher's lesson plans. Maps. The author's aim is to keep constantly before the
This book
pupil's mind the general movements in American history and their relative value in the development of our nation.
$1. diagrams.
New York
SAN FRANCISCO
BOSTON
CHICAGO
ATLANTA
. " This volume
etc.40
is distinguished from a large number of American text-books in that its main theme is the development of history the nation. This book is up-to-date not only in its matter and method. All
smaller movements and single events are clearly grouped under these general movements.
THE MACMILLAN COMPANY
64-66 Fifth Avenue. and a full index are provided. which put the main stress upon national development rather than upon military campaigns. supply the student with plenty of historical
narrative on which to base the general statements and other classifications made in the text. which have been selected with great care and can be found in the average high school library.
but in being fully illustrated with
many excellent maps.
Topics.
Cloth.
An exhaustive system of marginal references.
i2mo.
photographs.
diagrams. The book deserves the attention
of history teachers/'
Journal of Pedagogy.AMERICAN HISTORY
For Use
fa
Secondary Schools
By ROSCOE LEWIS ASHLEY
Illustrated.
is
an excellent example of the newer type of
school histories. | 677.169 | 1 |
The very nature of algebra concerns the generalization of patterns (Lee 1996). Patterning activities that are geometric in nature can serve as powerful contexts that engage students in algebraic thinking and visually support them in constructing a variety of generalizations and justifications (e.g., Healy and Hoyles 1999; Lannin 2005). In this article, we discuss geometric patterning tasks that engage students with wide-ranging levels of ability, interest, and motivation. This succession of tasks is likely to elicit recursive reasoning strategies to build mathematical sequences on previous terms' values or explicit formulas to determine any value in the sequence. The tasks are increasingly complex in terms of mathematical patterns, numeric computations, and visualization demands. | 677.169 | 1 |
Do you Need a Software to Boost Math Grades?
Typical math software applications involve the use of calculators and scientific calculators to solve math problems. However, you can achieve much better results with the aid of Mathlab Scientific Graphing Calculator. A graphing calculator that comes wit...
This Electronic Calculator Teaches Math with Little or no Supervision!
If you are interested in learning and studying math in an engaging and interactive context, the Mathlab Scientific Graphing Calculator is the perfect electronic device. An intelligent scientific calculator that does far more th... | 677.169 | 1 |
This section contains free e-books and guides on Basic Mathematics, some of the resources in this section can be viewed online and some of them can be downloaded.
The aim of this book has been to illustrate the use of mathematics in constructing
diagrams, in measuring areas, volumes, strengths of materials, in calculating
latitudes and longitudes on the earth's surface, and in solving similar
problems. One great branch of Practical Mathematics, that dealing with
electricity and magnetism, has not been included in this book.
This book is considered as a great reference book for
beginners. The chief purpose of the book is to help to bridge the gap
which separates many engineers from mathematics by giving them a bird's-eye view
of those mathematical topics which are indispensable in the study of the
physical sciences.
These are the sample pages
from the textbook, 'Mathematics Reference Book for Scientists and Engineers'.
Fundamental principles are reviewed and presented by way of examples, figures,
tables and diagrams. It condenses and presents under one cover basic concepts
from several different applied mathematics topics. | 677.169 | 1 |
An Introduction to Engineering Mathematics
Description
Originally published in 1936, this textbook was primarily designed for students of applied mathematics to provide a solid foundation for studies on the practical side of the subject. The book, whilst carefully keeping the groundwork of pure mathematics, covering key elementary aspects of the subject, such as quadratic equations, in the main explores the 'generally understood technical allusions, illustrations, and graded applications'. The book provides multiple useful and challenging exercises and is shaped and established around the syllabus at the time of publication. Chapters are broad in scope, detailed and clearly written; chapter titles include, 'Curved graphs', 'Trigonometry ratios and exercises' and 'Loci'. Aiming to achieve a 'logical sequence of leading theorems, clear deductions and concise, practical applications', this textbook will be of great value to students of engineering mathematics as well as to anyone with an interest in physics, materials science and the history of education. | 677.169 | 1 |
In this session, you will create lessons to help you learn more about your students' algebraic thinking and functions.
This session continues our work with algebraic thinking related to functions, focusing on the use of multiple representations. You will begin by reading chapter 7 of the Driscoll book, then you will complete a series of activities – some using technology – that provide some examples of activities that help students understand the relationship between different representations of functions. You will watch several video segments of a class discussing some of the problems and discuss the students' understanding. Finally, you will complete the next several steps of your final project.
This folder contains the resources linked to Session 7 of Building Algebraic Thinking through Pattern and Function - Professional Development Course for Mathematics Teachers in PDF and Word formats for your convenience and in case the links get broken. | 677.169 | 1 |
Pre-Calculus Study Guide (Speedy Study Guide)
Studying for Pre-Calculus is no joke, and it the beginning to the part of math where paying attention alone is not an option. In order for any student to get better in math and know their material, many different trials and errors must take place in addition to trying out the formulas learned to see how applying a formula is very different than just memorizing them. By teaching the student about formulas and how to approach pre-calculus best, the study guide is always considered to be the most useful asset that teachers use to help them get the message across to students for years to come in their professional | 677.169 | 1 |
ISBN 13: 9780070131651
Mathematics for Technicians
Mathematics for Technicians 6E remains the leading Australian text for students studying Mathematics, including Engineering Maths A and Engineering Maths B. This edition uses a building block approach with worked examples, banks of exercises for each section, and self-test questions ideal for revision and exam preparation. Answers to questions in the text are located at the back of the book. | 677.169 | 1 |
Compressed Zip File
Be sure that you have an application to open this file type before downloading and/or purchasing. How to unzip files.
31.61 MB | 200+ pages
PRODUCT DESCRIPTION
This activity bundle is designed for an algebra student who is learning exponential functions for the very first time. The main topics in this bundle include an introduction to exponential functions,
characteristics of exponential functions, graphing exponential functions with transformations, applications of exponential functions with regression modeling, and comparing linear functions and exponential functions.
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Mathematics is the science study of the number, structure, and spatial transformations. In other words, it is assumed that subjects' shape and number. "According to the official view, it is the study of the abstract structure defined from the axioms, using Logic (logic) and mathematical symbols. The other point of it is described in mathematical philosophy. Due to their wide applications in many science, mathematics is known as the "universal language". Experts in the field of mathematics known as mathematicians....
This book is a survey of abstract algebra with emphasis on algebra tinh.Do is online
for students in mathematics, computer science, and physical sciences.
The rst three or four chapters can stand alone as a one semester course in abstract
algebra. However, they are structured to provide the foundation for the program
linear algebra. Chapter 2 is the most di cult part of the book for group
written in additive notation and multiplication, and the concept of coset is confusing
at rst. Chapter 2 After the book was much easier as you go along....
The students and teachers refer you quiz "mathematics Assessment: Grade 3" below to systematize their knowledge and experience with the subject. Hope quiz will help you to achieve good results in the upcoming exam.
Invite you to consult "Student practice test booklet Grade 5: Mathematics", with this quiz will help you review the knowledge learned, had the opportunity to assess their capabilities. I wish you success in the upcoming exam.
Invite you to consult "Assessment mathematics: Grade 6", with this quiz will help you review the knowledge learned, had the opportunity to assess their capabilities. I wish you success in the upcoming exam.
Many times, the first question career-changers ask is, "Is this new
path right for me?" Our self-assessment quiz, coupled with the career
compasses at the beginning of each chapter, will help you to match your
personal attributes to set you on the right track. | 677.169 | 1 |
traditional text for the modern student--Pat McKeague's INTERMEDIATE ALGEBRA, 9E--is user-friendly for both students and instructors with concise writing, continuous review, and contemporary applications. Retaining its hallmark strengths of clarity and patience in explanation and concept development, this new edition contains new examples, applications, and a closer integration with Enhanced WebAssign. In a course in which many students enter with math anxiety, the author helps students connect mathematics to every day examples through the use of relevant applications and real data. In addition, Enhanced WebAssign, an online homework management system, is fully integrated with the new edition providing interactive, visual learning support with thousands of examples and practice exercises that reinforce the text's pedagogical approach.
Additional Product Information
Features and Benefits
Enhanced WebAssign! Easy to Use. Easy to Assign. Easy to Manage. Enhanced WebAssign is a user friendly way to assign, collect, grade, and record homework. Exercises come directly from the text and can be parameterized to ensure that every student has a parallel, but slightly different homework assignment. This complete learning system for students includes text-specific exercises, as well as tutorials, videos, and links to online tutoring, and eBook sections of the text.
"Getting Ready for the Next Section" exercises are found in every problem set. This grouping of exercises ensures students have the requisite skills to perform the concepts discussed in the next section.
Found throughout the text, "Using Technology" shows how calculators can be used to enhance learning and help solve problems. These boxes are easily identified and can be highlighted or skipped depending on instructor preference.
Chapter openers motivate students to learn the topics in each chapter. Each chapter opener illustrates a real-world application of one of the main chapter topics, and most present the information both graphically and numerically.
Section openers have been freshly redesigned and are linked to specific examples and applications found in the text.
Hundreds of new problems and examples throughout the text include more contemporary applications. Additionally, wherever real data is used, it has been updated to reflect current statistics.
Each problem set contains a variety of problems, including both skills and applications exercises, and great breadth from easy to challenging. In addition to opening skills based problems (paired and graduated), each problem set contains opportunities to hone analytic skills (asking students to practice estimating), provide more relevance ("Applying the Concepts" questions) and improve their quantitative literacy ("Maintaining your Skills" problems). More challenging critical thinking questions called "Extending the Concepts" are found at the end of the section problem sets.
What's New
"Preview" boxes provide a glimpse into what topics appear in the chapter and include key words and definitions.
"Chapter Challenge" problem gives students a scenario and require them to use geographical information to solve it through a series of steps.
"Ticket to Success" feature requires students to think critically about the section and write a response to the questions as their "ticket into class." This feature encourages students to read the section before coming to class.
"Moving Toward Success" feature starts each problem set with a motivational quote followed by a few questions to focus the students on their own success. Because good study habits are essential to the success of math students, this feature is prominently displayed prior to each problem set.
Alternate Formats
Choose the format that best fits your student's budget and course goals
Instructor Supplements
This CD-ROM (or DVD) provides the instructor with dynamic media tools for teaching. Create, deliver, and customize tests (both print and online) in minutes with ExamView® Computerized Testing Featuring Algorithmic Equations. Easily build solution sets for homework or exams using Solution Builder's online solutions manual. Microsoft® PowerPoint® lecture slides and figures from the book are also included on this CD-ROM (or DVD).
Solution Builder
(ISBN-10: 1133490662 | ISBN-13: 9781133490661)
This online instructor database offers complete worked solutions to all exercises in the text, allowing you to create customized, secure solutions printouts (in PDF format) matched exactly to the problems you assign in class. For more information, visit
Each section of the main text is discussed in uniquely designed Teaching Guides containing instruction tips, examples, activities, worksheets, overheads, assessments, and solutions to all worksheets and activitiesStudent Supplements
Go beyond the answers—see what it takes to get there and improve your grade! This manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text. This gives you the information you need to truly understand how these problems are solvedMeet the Author
Author Bio
Charles P. McKeague | 677.169 | 1 |
Error Coding for Arithmetic Processors
Kobo ebook | January 1, 1974 algebraic structures, linear congruences, and residues. Organized into eight chapters, this volume begins with an overview of the mathematical background in number theory, algebra, and error control techniques. It then explains the basic mathematical models on a register and its number representation system. The reader is also introduced to arithmetic processors, as well as to error control techniques. The text also explores the functional units of a digital computer, including control unit, arithmetic processor, memory unit, program unit, and input/output unit. Students in advanced undergraduate or graduate level courses, researchers, and readers who are interested in applicable knowledge on arithmetic codes alge... | 677.169 | 1 |
Calculus: Single Variable Part 2 - Differentiation (Coursera)Distinguishing features of the course include: 1) the introduction and use of Taylor series and approximations from the beginning; 2) a novel synthesis of discrete and continuous forms of Calculus; 3) an emphasis on the conceptual over the computational; and 4) a clear, dynamic, unified approach.
In this second part--part two of five--we cover derivatives, differentiation rules, linearization, higher derivatives, optimization, differentials, and differentiation operators.
Related courses Two: Sequences and Series is an introduction to sequences, infinite series, convergence tests, and Taylor series. The course emphasizes not just getting answers, but asking the question "why is this true?"Quantitative and model-based introduction to basic ideas in economics, and applications to a wide range of real world problems. This course provides a quantitative and model-based introduction to basic economic principles, and teaches how to apply them to make sense of a wide range of real world problemsControl of Mobile Robots is a course that focuses on the application of modern control theory to the problem of making robots move around in safe and effective ways. The structure of this class is somewhat unusual since it involves many moving parts - to do robotics right, one has to go from basic theory all the way to an actual robot moving around in the real world, which is the challenge we have set out to address through the different pieces in the course. | 677.169 | 1 |
This best-selling textbook for a second course in linear algebra is aimed at undergrad math majors and graduate students. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. The third edition contains major improvements and revisions throughout the book. More than 300 new exercises have been added since the previous edition. Many new examples have been added to illustrate the key ideas of linear algebra. New topics covered in the book include product spaces, quotient spaces, and dual spaces. Beautiful new formatting creates pages with an unusually pleasant appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
This text for a second course in linear algebra, aimed at math majors and graduates, adopts a novel approach by banishing determinants to the end of the book and focusing on has an eigenvalue. The book starts by discussing vector spaces, linear independence, span, basics, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite- dimensional spectral theorem. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. This second edition features new chapters on diagonal matrices, on linear functionals and adjoints, and on the spectral theorem; some sections, such as those on self-adjoint and normal operators, have been entirely rewritten; and hundreds of minor improvements have been made throughout the text.
This text for a second course in linear algebra is aimed at math majors and graduate students. The novel approach taken here banishes determinants to the end of the book and focuses on the central goal of linear algebra: (or an odd-dimensional real vector space) has an eigenvalue. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. Thus the text starts by discussing vector spaces, linear independence, span, basis, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite-dimensional spectral theorem. This second edition includes a new section on orthogonal projections and minimization problems. The sections on self-adjoint operators, normal operators, and the spectral theorem have been rewritten. New examples and new exercises have been added, several proofs have been simplified, and hundreds of minor improvements have been made throughout the text.
"Linear Algebra" is intended for a one-term course at the junior or senior level. It begins with an exposition of the basic theory of vector spaces and proceeds to explain the fundamental structure theorem for linear maps, including eigenvectors and eigenvalues, quadratic and hermitian forms, diagnolization of symmetric, hermitian, and unitary linear maps and matrices, triangulation, and Jordan canonical form. The book also includes a useful chapter on convex sets and the finite-dimensional Krein-Milman theorem. The presentation is aimed at the student who has already had some exposure to the elementary theory of matrices, determinants and linear maps. However the book is logically self-contained. In this new edition, many parts of the book have been rewritten and reorganized, and new exercises have been added.
This textbook on linear algebra includes the key topics of the subject that most advanced undergraduates need to learn before entering graduate school. All the usual topics, such as complex vector spaces, complex inner products, the Spectral theorem for normal operators, dual spaces, the minimal polynomial, the Jordan canonical form, and the rational canonical form, are covered, along with a chapter on determinants at the end of the book. In addition, there is material throughout the text on linear differential equations and how it integrates with all of the important concepts in linear algebra. This book has several distinguishing features that set it apart from other linear algebra texts. For example: Gaussian elimination is used as the key tool in getting at eigenvalues; it takes an essentially determinant-free approach to linear algebra; and systems of linear differential equations are used as frequent motivation for the reader. Another motivating aspect of the book is the excellent and engaging exercises that abound in this text. This textbook is written for an upper-division undergraduate course on Linear Algebra. The prerequisites for this book are a familiarity with basic matrix algebra and elementary calculus, although any student who is willing to think abstractly should not have too much difficulty in understanding this text.style sections have been added. Investigations of Euler's computation of ζ(2), the Weierstrass Approximation Theorem, and the gamma function are now among the book's cohort of seminal results serving as motivation and payoff for the beginning student to master the methods of analysis. Review of the first edition: "This is a dangerous book. Understanding Analysis is so well-written and the development of the theory so w ell-motiv ated that exposing students to it could well lead them to expect such excellence in all their textbooks. ... Understanding Analysis is perfectly titled; if your students read it, that's what's going to happen. ... This terrific book will become the text of choice for the single-variable introductory analysis course ... " — Steve Kennedy, MAA Reviews
As a newly minted Ph.D., Paul Halmos came to the Institute for Advanced Study in 1938--even though he did not have a fellowship--to study among the many giants of mathematics who had recently joined the faculty. He eventually became John von Neumann's research assistant, and it was one of von Neumann's inspiring lectures that spurred Halmos to write Finite Dimensional Vector Spaces. The book brought him instant fame as an expositor of mathematics. Finite Dimensional Vector Spaces combines algebra and geometry to discuss the three-dimensional area where vectors can be plotted. The book broke ground as the first formal introduction to linear algebra, a branch of modern mathematics that studies vectors and vector spaces. The book continues to exert its influence sixty years after publication, as linear algebra is now widely used, not only in mathematics but also in the natural and social sciences, for studying such subjects as weather problems, traffic flow, electronic circuits, and population genetics. In 1983 Halmos received the coveted Steele Prize for exposition from the American Mathematical Society for "his many graduate texts in mathematics dealing with finite dimensional vector spaces, measure theory, ergodic theory, and Hilbert space." | 677.169 | 1 |
POST-16 MATHEMATICS By the end of this presentation you should understand what options you have to continue you Mathematics education after GCSEs. You.
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Presentation on theme: "POST-16 MATHEMATICS By the end of this presentation you should understand what options you have to continue you Mathematics education after GCSEs. You."— Presentation transcript:
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POST-16 MATHEMATICS By the end of this presentation you should understand what options you have to continue you Mathematics education after GCSEs. You might also understand what each of these entail.
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1.What's it about? 2.Why would I do it? 3.What next? BTEC MONEY AND FINANCE FUNCTIONAL SKILLS What's a Direct Debit? How much income tax will I pay? What's the point in National Insurance? … What then is a Standing Order? Borrowing money and taking risk National and Global Money Matters Money Matters for Career Planning Working and Earning Saving and Spending Six Units… No EXAMS Managing Money Matters
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1.What's it about? 2.Why would I do it? 3.What next? Employers look for Mathematics ability FUNCTIONAL SKILLS Prove that you can solve problems! These courses are designed around things you'll definitely use! Almost 50 per cent of adults in Britain lack basic Maths skills (that's almost a half!!!). Show that you're above average! How would you answer the following interview question: "What evidence do you have that you can solve problems under pressure?"
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1.What's it about? 2.Why would I do it? 3.What next? If you hope to go to University, they can reject you if you don't have C in GCSE Mathematics RESIT FOUNDATION GCSE MATHEMATICS The world's your oyster, you've got you're "C"… That will never hold you back!
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STATISTICS Used in most degrees to aim with research! The application of methods to collect, analyse and interpret data.
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1.What's it about? 2.Why would I do it? 3.What next? A-LEVEL MATHEMATICS It will help your other A-Level subjects Mathematics Statistics Computer Science Physics Accountancy Engineering Geography Psychology, Sociology Sports Science Earth Sciences Medicine Chemistry, Biology Business On average people who have done A level Maths earn 10% more than those who haven't. Even those who don't get the top grades earn 8% more Logical Skills Analytical Skills Problem Solving It's AWESOME!!!!
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A-LEVEL FURTHER MATHEMATICS Further Mathematics is known to be amongst the most challenging A-Levels. You'll be one of a select few individuals who get's this: 6 more Modules!!! "Further Pure Mathematics Matrices Imaginary Numbers Polar Coord's | 677.169 | 1 |
Publisher's Summary
Math and science are, without a doubt, some of the most difficult subjects in school. These subjects require you to learn different methods, formulas, and terminologies, and most of the time they can be too much to handle. With the help of this audiobook, you will learn how to tackle the heavy mental workload required by these subjects in the most efficient manner. Using just 30 simple steps, you will learn how to breeze through math and science, and in the end you will also find out that studying can be fun.
A person who listens closely and actually applies these techniques will likely join the crowd at the upper end of any class (the person has a reasonable aptitude for). All other things being equal, students who study any subjects like this steadily will excel. The trick is in actually sticking with all of it; but no single part is frankly that hard. (It might seem so, for a student habituated to all the noise and distractions of the modern cluttered life.) As to math and science specifically, I did not find any new revelations here. It seems to me a generic set of tips, and subject to that, a fine set. Good studiers already do most all of this.
I feel robbed! There is absolutely nothing said in this book that you haven't already heard from your professors, parents, tutors or mentor. Just focus on your study material, READ your textbook, do your homework, eat right, exercise, get plenty of rest and relax when taking exams. That's about it.
I read other reviews about this book that said the exact same thing that I just typed, and was SURE that there was more to it than they had mentioned...But there really isn't. Save your money. Look elsewhere if you want actual study tips or tricks for doing better in math and science, cuz this isn't it!
This is a really good, short, and inexpensive book with a lot of really good tips about studying.
The author presents 30 tips about how to approach study. The focus is on math and science, but the tips can be applied to study and review of any type.
There were some really good ideas based on particular preferred styles of learning that aren't as well presented in other books, and I particularly enjoyed the part about how to read a math book. Mnemonics, study techniques, time management, organization, and preparation for study are also addressed, and the concepts are well presented.
Again, this is a good audiobook for study of any type and well worth the inexpensive price and the short time required to go through it. You will get a lot of bang for your buck out of this one.
This book is filled with very creative ways of learning truly any subject. Although, it's main focus is math and science the skills taught and once learned will be extremely helpful across the educational criculum. Ryan E. Strickland Author of Is Your Mind Shackle Free! | 677.169 | 1 |
How could an ancient king be tricked into giving his servant more than 671 billion tons of rice? It's all due to a simple but powerful calculation involving the sum of geometric progression – an important concept in number theory and just one of the fascinating concepts you'll encounter in An Introduction to Number Theory. Taught by veteran Teaching Company instructor Edward B. Burger, this 24-lecture course offers an exciting adventure into the world of numbers.
The majority of students who take courses in number theory are mathematics majors who will not become number theorists. Many of them will, however, teach mathematics at the high school or junior college level, and this book is intended for those students learning to teach, In addition to a careful presentation of the standard material usually taught in a first course in elementary number theory, this book includes a chapter on quadratic fields which the author has designed to make students think about some of the "obvious" concepts they have taken for granted earlier. | 677.169 | 1 |
A.P. Calculus (12) (ELCA)
Contents
This scope and sequence has been officially authorized by the College Board for use in my Calculus AB classroom and will provide the student with AP status.
The following is an outline of the topics covered in AP Calculus AB. An approximate number of days are allotted to the topics since the actual time varies from year to year depending upon the richness of class discussions that are generated.
Students use graphing calculators on a routine basis to explore, discover, and re-enforce the concepts of calculus. Students may use the graphing calculators on some, but not all assessments. It is stressed to students that they must know how to use a graphing calculator to do the following four required features: finding a root, sketching a function in a specified window, numerically calculating the derivative of a function at a value, and numerically calculating the value of a definite integral. Students are taught by the rule of four: analytically, graphically, numerically, and verbally. Communication in this course is important, as students are expected to give written justification for many of the processes used throughout the course. Students do best when they have an understanding of the conceptual nature of calculus, therefore stressing the reasons why behind the major ideas is essential. Quizzes are given routinely and students can expect a minimum of two tests every six weeks. AP questions and resources from previous years are used extensively throughout the course.
During this time we take a curve and point of tangency and use multiple delta x's close by and calculate the secant slopes and watch them become the tangent value.
Find derivative using limit of the difference quotient
Know the relationship between differentiability and continuity
Derivative at a Point
Power Rule, Product and Quotient Rules, Chain Rule
Slope of a curve at a point, Tangent and Normal lines to a curve at a point
Local linear approximation and differentials to estimate the tangent lines to a curve at a point
Instantaneous rate of change of a function from a graph or table of values
Instantaneous rates of change versus average rates of change from graphs or tables
Discussion on vertical tangent lines, cusps, and corners along with differentiability
Higher-Order Derivatives and Implicit Differentiation
Locations of vertical and horizontal tangent lines to curves
Derivative of a Function
Relate graph of function and derivative graph
Rolle's Theorem and Mean Value Theorem for Derivatives
Identification of critical numbers
Relationship between sign of derivative and increasing/decreasing nature of function
Find relative and absolute maxima and minima
Second Derivative
Relate graph of function, first derivative graph, and second derivative graph
Relationship between concavity and sign of second derivative
Find points of inflection
Applications of Derivatives (Six Weeks)
Curve Sketching
Sketch a curve using first and second derivatives, analyze critical points
Use points of inflections and concavity
Use asymptotes
Use symmetry
Optimization and Related Rates
Solve optimization problems and related rates problems
More applications of Derivatives
Solve Rectilinear Motion problems
L'Hopital's Rule
Newton's Method
Integral Calculus (Five Weeks)
Riemann Sums
Find the sum of a region using left, right, midpoint evaluations, and Trapezoidal Rule
During this time students sketch a region bounded by a nonnegative curve and use the regular partitions to fill the space as well as try to get as close to the actual value using any size and any number of rectangles (or other figures as well). I share with them the actual answer for comparison.
Summation formulas through the cubes
Using summation and limit process to generate an exact value for a sum
Discussion on the difference between the definite integral and indefinite integral
Integrate using Power Rule and U-Substitution
Find particular antiderivatives using initial conditions
Integration by Parts
Some Applications of Integral Calculus
Mean Value Theorem for Integrals
Average value of a function
Rectilinear Motion
Total distance versus displacement and the use of the integral
Average velocity using the integral as compared to change in position divided by change in time
Slope Fields
Visual interpretation of a differential equation
The TI-89 and presenter are used to observe various slope fields and particular solutions on the calculator. We discuss asymptotic behavior, how to sketch a slope field efficiently by hand, etc.
Discuss how to justify increasing/decreasing behavior as well as max/min and concavity at a point using the differential equation
Discussion of isoclines, nullclines, and equilibrium
Transcendental Functions (Five Weeks)
Differentiation and Integration of the Natural Logarithmic Function
Definition of natural log in terms of an integral function
Use slope field on TI-89 to tap into visual aspect
Discussion on domain and necessity of absolute value
Completion of the list of derivatives of all of the trig functions
Differentiation and Integration of the Exponential Functions
Base e
Bases other than e
Applications
Differential Equations
Slope Fields as visual interpretations of the differential equations
Separation of Variables
Growth and Decay Applications
Differentiation and Integration of Inverse Trigonometric Functions
Derivatives of Inverse Functions
From original function
Implicitly
From a table using formula
Applications of Integration (Three Weeks)
Area of a Region with finite boundaries
Volume
Find the volume of a solid with a known cross-section
Find the volume of a solid of revolution by the Disc and Washer Method as well as the Shell Method
More on the Integral as an accumulator
This schedule leaves 3 weeks per semester for flexibility with teaching, learning, and review of material. At numerous times during the course students are required to communicate verbally through written explanation as well as orally in class. Students are also given a 50-60 question take-home test each semester where they are encouraged to do it individually and then get together in a group and discuss the answers. This is a very effective tool used to increase their communication skills.
During the topics of slope fields and rectilinear motion I have to supplement the textbook heavily. I do so with worksheets of my own creation which are written specifically for the way I want to develop those topics and include links to other concepts as well.
To provide students with the academic rigor and college-level experience that is associated with an Advanced Placement course and to help students perform well on the AP EXAM while bringing glory to GOD.
The textbook is utilized as a resource tool but other rich AP resources will be used. I will refer to potential content on the AP EXAM through the use of past AP EXAMS as we progress through the concepts. The approach to problem-solving will involve the "rule of four" numerically, graphically, analytically, and verbally. Tests will vary in format and students will be expected to approach problems in various ways with/without the aid of a graphing calculator. There will be a minimum of two tests within a 6-week period.
It is imperative that students make the effort to understand the "why" behind the concepts because the AP rigor requires deep thinking and understanding. There will be times when rote skill is necessary but an emphasis on the conceptual nature of calculus will help the student comprehend the material thoroughly enough for college courses and the AP EXAM | 677.169 | 1 |
In sixth gradeCourse 2, students develop the concepts needed to prepare for Pre-Algebra. The Book 2 curriculum focuses on rational numbers, their operations, and their algebraic representations. It focuses on problem-solving situations and the use of estimation to check reasonableness.Measurement conversions, area, averages, data analysis, and data display are also stressed. There is focus on equations and functions, geometry, square roots, and probability. The curriculum also prepares students for standardized tests while meeting Pennsylvania State Standards and Guidelines.
Students enrolled in this course will receive a solid foundation in comparing, ordering, adding, subtracting, multiplying and dividing, decimals, fractions,and integers,including the necessary concepts needed to be successful in pre-algebra.A smaller class setting is utilized to provide more individualized instruction and remediation. Additional attention is given to assist students in the acquisition of more abstract topics. The purpose of this course is to meet the needs of students requiring more assistance.
Academic Course 2
Students enrolled in this course will receive a solid foundation in comparing, ordering, adding, subtracting, multiplying and dividing, decimals, fractions, and integers, including the necessary concepts needed to be successful in Pre-Algebra.
Advanced Course 2
This course is designed to lead into the advanced Pre-Algebra 1. Consequently, the course content will be substantial and more rigorous than the academic course.The pacing of the course will also be faster than the academic course.The homework assignments will be longer and involve additional problem-solving applications. There will also be additional content taught, (stem-and-leaf plots, box-and-whisker plots, graphing functions, slope, scale drawings & models, transformations in the coordinate plane, square roots, approximating square roots, the Pythagorean Theorem, classifying & sketching solids,finding surface area and volume of rectangular prisms and cylinders, permutations and combinations, disjoint events).Finally, assessments will be designed to measure students' mastery of the material at the synthesis and application levels.
Mathematics — Pre-Algebra, Grade 7
Full Year/Full Time
In seventh gradePre-Algebra, students develop the concepts needed to prepare for Algebra 1.The Pre-Algebra curriculum provides a strong foundation in algebra while also preparing students for future study in areas of geometry, probability, and data analysis.The curriculum has a strong focus on algebraic concepts and reasoning, solving of equations and inequalities, and geometry. The curriculum also prepares students for standardized tests while meeting Pennsylvania State Standards and Guidelines.
Specific topics of study will include: writing and evaluating algebraic expressions, simplifying expressions, writing, solving and graphing equations and inequalities, writing and solving proportions, simplifying ratios and finding probabilities, solving percent problems, applying rules of exponents and properties of square roots, writing and graphing linear functions and system of linear equations, identifying and applying properties of triangles and quadrilaterals, exploring right triangles using the Pythagorean Theorem, exploring angle relationships, similarity and congruence of geometric figures, finding perimeter, circumference, area, surface area and volume of geometric figures, finding the slope of a line, graphing various data displays, and evaluating formulas.
Essentials of Pre-Algebra
Students enrolled in this course will receive a solid foundation in algebraic concepts Pre-Algebra
Students enrolled in this course will be given a strong exposure to algebraic concepts.Emphasis will be given to topics that are considered to be the foundation for an academic Algebra 1 course.
Advanced Pre-Algebra
This course is designed to lead into the advanced Algebra 1 course and later into the honors and advanced placement sequence in mathematics at the high school level.Consequently, the course content will be substantial and more rigorous than the academic course.The pacing of the course will also be considerably faster than the academic course.The homework assignments will be longer and involve additional problem-solving applications. There will also be additional content taught (trigonometric ratios, distance and midpoint formulas, special right triangles, simplifying square roots, and polynomials).Finally, assessments will be designed to measure students' mastery of the material at the synthesis and application levels.
Mathematics, Grade 8
Full Year/Full Time
In eighth grade Algebra 1, the content of is organized around families of functions, with special emphasis on linear and quadratic functions.As students learn about each family of functions, they will learn to represent them in multiple ways – as verbal descriptions, equations, tables, and graphs.They will also learn to model real-life situations using functions in order to solve problems arising from those situations.In addition to its algebraic content, Algebra 1 includes lessons on probability and data analysis as well as numerous examples and exercises involving geometry.These math topics often appear on standardized tests, so maintaining students' familiarity with them is important.
Students enrolled in this course will receive a solid foundation in algebra Algebra 1
Students enrolled in this course will be given a strong exposure to algebra. Emphasis will be given to topics that are considered to be the foundation for an academic high school mathematics program.
Advanced Algebra 1
This course is designed to lead into the honors and advanced placement sequence in mathematics at the high school level.Consequently, the course content will be substantial and more rigorous.The pacing of the course will be considerably faster than the academic course.The homework assignments will be longer and involve additional problem-solving applications.Finally, assessments will be designed to measure students' mastery of the material at the synthesis and application levels. | 677.169 | 1 |
chief markers and experienced teachers, this book provides an authoritative but practical approach to revising for the KS3 National Tests. It is divided into short revision sessions organized by level so students can use the book wherever they are in their KS3 course.
Author Biography
Kevin Evans is Head of Mathematics at Abbey Grange C of E High School, Leeds. He is a Senior Moderator and Senior Examiner for GCSE Mathematics for a major examining board and a Senior Marker for Key stage 3 Mathematics. Keith Gordon is Head of Mathematics at Wath Comprehensive School, Rotherham and is Principal Examiner for GCSE Mathematics for a major examining board. | 677.169 | 1 |
Algebra Regents Prep
Use these free video lessons and practice tests to get ready for your Algebra Regents exam. Check also other Regents prep lessons.
Algebra 1
The NYS Regents Algebra exam consists of four sections with, in total, 37 questions. All questions in this exam must be answered.Algebra Part I – This section contains 24 multiple choice questions. For each question or statement you need to choose best answer or statement. Write your answers on the separate answer sheet that is provided.
For these sections you must clearly indicate all necessary steps, including diagrams, appropriate formula substitutions, charts, graphs, and so on. You should use the provided information on each question to come up with your answer.
The formulas you could need for answering some of the questions in this exam can be found at a perforated sheet at the end of the exam. Remove it from the booklet and use it when needed.
What does the Algebra Regents exam include?
Algebra 1 is actually the 2nd math course at high school and in this part you will be led through things like systems of equations, expressions, functions, inequalities, real numbers, exponents, radical & rational expressions, and polynomials.
Algebra 1 is primarily teaching students how to consider real world situations in a mathematical way. Word problems are making up a large portion of this section's curriculum, and almost every unit includes a form of word problem interpretation. Exponents, something already taught before Algebra 1, are now reviewed and getting applied to various equations. Students are taught exponents' mathematical laws, and they are now used in exponential equations. The concepts of zero and negative exponents are also explained and applied.
Algebra 1 is actually the first math class where the concepts of scientific notations as substitute formats for very small or very large numbers are explained and applied. During this part, also the Pythagorean theorem is explained and applied. This provides the mathematical basis for various geometrical concepts that will be taught in later classes. Most curricula also include probability and statistics as supplemental units, but these subjects are usually only addressed briefly.
Algebra Regents Exam – How to use an algebra formula sheet?
You can use an algebra formula sheet that's provided in the booklet. The formula sheet will be helpful to remind you of the most common algebraic formulas, for example the point-slope form of a line. You can use these while you're dealing with the given math problems.
You can best use a formula sheet is when you write as concisely and clearly as possible, and when you're only including the least possible information to be reminded of the equations and formulas that you need to look up to deal with your exam.
To see what information your formula sheet in including, go through a few practice tests and check also the section of your text book that deals with the exam itself. Try to learn all the formulas and equations that are also used in your text book's sample questions, and also those formulas that you need to use for other practice tests. This will save you time on the actual test.
There are times and situations when you are required to memorize the equations and formulas, while at other times you are allowed to use a formula sheet during a test. If you need to memorize a lot of things, you best do so by writing a lot of problems and solutions by hand, and make sure that you fully write out the equations and formulas, as well as every step you took to get to solution, and that each time! If you're in a situation that you're left with just only limited time to learn, and if you're allowed to use a formula sheet at the exam, it could be very helpful to include a few examples of how you solved some problems. | 677.169 | 1 |
Algebra II is the crucial high school mathematics course in preparing for a college education. College students who have not mastered Algebra II find themselves taking a remedial, noncredit course called Intermediate Algebra. If you are struggling with Algebra II or are preparing to enter college or are currently in college and are struggling with Intermediate Algebra, then this Algebra II video series is for you. Or if you are an adult who has always wanted to master the beauty of mathematics and now wants the opportunity to do so, this course can fulfill that dream.
A Man with a Plan
Dr. Murray H. Siegel earned his Ph.D. in Mathematics Education. Kentucky Educational Television named him "The Best Math Teacher in America," and he has received the Presidential Award for Excellence in Math Teaching. He has been teaching mathematics for decades to students of all ages, and his methods have been tested and proven in real classrooms.
This course is designed, in its sequencing of lessons and in each subject taught, to produce true understanding—the kind of confident knowledge that eliminates the anxiety many have in mathematics.
In this course, Dr. Siegel not only explains the key concepts and methods of Algebra II, he also takes you "behind the scenes." The 30 half-hour lessons show both how as well as why the methods work, using real-world applications to answer the age-old question in mathematics classes: When will I ever use this? With this series you learn Algebra II in a way you have never learned it before.
The series begins by relating the foundation of algebra—polynomials—to the numbers with which you are already familiar. You learn to operate with them, graph them, and solve polynomial equations.
The course then progresses to another essential algebraic concept—the function. Functions of all types are investigated: linear, polynomial (including quadratic), rational, and recursive.
Even the aspects of Algebra II that are usually considered difficult to teach are treated in a user-friendly manner so that all learners can truly understand these topics. Included in this list are matrices and determinants, imaginary numbers, sequences and series, and logarithms. (Yes, with Dr. Siegel as your guide, even logarithms will make sense!)
The final part of the course includes an introductory lesson in Trigonometry.
One important difference not found in most mathematics lessons is that serious learning is punctuated with humor. Dr. Siegel is not a stand-up comedian (although he was a radio talk show host long ago), but his students have never found him boring.
If you want to succeed in mathematics and have concerns that you can be successful in an Algebra course, then this course is for you. Your efforts will be rewarded and you will thank Dr. Siegel for opening the door to mathematical knowledge and understanding. [свернуть] The Teaching Company The Teaching Company was founded in 1990 by Thomas M. Rollins, former Chief Counsel of the United States Senate Committee on Labor and Human Resources.
Years earlier, as a Harvard Law School student, Rollins had an unforgettable experience that opened his eyes to the extraordinary power of a great lecturer captured on tape.
Rollins was facing an important exam in the Federal Rules of Evidence but was not well prepared. He managed to obtain videotapes of 10 one-hour lectures by a noted authority on the subject, Professor Irving Younger.
"I dreaded what seemed certain to be boring," Rollins says. "I thought that few subjects could be as dull as the Federal Rules of Evidence. But I had no other way out."
Rollins planted himself in front of the TV and played all 10 hours nearly non-stop. The lectures, he says, "were outrageously insightful, funny, and thorough." Watching Professor Younger's lectures was one of Rollins's best experiences as a student.
Rollins made an "A" in the course. And he never forgot the unique power of recorded lectures by a great teacher.
After many years of government service, Rollins founded The Teaching Company in 1990 to ignite people's passion for lifelong learning by offering great courses taught by great professors. | 677.169 | 1 |
Fast Track
Fast Track Math Assessment Prep Workshops
These 8-day workshops are designed to provide students with an intensive review of
elementary and intermediate algebra. Instruction on important algebra topics, one-on-one
help, and workbooks with extra practice are all provided. These workshops prepare
qualified students to potentially place into a higher-level math course, getting them
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books.google.com problems deducible from the first six books of Euclid, arranged and solved. To which is added, an appendix containing the elements of plane trigonometry | 677.169 | 1 |
Differential Equations and Linear a mathematician/engineer/scientist author who brings all three perspectives to the book. This volume offers an extremely easy-to-read and easy-to-comprehend exploration of both ordinary differential equations and linear algebra--motivatedMore...
Written by a mathematician/engineer/scientist author who brings all three perspectives to the book. This volume offers an extremely easy-to-read and easy-to-comprehend exploration of both ordinary differential equations and linear algebra--motivated throughout by high-quality applications to science and engineering. Features many optional sections and subsections that allow topics to be covered comprehensively, moderately, or minimally, and includes supplemental coverage of Maple at the end of most sections. For anyone interested in Differential Equations and Linear Algebra.
Preface
Introduction to Differential Equations
Definitions and Terminology
Linear First-Order Equations
Homogeneous Case
Solution by Separation of Variables
Nonhomogeneous Case
Applications of Linear First-Order Equations
General First-Order Equations
Separable Equations
Existence and Uniqueness
Exact Equations and Ones That Can Be Made Exact
Additional Applications
Linear and Nonlinear Equations Contrasted
Vectors and n-Space
Geometrical Representation of Arrow Vectors n-Space
Dot Product, Norm, and Angle for n-Space
Gauss Elimination
Span
Linear Dependence and Independence
Vector Space
Bases and Expansions
Matrices and Linear Algebraic Equations
Matrices and Matrix Algebra
The Transpose Matrix
Determinants
The Rank of a Matrix
Inverse Matrix, Cramer's Rule, and Factorization
Existence and Uniqueness for the System Ax= c
Vector Transformation (Optional)
Linear Differential Equations of Second Order and Higher
The Complex Plane and the Exponential, Trigonometric, and Hyperbolic Functions | 677.169 | 1 |
This textbook is for the standard, one-semester, junior-senior course that often goes by the title "Elementary Partial Differential Equations" or "Boundary Value Problems". The audience consists of students in mathematics, engineering, and the physical sciences. | 677.169 | 1 |
97804864427 Study and Difficulties of Mathematics (Dover Books on Mathematics)
One of the twentieth century's most eminent mathematical writers, Augustus De Morgan enriched his expositions with insights from history and psychology. On the Study and Difficulties of Mathematics represents some of his best work, containing points usually overlooked by elementary treatises, and written in a fresh and natural tone that provides a refreshing contrast to the mechanical character of common textbooks. Presuming only a knowledge of the rules of algebra and Euclidean theorems, De Morgan begins with some introductory remarks on the nature and objects of mathematics. He discusses the concept of arithmetical notion and its elementary rules, including arithmetical reactions and decimal fractions. Moving on to algebra, he reviews the elementary principles, examines equations of the first and second degree, and surveys roots and logarithms. De Morgan's book concludes with an exploration of geometrical reasoning that encompasses the formulation and use of axioms, the role of proportion, and the application of algebra to the measurement of lines, angles, the proportion of figures, and surfaces | 677.169 | 1 |
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Summary
For introductory courses in Differential Equations. This text provides the conceptual development and geometric visualization of a modern differential equations course while maintaining the solid foundation of algebraic techniques that are still essential to science and engineering students. It reflects the new excitement in differential equations as the availability of technical computing environments likeMaple, Mathematica, and MATLAB reshape the role and applications of the discipline. New technology has motivated a shift in emphasis from traditional, manual methods to both qualitative and computer-based methods that render accessible a wider range of realistic applications. With this in mind, the text augments core skills with conceptual perspectives that students will need for the effective use of differential equations in their subsequent work and study. | 677.169 | 1 |
Matrix Techniques, Trigonometry, and Analytic Geometry
Download a free gameboard of math jeopardy! You do not have to give it angles to search within, or number of solutions you want. And I want to find the area of this, for a and b, in terms of a and b. Interactive College Algebra course designed to ensure engaging, self-paced, and self-controlled e-learning process and help students to excel in their classes. Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student.
Basic algebra for kids online tutorial, quickest way to factor a number, math trivia questions for second year, answers to all math answers solving linear equations by using substitution, physics book problem answers, inequalities worksheet A Drill-Book In Trigonometry. And you discover that these techniques are ultimately connected in a beautiful way. Perceiving these connections helps you choose the best tool for a given problem: Algebraic functions: Including polynomial functions and rational functions, these equations relate the input value of a variable to a single output value, corresponding to countless everyday situations in which one event depends on another Study Guide With Practice Tests For Gustafson and Frisk's Intermediate Algebra 5th. Practise drawing that part for each of the curves, marking in -1 and 1 on the y-axis and 0, 90, 180, 270 and 360 on the x-axis. Now that you know what y = sin x° and y = cos x° look like, suppose you were asked to draw graphs for: COM, program ti 84 for radical equations, how to solve log base 2 in ti 89, free fourth grade math papers with reducing fractions, CONVERT FROM POLAR TO RECTANGULAR FORM/TI-83, how do you graph the equation x squared + 5 = 0, rational expression math games. Ti-84 radical equations, how to convert 7/20 into a decimal, Factor+trees+elementary+worksheets, The difference of a square, trinomial calculator free, problems for transforming formulas, free worksheets on solving problems with inequalities Algebra: Structure and Method (Book 1).
Student's Solutions Manual for Trigonometry: A Right Triangle Approach
Students apply their understanding of these concepts to solve problems and communicate about mathematics in their world A Treatise on Spherical Trigonometry: And Its Application to Geodesy and Astronomy, with Numerous Ex. Adding subtracting multiplying dividing integers worksheets, graphing square root picture on ti 83, worksheets for graphing systems of equations, answers for prentice hall algebra 1 Natural Trigonometric Functions to Seven Decimal Places for Every Ten Seconds of Arc Together with Miscellaneous Tables (Second Edition, Ninth Printing). Babylonian and Egyptian astronomers were able to measure the altitude and lateral displacement of heavenly objects from a particular direction by using a Merkhet, thus giving the earliest ideas of turning, or angle. A pair of merkhets were used to establish a North-South direction by lining them up, one behind the other, with the Pole Star Elements of Trigonometry, and Trigonometrical Analysis, Preliminary to the .... The Mathematics Teacher. 87, no. 5, (May 1994): 372-375. A 3-day introductory trigonometry … this discussion of accuracy to motivate the study of trigonometry. By using the trigonometry we can calculate results with far more accuracy … Who am I? Find A Polynomial From Its Roots - Exponential Growth and Decay - … Trigonometry 12 Mandatory Package College Algebra with Trigonometry with Smart CD (MAC). The Millennium Mathematics Project (MMP) is a maths education and outreach initiative for ages 3 to 19 and the general public pdf. Finding the square - easy, differential equation calculator online, calculator to divide polynomials, decimal to mixed fraction. Problem solving in mathematics-simplifying a problem, permutation and combination for dummies, kirchhoff calculator equation. Steps to solve by graphing, english aptitude questions, trivia in polynomial function, algebra 2 answers vertex formula, interactive pre-algebra simplifying square roots, intermediate algebra homework help, prime factorization life example Plane Trigonometry and Complex Numbers.
Algebra and Trigonometry
Plane and spherical trigonometry in three parts
Modern Trigonometry
Schaum's Outline of Theory and Problems of Continuum Mechanics
College algebra and trigonometry
Geometrical problems deducible from the first six books of Euclid, arranged and solved. To which is added, an appendix containing the elements of plane trigonometry
Trigonometry: With The Theory And Use Of Logarithms (1914)
College trigonometry
Logarithmic and Trigonometric Tables
Prep-Course: Trigonometry: A general review on Algebra and an overview of what is most important to retain from Trigonometry in order to be successful in future courses. (Volume 2)
Elements of Geometry and Trigonometry, from the Works of A.M. Legendre
Algebra, Geometry and Trigonometry in Science, Engineering and Mathematics (Mathematics and its Applications)
Plane And Spherical Trigonometry
Spherical trigonometry, for colleges and secondary schools
An Elementary Treatise On Plane & Spherical Trigonometry: With Their Applications to Navigation, Surveying, Heights, and Distances, and Spherical ... of Bowditch's Navigator, and the Nautical
Introduction to algebraic expressions and equations, rational expressions, exponents, functions and their graphs, trigonometric functions on the unit circle, system of equations and their associated matrices Insider's Guide to Teaching With the Hornsby/Lial/Rockswold Graphical Approach Series for A Graphical Approach to College Algebra Fifth Edition, Algebra and Trigonometry Fifth Edition, Precalculus With Limits A Unit Circle Approach Fifth Editio. Animation shows what happens to perimeters, areas, and volumes under scaling, with various applications from real life The Theorem of Pythagoras Several engaging animated proofs of the Pythagorean theorem are presented, with applications to real-life problems and to Pythagorean triples. The theorem is extended to 3-space, but does not hold for spherical triangles A Drill-Book In Trigonometry. Kansas State University Published in 2011, 435 pages Published in 1913, 364 pages Published in 1902, 348 pages John C. Air Force Publication Published in 2006, 58 pages Curtis T. Princeton University Press Published in 1994, 219 pages Lewis Carroll Published in 1885, 192 pages Pete L. University of Georgia Published in 2011, 264 pages Henri Darmon, Shou-Wu Zhang The student applies the mathematical process standards and algebraic methods to write, solve, analyze, and evaluate equations, relations, and functions Matrix Techniques, Trigonometry, and Analytic Geometry online. In other words, the curve represents an ellipse with its centre at the point (3, 1) in the original coordinate system Trigonometry: Revised Third Edition by Baley, John, Sarell, Gary [McGraw-Hill Science/Engineering/Math,2002] (Paperback) 3rd edition [Paperback]. Below are a number of worksheets covering trigonometry problems. Trigonometry, at it's most basic level, is concerned with the measurement of triangles - calculations of unknown lengths and angles. The trigometric functions have a number of practical applications in real life and also help in the solutions of problems in many branches of mathematics By Ron Larson - Trigonometry Advanced Placement: 7th (seventh) Edition. From our calculator we find that tan 60° is 1.733, so we can write which comes out to 26, which matches the figure above Algebra and Trigonometry 4th (Fourth) Edition byBittinger. Early History of Mathematics This video traces some of the landmark developments in the early history of mathematics, from Babylonian calendars on clay tablets produced 5000 years ago, to the introduction of calculus in the seventeenth century. Teachers Workshop This 28-minute tape contains excerpts from a two-day workshop held in 1991 for teachers who have successfully used project materials in their classrooms St Sol Ml/Trigonometry. I wish every math or science book was written the way this one is. The writers have not over complicated the instructions and the examples in the book actually help you with the problems that are at the end of each section Standard Field tables and Trigonometric Formulas. To see some of the other products available on zazzle.com, check out our main trigonometry page. The Trigonometry examination covers material usually taught in a one-semester college course in trigonometry with primary emphasis on analytical trigonometry online. It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE. [68] Mahavira Acharya (c. 800–870) from Karnataka, the last of the notable Jain mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha College Mathematics, Basic Course in Essentials of Algebra, Trigonometry, a Nalytic Geometry, Calculus With Exercises, Examinati. The lower section contains pictures of star gods or demons. They represent some of the most important days of the year. The chart is largely symbolic and functional but does contain pictures of some significant groups of stars. Observations of celestial bodies by the Babylonians from about 1,800 BCE gave rise to the eventual division of the circle into 360 degrees, and by about 500 BCE, the division of the heavens into twelve regions of 30 degrees each, often referred to as the 12 houses of the zodiac download Matrix Techniques, Trigonometry, and Analytic Geometry pdf. | 677.169 | 1 |
PRODUCT DESCRIPTION
This activity is designed for Calculus 1, AP Calculus, and Calculus Honors. It is part of Unit 4, Integration.
Included in the Lesson:
•Task Cards: There are 20 Task Cards which increase in difficulty numerically. Included types are arctan, arcsin, and arcsec. #1- 14 are indefinite integrals; 15 and 16 are definite integrals; and 17 – 20 are applications. Students should not need a calculator for this activity.
•Master List of Questions which can also be used as an assessment or class worksheet
•Student response sheet
•Answer keys
• Additional Handout with ten questions and room for students to show work. The handout can be used as enrichment, homework, group work, or as an assessment.
•Graphic Organizer with section formulas. For my classes I printed this on neon paper, cut them up, and had my students glue it into their notebooks for easy reference.
Please look at my very popular •Ultimate Calculus Survival Kit, an incredible 70 page resource for PreCalculus and Calculus teachers, and a great value. It includes the derivative and Integration Flash Card kits, and a "cheat sheet" just for this unit.
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A
QUICKRECAP
B
SOME OTHER IMPORTANT DEFINITIONS!
IDENTITY MATRIX: Any square matrix that has 1's on its main
diagonal and 0's elsewhere.
INVERSE MATRIX: If A is the inverse of B,
then AB = BA = identity matrix!
Verify that A and B are inverses!
FINDING PO
TASKSTODAY
*StartofChapter3
*MatricesIntroduction:Adding,Subtracting,
Multiplying,Determinants,InversesusingtheTI
Nspires
Tests are not ready.make sure to come talk to
me/Mr.Rodrigo if you need to make up this test.
Turned in:
*BLUE REVIEW PACKET
*HW 5.5B
HW 5.5B Questions: p268 #8, 11-20
5.6:BinaryTrees,ExpressionTrees,andTraversals
Rooted Trees: A directed tree in which every vertex except the
root has an indegree of 1, while the root has an indegree of 0.
(NOTE: Since all edges are directed away from th
WARM-UP (Due in 10 min)
WithONEpartnerandONEsheetofpaper!
COMPLETE#1aeonp244
And NO - they are not ready!
SMARTDocumentCamera
EnsurethataSMARTDocumentCameraisconnectedandisn'tinuseinan
otherapplication.
Just to reiterate why brute force is not the best fo | 677.169 | 1 |
Linear Algebra
Linear Algebra is a branch of mathematics that seeks to describe lines and planes using structures like vectors and matrices.
Vector
Vectors in geometry are 1-dimensional arrays of numbers or functions used to operate on points on a line or plane. Vectors store the magnitude and direction of a potential change to a point. A vector with more than one dimension is called a matrix.
Vector Notation
There are a variety of ways to represent vectors:
Or more simply:
v→=(v1,v2){\displaystyle {\vec {v}}=(v1,v2)}
How Vectors Work
Vectors typically represent movement from a point. They store both the magnitude and direction of potential changes to a point. This vector says move left 2 units and up 5 units.
v→=(−2,5){\displaystyle {\vec {v}}=(-2,5)}
A vector can be applied to any point on a plane:
The vector's direction equals the slope created by moving up 5 and left 2. Its magnitude equals the length of the hypotenuse (the long side in a right angle triangle).
Unit Vectors
A unit vector is a vector with a magnitude of 1. Unit vectors can have any slope (move in any direction), but the magnitude (length of vector) must equal 1. They are useful when you care only about the direction of the change, and not the magnitude. Unit vectors are used by directional derivatives.
Example
Given the vector (3, 4), the magnitude is 5 (hypotenuse). This is not a unit vector. To find the unit vector of this vector, we divide each value by the magnitude of the vector. The new vector (3/5, 4/5) points in the same direction and has a magnitude of 1.
Vector Fields
A vector field is a diagram that shows how for a given point in space (a,b), where that point would move in your if you apply a vector function to it. Given a point, the vector field shows the "power" and "direction" of our vector function. Here is an example vector field:
The difference is because the second vector contains only scalar numbers. So from any point we always move over 2 and up 5. The first vector on the other hand contains functions. So for each point, we derive the direction by inputing the coordinates into a function. For non-linear functions, things can become very fancy indeed.
Matrix
A matrix is a rectangular grid of numbers. Like an Excel spreadsheet. We describe the dimensions of a matrix as Rows x Columns. There are a variety of matrix operations, but we will focus on multiplication as it is the most relevant to deep learning.
Matrix Multiplication
Matrix multiplication specifies a set of rules for multiplying matrices to produce a new matrix.
Why is it Useful?
It turns complicated problems into simple, more efficiently calculated problems. It's used in a number of fields including machine learning, computer graphics, and population ecology.
Source
Rules
Not all matrices are eligible for multiplication. In addition, there is a requirement on the dimensions of the matrix product. Source.
The number of columns in the first matrix must equal the number of rows in the second
The product of an M x N matrix and an N x K matrix is an M x K matrix. The new matrix takes the rows of M1 and columns of M2.
Steps
Matrix multiplication uses Dot Product to multiply various combinations of rows and columns to derive its product. In the image below, each entry in Matrix C is the dot product of a row in matrix A and a column in matrix B.
In the image above, a1{\displaystyle a_{1}} represents the vector (1,7){\displaystyle (1,7)} and b1{\displaystyle b_{1}} represents the vector (3,5){\displaystyle (3,5)}. When we see a1⋅b1{\displaystyle a_{1}\cdot b_{1}}, it really means we take the dot product of the first row in matrix A and the first column in matrix B. | 677.169 | 1 |
Move the Monster - Exercises in Graph Transformations STUDENT EDITION
305 Downloads
Presentation (Powerpoint) File
Be sure that you have an application to open this file type before downloading and/or purchasing.
1.1 MB | 22ish pages
PRODUCT DESCRIPTION
This is the student edition of my Move the Monster Presentation where I have removed the solution slides and notes from the slides so it is ready to push out to students or you could print these as handouts and have students work through them or follow along as you move through the Move the Monster presentation.
**UPDATE** I changed the tables to be in ordered pair notation. Previously, students were having trouble understanding that the tables were tables of points.
**UPDATE** I added a blank, "Classify the Transformation:" so students can practice connecting words like stretch, reflect, vertical, horizontal, shift, etc. to the actions of the graph. Also, I added a blank for identifying the domain and range of each function.
The first slide contains one graph of a monster (f(x)) and a table of values for points on that monster: (x,f(x)) On each succeeding slide, the table appears with the values you (hopefully) discovered with your students for x and f(x) and a new column for the transformation function. All the basic transformations are covered, including key slides that have the monster already transformed and values entered into the | 677.169 | 1 |
SMP 11-16 Teacher's Guide to Books A4, A5, and A6 (School
This decomposition is also known as expanded notation and would look like this: 7,524 could be expanded and written as [7,000 + 500 + 20 + 4]. Plus is an internet magazine which is free for educational and non-commercial purposes. The wealth and depth of resources provided support the realization of West Virginia's goal to prepare students to be successful in tomorrow's world. The video and PowerPoint weren't great, but the dividing idea seemed to make sense to them.
Pages: 112
Publisher: Cambridge University Press (October 3, 1996)
ISBN: 052147552X
Mathematical Structure (School Mathematics Project 16-19)
Note that the student s do not receive partial credit for partially correct responses on mastery-based assessments. • The course is taught in a self-paced lab format. Students cannot continue to the next module in the course until they pass the previous module with an 80% on the end-of-module assessment SMP 11-16 Teacher's Guide to Books A4, A5, and A6 (School Mathematics Project 11-16) online. The print textbooks teach each lesson clearly and succinctly. But if anything is unclear, or if you think you might like some dynamic visuals and auditory input to help you understand better: slide in the CD, watch and listen. And no one will think the worse of you if you replay a portion of a lecture or study a problem solution several times over Gcse Mathematics for Ocr Modular Two Tier Gcse M5 Homework Book. Calculus requires a solid foundation in mathematics for students to grasp the various concepts Math Expressions: Teacher Edition, Level 1, Vol 1 2009. Research shows that some conduct disorders, such as EBD, can be linked to poor reading performance, although the causal relationship has not been determined. More Sing Along and Learn: Early Math: Easy Learning Songs and Instant Activities That Teach Key Math Skills and Concepts. Teaching with Data Simulations: Teaching with data simulations means giving students opportunities to simulate data in order to answer a particular research question or solve a statistical problem. Compiled by Danielle Dupuis and Joan Garfield of the University of Minnesota. Teaching With Equations: Teaching students to make connections between physical world, words and equations is a fundamental skill for geoscientists Basic Math G.A.M.E.S., Grade K: Games, Activities, and More to Educate Students.
Using a fictional Grade 4 classroom as the setting for this example, you are provided with a framework of the RTI identification process, along with frequent opportunities to check your comprehension of the information presented Mathematics Education: Models and Processes. For others, the process is fraught with frustration and failure. No responsible adult can turn a deaf ear to con- cerns about how reading is taught, especiallyconcerns that some teaching methods increase the chances that children will fail to become competent readers in the early elementary years. But the proper place for these discussions is at meetings of boards of education, in school administrative offices, in classrooms with principals or teachers, and especially in the colleges and universities where future teachers are being trained Great Source Mathstart: Teacher's Guide Game Time! 2001.
Best Practices for Teaching Mathematics: What Award-Winning Classroom Teachers Do
Basic Mathematics: Addition and Subtractions of Fractional Numbers and Multiplication and Division of Fractional Numbers
The ladder logic instructions are commonly called AND, ANDA, ANDW, OR, ORA, ORW, XOR, EORA XORW. As we saw with the MOV instruction there are generally two common methods used by the majority of plc makers. The first method includes a single instruction that asks us for a few key pieces of information pdf. In recent years, topics have included Fourier analysis in Euclidean spaces, groups, and symmetric spaces. May be repeated for credit with consent of adviser as topics vary. Prerequisites: Math 240C, students who have not completed Math 240C may enroll with consent of instructor. Prerequisites: graduate standing or consent of instructor. (S/U grades only.) Convex sets and functions, convex and affine hulls, relative interior, closure, and continuity, recession and existence of optimal solutions, saddle point and min-max theory, subgradients and subdifferentials Little Books 1 2 3. Eventually, you will become a professional in an honored profession. As a professional you will be engaged in a lifelong process of learning on the job, maintaining and increasing your professional competence. In this process you will make decisions for yourself about what you need/want to learn, how you will go about gaining relevant knowledge and skills, and how you will use your new knowledge and skills download SMP 11-16 Teacher's Guide to Books A4, A5, and A6 (School Mathematics Project 11-16) pdf. They were able to participate far more than before and learned more than ever before Keys to Math Success, Grades K - 1: "FUN" Standard-Based Activities to Boost the Math Skills of Struggling and Reluctant Learners. Manipulative materials must be selected that are appropriate for the developmental level of the students. Encourage students to share what they have learned orally or in written form. Most students benefit from presenting this explanation as it helps organize their thoughts. You can also assess the use of the manipulative based on the student's explanation Teaching Secondary Mathematics. Rationale: For multiple-choice items, this section provides the correct option and demonstrates one method for arriving at that response. For constructed response items, one possible approach to solving the item is shown followed by the scoring rubric that is specific to the item WordsWorth Times (Play Together, Learn Together). Of course students differences in many socio-economic, cultural and language area are important and problematic Advanced Mathematical Thinking: A Special Issue of Mathematical Thinking and Learning (Special Issue of Mathematical Thinking & Learning S). CT4ME's section on Professional Development includes a variety of resources to assist you with becoming more knowledgeable about the mathematics content you teach and how to enhance your teaching skills. Our section on Standardized Test Preparation provides solid advice and resources HSP Math: Time-Saver Lesson Resources with Resource Management System Grade 4. Check out local homeschool co-ops and tutoring centers such as the one I teach at in Tulsa, OK, called Cornerstone Tutorial Center English for the English: a Chapter on National Education / by George Sampson. Sixty-three students in the TAI classes and 50 in the control classes were receiving special education services and were designated as being academically handicapped. The outcomes studied were the Mathematics Computation and Concepts & Applications subscales of the CTBS The Complete Book of Numbers and Counting (The Complete Book Series). Here are some unsolicited comments: I ordered your book in June and it has been a real benefit for the time I invested studying it. I am now teaching precalculus (both plain jane and goosed-up versions) and calculus and I am able to use more precise language and to express concepts that I otherwise would have used improvised language. The $42 price of your self-published text was a real investment Math Games, Grade 7. | 677.169 | 1 |
Summary and Info
This book has evolved over many years from lecture notes that accompany cer- certain upper-division courses in mathematics and computer sciences at our university. These courses introduce students to the algorithms and methods that are commonly needed in scientific computing. The mathematical underpinnings of these methods are emphasized as much as their algorithmic aspects. The students have been diverse: mathematics, engineering, science, and computer science undergraduates, as well as graduate students from various disciplines. Portions of the book also have been used to lay the groundwork in several graduate courses devoted to special topics in numerical analysis, such as the numerical solution of differential equations, numerical linear algebra, and approximation theory. Our approach has always been to treat the subject from a mathematical point of view, with attention given to its rich offering of theorems, proofs, and interesting ideas. From these arise many computational procedures and intriguing questions of computer science. Of course, our motivation comes from the practical world of scientific computing, which dictates the choice of topics and the manner of treating each. For example, with some topics it is more instructive to discuss the theoretical foundations of the subject and not attempt to analyze algorithms in detail. In other cases, the reverse is true, and the students learn much from programming simple algorithms themselves and experimenting with them-although we offer a blanket admonishment to use well-tested software, such as from program libraries, on problems arising from applications.
More About the Author
Dave Kincaid (born March 21, 1957) co-founded the New York band The Brandos with Ernie Mendillo in 1985. Besides playing with The Brandos, Kincaid has also released two albums of Irish music under the name David Kincaid. | 677.169 | 1 |
Item is available through our marketplace sellers and in stores.
Overview
Developed for test-takers who need a refresher, Foundations of Math provides a user-friendly review of basic math concepts crucial for GMAT success.
Manhattan GMAT''s Foundations of Math book provides a refresher of the basic math concepts tested on the GMAT. Designed to be user-friendly for all students, this book provides easy-to-follow explanations of fundamental math concepts and step-by-step application of these concepts to example problems. With ten chapters and over 700 practice problems, this book is an invaluable resource to any student who wants to cement their understanding and build their basic math skills for the GMAT. Purchase of this book includes one year of online access to the Foundations of Math Homework Banks consisting of over 400 extra practice questions and detailed explanations not included in the book.
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Meet the Author
In 2000, Teach for America alumnus and Yale graduate Zeke Vanderhoek had a radical idea: students learn better from better teachers. His vision of what test prep could be if written and taught by great educators led him to start Manhattan Prep. Since we began, Manhattan Prep has grown from a boutique tutoring company to one of the world's leading test prep providers, offering GMAT, GRE, LSAT, ACT, and SAT courses and tutoring worldwide.
We believe test prep should be real education. From our instructors to our materials, we work to teach you the skills you'll need to succeed on the test, in school, and beyond. | 677.169 | 1 |
Algebra II is the crucial high school mathematics course in preparing for a college education. College students who have not mastered Algebra II find themselves taking a remedial, noncredit course called Intermediate Algebra. If you are struggling with Algebra II or are preparing to enter college or are currently in college and are struggling with Intermediate Algebra, then this Algebra II video series is for you.
The Algebra 2 Tutor is a 6 hour course spread over 2 DVD disks that will aid the student in the core topics of Algebra 2. This DVD bridges the gap between Algebra 1 and Trigonometry, and contains essential material to do well in advanced mathematics. Many of the topics in contained in this DVD series are used in other Math courses, such as writing equations of lines, graphing equations, and solving systems of equations | 677.169 | 1 |
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E-book, Lesson Plan
Description:
This lesson develops an intuitive and reason-based approach to the method of solving multi-step linear equations using the additive and multiplicative properties of equality.
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S2486839: Texas Essential Knowledge and Skills for Mathematics
model and solve one-variable, one-step equations and inequalities that represent problems, including geometric concepts10-02.
Component Ratings:
Technical Completeness: 3 Content Accuracy: 3 Appropriate Pedagogy: 3
Reviewer Comments:
A fine lesson on the uses of the distributive property in algebra and in real life.
Not Rated Yet.
Summary:
If the skill of multi-step equations is difficult then perhaps multiple days worth of class are merited in practicing this all-important skill in Algebra 1.
Essential Question(s):
How do you solve a multi-step linear equation?
What four consecutive numbers have a sum of 2010?
How are equations like onions?
When will the water level be back down to normal? | 677.169 | 1 |
When Topology Meets Chemistry by Erica Flapan
Book Description
The applications of topological techniques for understanding molecular structures have become increasingly important over the past thirty years. In this topology text, the reader will learn about knot theory, 3-dimensional manifolds, and the topology of embedded graphs, while learning the role these play in understanding molecular structures. Most of the results that are described in the text are motivated by questions asked by chemists or molecular biologists, though the results themselves often go beyond answering the original question asked. There is no specific mathematical or chemical prerequisite; all the relevant background is provided. The text is enhanced by nearly 200 illustrations and more than 100 exercises. Reading this fascinating book, undergraduate mathematics students can escape the world of pure abstract theory and enter that of real molecules, while chemists and biologists will find simple, clear but rigorous definitions of mathematical concepts they handle intuitively in their work.
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Books By Author Erica Flapan
Number Theory: A Lively Introduction with Proofs, Applications,and Stories, is a new book that provides a rigorous yetaccessible introduction to elementary number theory along withrelevant applications. Readable discussions motivate new concepts and theorems beforetheir formal definitions and statements | 677.169 | 1 |
The majority of students who take courses in number theory are mathematics majors who will not become number theorists. Many of them will, however, teach mathematics at the high school or junior college level, and this book is intended for those students learning to teach, In addition to a careful presentation of the standard material usually taught in a first course in elementary number theory, this book includes a chapter on quadratic fields which the author has designed to make students think about some of the "obvious" concepts they have taken for granted earlier. The book also includes a large number of exercises, many of which are nonstandard | 677.169 | 1 |
Step inside any high-school math class in the United States, and chances are you'll find students staring down at their Texas Instrument calculators, nimbly typing commands into those $100 pocket computers. Calculators are so commonplace in modern American education that a TI-84 or -89 can be found stashed away in many homes, mementos from taking the SAT or computing integrals on the Advanced Placement calculus exam.
Still, college professors remain divided on the use of calculators in their classes. When I took my freshman math courses at McGill University in Montreal last school year, I had to revert back to pencil and paper, clumsily lining up columns to do addition and long-multiplication problems at my professor's request. This isn't an unusual predicament: According to a 2010 national survey by the Mathematical Association of America, nearly half of Calculus 1 college instructors prohibit students from using graphing calculators on exams.
The debate over the use of calculators in math classrooms has ensued for more than four decades—nearly as long as the contemporary calculator has been around. Although the abacus has been in use since the time of the Sumerians and Ancient Egyptians, it wasn't until 1958 when the Texas Instrument engineer Jack Kilby invented the integrated circuit, which paved the way for the cheap and compact computer chips used in most electronic devices today. (Kilby later won the Nobel Prize in physics.) A decade later, calculators would no longer be stored in gigantic cabinets with a price tag of over $700,000; they would substantially diminish in size and gradually become more affordable. Today, prices for graphing calculators hover around $80.
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By the mid 1970s, 11 percent of Americans owned a calculator. Four-function calculators—those that only perform addition, subtraction, multiplication, and division—gradually entered the classrooms, dividing educators and parents alike. Debates over the role of calculators in the classroom quickly emerged, and arguments for and against their use have hardly changed since then. Proponents of the calculator argued that machines could help students make sense of abstract mathematical notations through real-life problems, making math fun and interesting. Opponents worried that students would become over-dependent on calculators, losing the ability to do simple arithmetic operations and exercise a solid sense of numbers.
In 1986, Connecticut became the first state to mandate the use of calculators on state tests, signaling the beginning of a calculator-dependent generation. But the most consequential move came three years later from the National Council of Teachers of Mathematics (NCTM), which advocated for the use of calculators from kindergarten through grade 12. The guidelines set by the NCTM were soon adopted into many local and state curricula. In 1994, the College Board made substantial changes to the SAT's math section to allow the use of calculators. The 1995 Advanced Placement calculus exams were the first to require the use of graphing calculators, a powerful electronic aid that is still used in most high-school math classes today.
For W. Stephen Wilson, a math and education professor at Johns Hopkins University, using a calculator is akin to relying on a crutch when one doesn't have a bad leg. "I have not yet encountered a mathematics concept that required technology to either teach it or assess it. The concepts and skills we teach are so fundamental that technology is not needed to either elucidate them or enhance them. There might be teachers who can figure out a way to enhance learning with the use of technology, but it's absolutely unnecessary," Wilson wrote in the journal Education Studies in Mathematics.
Proponents of using technology in classrooms argue that graphing calculators, particularly those equipped with programs that can compute algebraic symbols, would reduce the need for students to memorize formulas and perform time-consuming computations. But Wilson fears that students who depend on technology will fail to understand the importance of mathematical algorithms.
Yes, a calculator could effortlessly churn out the derivative of an equation, but would students understand how to find the answer using the fundamental theorem of calculus or the definition of the derivative? The idea, Wilson says, is not to have students mindlessly perform mechanical operations, but for students at all levels to apply linear thinking in understanding the beauty of efficient algorithms. If a student can't master long division, how can she grasp derivatives and integrals?
Wilson says he has some evidence for his claims. He gave his Calculus 3 college students a 10-question calculator-free arithmetic test (can you multiply 5.78 by 0.39 without pulling out your smartphone?) and divided the them into two groups: those who scored an eight or above on the test and those who didn't. By the end of the course, Wilson compared the two groups with their performance on the final exam. Most students who scored in the top 25th percentile on the final also received an eight or above on the arithmetic test. Students at the bottom 25th percentile were twice as likely to score less than eight points on the arithmetic test, demonstrating much weaker computation skills when compared to other quartiles.
It's worth noting that calculators are also more likely to be barred in math exams at research universities than at two-year colleges and regional public universities. Out of the 50 national universities ranked at the top by U.S. News and World Report, only four schools had policies allowing electronic devices on Calculus 1 exams. One explanation is that selective institutions are less likely to offer remedial math courses and generally accept students who possess a strong math background.
Why aren't high schools taking their cue from math professors at Harvard and MIT? Because most college students won't major in STEM subjects and won't need advanced math knowledge for much of their work. Dan Kennedy, a high-school teacher at Baylor School, argues that to set a reasonable expectation for all students, calculators should be used because many real-world problems cannot be solved without technology. Students, he says, would be better served by learning probability, statistics, computer literacy, financial mathematics, and matrix algebra—the kind of math that requires the use of graphing calculators—not the kind of theoretical math that dominates math competitions.
David Bressoud, a math professor at Macalester College in Minnesota, has a different theory: He thinks that large research universities typically ban calculators because the devices are essentially obsolete there. "The larger universities have traditionally had computer-lab resources, [and now] it is easier to expect that all students have access to a computer," Bressoud said. Computers, Bressoud says, are a much better tool for teaching calculus because they are more flexible and faster than calculators.
The calculator debate also plays into a larger discussion of whether colleges should be less theoretical and more practical.
At Macalester, first-year calculus is known as "Applied Multivariable Calculus 1." Computers are heavily encouraged in class, and professors aren't slowly chalking away proofs and theorems on the blackboard. Unlike those at traditional college math classes, Macalester professors take the word "applied" seriously: A lecture on functions, for example, is demonstrated using the Body Mass Index, a function of height and weight used to determine whether a person is obese. Students in their first year of calculus also learn differential equations, a topic that is generally covered only when students have three semesters' worth of calculus under their belts. The aim is that, by introducing differential equations early on, students understand how mathematical models are generated. Why? Because these models are used in many fields, including, but not limited to, economics, environmental science, psychology, and medicine.
The calculator debate also plays into a larger discussion of whether colleges should be less theoretical and more practical. For technology advocates, an increased emphasis on technology is often seen as a way to prepare students for the real world. Calculator opponents tend to see it differently. The goal of university education, they contend, is for students to get a good grasp of the theoretical foundation of a subject, not to master calculators or computers. After all, today's technology might become drastically different 20 years from now, but a good foundation will always last.
Even Socrates once quipped that a reliance on writing would lead to the deterioration of memory. And many of the best practices in pedagogy teach that memorization does have its merits when it comes to education, despite the invention of the internet and search engines. Drawing the line between the use of and barring of calculators could prove difficult, says Jon Star, an education professor at Harvard University. "That line is also moving as time goes on, at different levels. There might be students who bring different skills who put the line in different places," Star noted. "I'm very wary of anyone who says it should be at one extreme or the other."
Many schools opt for the middle way. While calculators might not be allowed on tests and exams, colleges know that tech-savvy students will utilize programs such as Wolfram Alpha, a powerful web-based computational tool, to aid with calculus assignments. Homework problems often require calculator use, asking for solutions that involve cumbersome values or algebraic symbols that are too tedious to compute by hand. Will future professors who were born in a generation of smartphones and tablets change the course of this discussion? Only | 677.169 | 1 |
An understanding of developments in Arabic mathematics between the IXth and XVth century is vital to a full appreciation of the history of classical mathematics. This book draws together more than ten studies to highlight one of the major developments in Arabic mathematical thinking, provoked by the double fecondation between arithmetic and the algebra of al-Khwarizmi, which led to the foundation of diverse chapters of mathematics: polynomial algebra, combinatorial analysis, algebraic geometry, algebraic theory of numbers, diophantine analysis and numerical calculus. Thanks to epistemological analysis, and the discovery of hitherto unknown material, the author has brought these chapters into the light, proposes another periodization for classical mathematics, and questions current ideology in writing its history. Since the publication of the French version of these studies and of this book, its main results have been admitted by historians of Arabic mathematics, and integrated into their recent publications.
This book is already a vital reference for anyone seeking to understand history of Arabic mathematics, and its contribution to Latin as well as to later mathematics. The English translation will be of particular value to historians and philosophers of mathematics and | 677.169 | 1 |
By uniquely combining concepts and practical applications in computer graphics, four well-known authors provide here the most comprehensive, authoritative, and up-to-date coverage of the field. The important algorithms in 2D and 3D graphics are detailed for easy implementation, including a close look at the more subtle special cases. There is also a thorough presentation of the mathematical principles of geometric transformations and viewing.
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Download Computer Graphics: Theory Into Practice - Jeffrey J. McConnell
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Algebra Beginning & Intermediate
Author:Richard N. Aufmann - Joanne Lockwood
ISBN 13:9780618820726
ISBN 10:618820728
Edition:2
Publisher:Cengage Learning
Publication Date:2007-01-22
Format:Hardcover
Pages:834
List Price:$319.95
 
 
Algebra: Beginning and Intermediate by Richard Aufmann and Joanne Lockwood is a combination of this author's team's bestselling hardback textbooks. With a clear writing style, emphasis on problem-solving strategies and an objective-based interactive approach, this text will ensure your mastery of the material and preparation for subsequent math courses. | 677.169 | 1 |
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