Simplifying Expressions Involving Radicals
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Simplification of Radical Expressions
Simplification of Radical Expressions by B. F. Caviness (I) Computer Science Department lllinois Institute of Technology Chicago, lllinois and R. J. Fateman Computer Science Division Department of Electrical Engineering and Computer Sciences University of California at Berkeley Berkeley, California Abstract As is the case with all simplification problems there are three desirable attributes for In this paper we discuss the problem of any solution. First it is desirable to have a simplifying unnested radical expressions. We clean theoretical solution to the problem. Second describe an algorithm implemented in ~CSYMA that the algorithms to carry out the simplifications simplifies radical expressions and then follow should be as efficient as possible, and finally the this description with a formal treatment of the solution should have good human engineering features, problem. Theoretical computing times for some of i.e., the use and actions of the simplification the algorithms are briefly discussed as is related procedures should be as natural and transparent as work of other authors. possible. Unfortunately these three aspects of a solution are not usually compatible with each other I. Introduction and compromises and choices must be made. This is certainly the case for the simplification of radical In this paper we discuss the problem of expressions. Standard techniques from the theory simplifying unnested radical expressions. By of algebraic field extensions can be used to get a radical expressions we mean, roughly speaking, nice theoretical solution to the problem. However sums, differences, products, and quotients of the nice theoretical solution neither leads to fast rational functions raised to rational powers. algorithms nor possesses nice human engineering Thus we will be discussing algorithms for carrying features. -
Phi)Ve Is a Magic Number
(Phi)ve is a Magic Number James J. Solberg Professor Emeritus, Purdue University Written for the 13th Gathering for Gardner, 2018 Martin Gardner would surely have noted the centennial significance of January 6 of this year. On that day, the digits of the date 1/6/18 matched 1.618 – the leading digits of the famous golden ratio number Phi. If you wanted three more digits, you could have set your alarm to celebrate at 3:39 am or at a more reasonable 3:39 pm when the eight digits of that moment aligned to 1.6180339. I am sure that Martin would have also noticed the puns in my title. The first word refers to both Phi and five, and the fourth word refers to magic squares. My goal is to reveal several surprising connections between the two values and their powers using both Fibonacci numbers and five-by-five magic squares. A warning: if ordinary word puns tend to make you numb, then my mathematical puns will make you number. As you presumably know, the golden ratio number Phi (or �) shows up in nature, in art, in architecture, and even beauty salons, as well as mathematics. If you are not familiar with �, you are in for a treat. There are plenty of books and websites that explain the fascinating properties and ubiquitous nature of this mysterious number. I want to use a few of the many identities, so here is a short summary. One of the ways to determine the golden ratio is to find the place to divide a line so that the ratio of the length of the longer segment to the shorter is the same as the ratio of the whole line to the longer segment. -
Radical Expressions and Equations”, Chapter 8 from the Book Beginning Algebra (Index.Html) (V
This is “Radical Expressions and Equations”, chapter 8 from the book Beginning Algebra (index.html) (v. 1.0). This book is licensed under a Creative Commons by-nc-sa 3.0 (http://creativecommons.org/licenses/by-nc-sa/ 3.0/) license. See the license for more details, but that basically means you can share this book as long as you credit the author (but see below), don't make money from it, and do make it available to everyone else under the same terms. This content was accessible as of December 29, 2012, and it was downloaded then by Andy Schmitz (http://lardbucket.org) in an effort to preserve the availability of this book. Normally, the author and publisher would be credited here. However, the publisher has asked for the customary Creative Commons attribution to the original publisher, authors, title, and book URI to be removed. Additionally, per the publisher's request, their name has been removed in some passages. More information is available on this project's attribution page (http://2012books.lardbucket.org/attribution.html?utm_source=header). For more information on the source of this book, or why it is available for free, please see the project's home page (http://2012books.lardbucket.org/). You can browse or download additional books there. i Chapter 8 Radical Expressions and Equations 1256 Chapter 8 Radical Expressions and Equations 8.1 Radicals LEARNING OBJECTIVES 1. Find square roots. 2. Find cube roots. 3. Find nth roots. 4. Simplify expressions using the product and quotient rules for radicals. Square Roots The square root1 of a number is that number that when multiplied by itself yields the original number. -
Surface Fitting and Some Associated Confidence Procedures
SURFACE FITTING AND SOME ASSOCIATED CONFIDENCE PROCEDURES by ARNOLD LEON SCHROEDER, JR . A THESIS submitted to OREGON STATE UNIVERSITY in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE June 1962 APPROVED: Redacted for Privacy Associate Professor of Statistics In Charge of Major Redacted for Privacy Chaii n of the Departrnent of Statistics Redacted for Privacy Chairrnan Cradua 'e Cornrnittee Redacted for Privacy of Graduate Date thesis is presented May 15, 1962 Typed by Jolene 'Wuest TABLE OF CONTENTS CHAPTER PAGE 1 INTRODUCTION .. ••.•... • . 1 2 REGRESSION ANALYSIS APPLIED TO ORE DISTRIBUTION ..• .... 4 3 FITTING DATA TO REGRESSION MODELS . ....•.. • 7 4 CONFIDENCE LIMITS OF REGRESSION COEFFICIENTS .... • •..... ... 13 5 CONFIDENCE PROCEDURES FOR THE DISCRIMINANT . 16 6 CONFIDENCE LIMITS FOR LOCATION OF MAXIMUM . .. 19 7 EXAMPLE ANALYSES 25 BIBLIOGRAPHY 32 APPENDIX . ...... 34 SURFACE FITTING AND SOME ASSOCIATED CONFIDENCE PROCEDURES CHAPTER 1 INTRODUCTION The use of statistics in geology has expanded rapidly, until today we have statistical analysis being applied in many branches of geology. One of the main reasons for this expansion is the enormous increase in the amount of available numerical data, especially in the field ofmining where thousands of sample assays are often taken in the course of mining operations. This thesis is one approach to the problem ofhow to analyze these vast amounts of sample data and is specifically a des crip tion of the analysis of the distribution of ore in the veins of two large mixed-metal ore deposits . However, the statistical techniques pre sented are applicable to geological data in general, as well as to the specific problem of analyzing ore distribution in metalliferous veins. -
The Collage of Backgrounds and the Golden Ratio
The Collage of Backgrounds and the Golden Ratio Supplemental Information about Sacred Geometry and the Art of Doug Craft All Materials and Art © 2008 Doug Craft Doug Craft [email protected] http://www.DougCraftFineArt.com The Collage of Backgrounds and the Golden Ratio © 2008 Doug Craft Introduction While my earlier collages were figurative with a clear visual distinction between foreground objects and background, my work since 1997 has been exploring a new concept that I call The Collage of Backgrounds. This technique basically arranges natural abstract images - what would be considered backgrounds in my figurative collages - according to simple geometric rules based on Sacred Geometry and the Golden Ratio. This idea can be seen in the collage to the right, AIR Elements Proportional Offset Square 2004-003. The following is an explanation and visual guide to the Golden Ratio and the shapes and geometric forms that I use in my collages, montages, and artwork in other media. Sacred Geometry and The Golden Ratio, Φ When I refer to Sacred Geometry, I am talking about geometry that is derived from or directly related to the structure of nature. Our universe is structured in a highly complex yet sublimely ordered manner. This is a truth that is readily felt by sensitive people, and has also been demonstrated by science and mathematics. Structural forms seen at the microscopic level are repeated at other scales, and the laws of fractional symmetry appear to apply throughout. So, geometry that refers to the structural unity of nature is a powerful metaphor for the mystery of life, and thus sacred. -
Pi Being Sent Forth Into the World, Via an Art Deco Archer ★ (A Bronze Piece by Alexandre Kelety, 1920) ★
The symbolic release of JainPi being sent forth into the World, via an Art Deco Archer ★ (A bronze piece by Alexandre Kelety, 1920) ★ The Book Of Phi, volume 8 The True Value of Pi = JainPi = 3.144605511029693144… 1 THE BOOK OF PHI VOLUME 8 THE TRUE VALUE OF PI JAIN PI = 3.144605511… H © JAIN 108 2014 ISBN = 978-0-9872543-4-4 = 3.14460551102969314427823434337183571809248823135089… to 50 Decimal Places… “…This reiterates what I had stated before, that we can now express Pi in terms of Phi, clearly meaning that Pi is redundant, meaning that we can express everything in the universe based on Phi and its Powers. It could be that Pi is a mere incarnation of the Avatar Phi…” Jain 108 The Book Of Phi, volume 8 The True Value of Pi = JainPi = 3.144605511029693144… 2 Pi-oneering… the re-introduction of the True Value of Pi at Gaia’s deepest request as the timing is appropriate to recalibrate the finest frequencies inpouring into humanity’s consciousness to override the density of war, need & greed as we begin to relearn to honour the trees and the skies and one another, making education of the people our sacred duty to teach the transcendental Laws of the Spirals setting our pathways to open & embrace the exciting gifts of the physics of blackholes - wormholes space travel embarking on a new journey where technology complements Nature, where sacred geometry is known and understood by all children at the deepest level of enquiry. This is our gift to the next generation of souls. JainPi in a Spiral having 108 decimal places… The Book Of Phi, volume 8 The True Value of Pi = JainPi = 3.144605511029693144… 3 Introducing Jain Pi: a Phi-Pi Relationshift. -
Simplifying Square Roots of Square Roots by Denesting 63
4 Simplifying Square Ro ots of Square Ro ots by Denesting David J. Je rey Alb ert D. Rich The UniversityofWestern Ontario Soft Warehouse, Inc. Abstract: We discuss why it is imp ortant to try to simplify the square ro ot of an expression containing other square ro ots, and we give rules for doing this when it is p ossible. The square ro ot of an expression containing nth ro ots is also considered brie y. This article, in addition to treating a sp eci c mathematical topic, shows some of the steps that develop ers must take when writing computer algebra systems. 4.1 Intro duction p p 3 1 2 and 5 are called surds [Chrystal64, Hall88]. The more general Numb ers suchas terms radical and algebraic number are also used, but surd is the most sp eci c. It is 2 avenerable term that has b een revived for use in computer systems . Given p ositive n integers n and k , the unique p ositive real ro ot of the equation x = k will b e denoted p n k and called a surd. Most b o oks [Chrystal64, Hall88]allow k to b e a p ositive rational numb er, and there is no great di erence in principle if we accept that generalization; in this article, however, all the examples use integers. Another p ointofvariation is p p 8or2 2; here we the treatment of p erfect-p ower factors, i.e., whether one writes use whichever form is more convenient. The term radical may b e familiar to some readers from the common phrase `solvable in terms of radicals', which is used whenever a textb o ok discusses the ro ots of a p olynomial [Dickson26]. -
Continued Radicals Jamie Johnson Western Kentucky University
Western Kentucky University TopSCHOLAR® Masters Theses & Specialist Projects Graduate School 2005 Continued Radicals Jamie Johnson Western Kentucky University Follow this and additional works at: http://digitalcommons.wku.edu/theses Part of the Geometry and Topology Commons Recommended Citation Johnson, Jamie, "Continued Radicals" (2005). Masters Theses & Specialist Projects. Paper 240. http://digitalcommons.wku.edu/theses/240 This Thesis is brought to you for free and open access by TopSCHOLAR®. It has been accepted for inclusion in Masters Theses & Specialist Projects by an authorized administrator of TopSCHOLAR®. For more information, please contact [email protected]. CONTINUED RADICALS A Thesis Presented to The Faculty of the Department of Mathematics Western Kentucky University Bowling Green, Kentucky In Partial Fulfillment Of the Requirements for the Degree Master of Science By Jamie Johnson May, 2005 CONTINUED RADICALS Date Recommended _____12/10/04_____ Dr. Tom Richmond, Chair Dr. John Spraker Dr. Daniel C. Biles Elmer Gray, Dean, Graduate Studies and Research, May 4, 2005 Contents Title page i Signatures ii Table of contents iii Abstract iv Chapter 1. Foundations and Motivations 1 Chapter 2. Introduction 5 Chapter 3. Continued radicals with nonzero terms 9 Chapter 4. Representation allowing 0 as a term 16 Chapter 5. Continued radicals whose terms assume only two values 24 Chapter 6. Further Results 26 6.1. Periodic Representations 26 6.2. A Pattern 28 6.3. Computer Generated Partial Representations 29 Bibliography 35 iii CONTINUED RADICALS Name: Jamie Johnson Date: May, 2005 Pages: 35 Directed by Dr. Tom Richmond Department of Mathematics Western Kentucky University Abstract √ If a1, a2, . , an are nonnegative real numbers and fj(x) = aj + x, then f1 ◦f2 ◦· · ·◦ fn(0) is a nested radical with terms a1, . -
Fibonacci Series and Number Five
mark h lane www.biblenumbersforlife.com CREATION NUMBERS CONNECTION OF FIBONACCI SERIES TO NUMBER FIVE INTRODUCTION The spiritual number 5 means LIFE1. “And God said ‘Let the water teem with LIVING creatures, and let birds fly across the expanse of the sky. So God created the great creatures of the sea and every LIVING and moving thing with which the water teems, according to their kinds, and every winged bird according to its kind. And God saw it was good. God blessed them and said ‘Be fruitful and increase in number and fill the waters of the seas, and let the birds increase on the earth. And there was evening, and there was morning – the FIFTH day.” (Genesis 1:20-23) We see the Fibonacci patterns all through creation. Is there any connection between the Fibonacci number series and the number FIVE? Yes! There are a tremendous number of connections. The number FIVE is embedded in the heart of the Fibonacci number pattern. This suggests the author of the Holy Bible is the same as the Creator of the Universe, the Lord God Almighty. He has placed His spiritual number signature on LIFE so we would know all this was His doing. FIBONACCI NUMBERS The Fibonacci series starts with one, adds one to give two, and from then on the following number is the sum of the previous two numbers in the series. For example: 3 = 1 +2, 5 = 3 +2, 8 = 5 + 3, etc. Fibonacci Numbers: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. The ratio of each Fibonacci number with the preceding Fibonacci number is called the Golden Ratio or Phi. -
Ramanujan's Route to Roots of Roots
Ramanujan's route to roots of roots Talk in IIT Madras on the occasion of s r q p Ramanujan's 1 + 119 1 + 120 1 + 121 ¢ ¢ ¢-th birthday B.Sury Stat-Math Unit Indian Statistical Institute Bangalore Introduction The fancy title points out to a subject which originated with Ramanujan or, at least, one to which he gave crucial impetus to. We can't help but feel a sense of bewilderment on encountering formulae such as q s s s 3 p 1 2 4 3 2 ¡ 1 = 3 ¡ 3 + 3 ; 9 9 9 q p p 1 p p 3 28 ¡ 3 27 = ¡ (¡ 3 98 + 3 28 + 1); 3 q p p 1 p p p 3 5 ¡ 3 4 = (¡ 3 25 + 3 20 + 3 2); 3 s s s s p 2¼ 4¼ 8¼ 3 5 ¡ 3 7 3 cos + 3 cos + 3 cos = ; 7 7 7 2 q s s 6 p 5 2 7 3 20 ¡ 19 = 3 ¡ 3 ; 3 3 r q p p 8 ¡ 8 + 8 ¡ ¢ ¢ ¢ = 1 + 2 3 sin 20±; s r q p p 23 ¡ 2 23 + 2 23 + 2 23 ¡ ¢ ¢ ¢ = 1 + 4 3 sin 20±; s p p e¡2¼=5 e¡2¼ e¡4¼ e¡6¼ 5 + 5 5 + 1 ¢ ¢ ¢ = ¡ : 1+ 1+ 1+ 1+ 2 2 The last expressions for the so-called Rogers-Ramanujan continued fraction appeared in Ramanujan's ¯rst letter to Hardy. These formulae are among some problems posed by Ramanujan in the Journal of the Indian Mathemat- ical Society. As was usual with Ramanujan, he had a general formula hidden in the background and, singled out striking special cases in these problem sections. -
Minnesota State Standards Alignments Grades One Through Eleven
Minnesota State Standards Alignments Grades One through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other countries. P.O. Box 8036 • Wisconsin Rapids, WI 54495-8036 Phone: (800) 338-4204 • Fax: (715) 424-4242 www.renlearn.com Minnesota State Standards Alignments Standards list with Aligned Product Skills The Standards list with Aligned Product Skills Report is a standards- oriented document showing the entire list of standards for the subject and grade on the left side of the report with the aligning product objectives on the right side. This alignment report shows the breadth of standards coverage for the purpose and focus of this product. Note to Educator .....................................................iii Grade 1 ..................................................... 1 Grade 2 ..................................................... 6 Grade 3 ....................................................12 Grade 4 ....................................................18 Grade 5 ....................................................25 Grade 6 ....................................................34 Grade 7 ....................................................44 Grade 8 ....................................................55 Grades 9 - 11 ....................................................64 © 2009 by Renaissance Learning, Inc. All rights reserved. No portion of this document may be reproduced, by any process or technique, without the express written consent -
Perfect Squares and Square Roots
Mathematics Enhanced Scope and Sequence – Grade 8 Perfect Squares and Square Roots Reporting Category Computation and Estimation Topic Working with perfect squares and square roots Primary SOL 8.5 The student will a) determine whether a given number is a perfect square; and b) find the two consecutive whole numbers between which a square root lies. Related SOL 8.2 Materials • Squares tiles in different colors (actual or heavy paper) • Discovering Squares and Squares Roots activity sheet (attached) • I Have. Who Has? Cards (attached) Vocabulary perfect squares, square root (earlier grades) consecutive (8.5) Student/Teacher Actions (what students and teachers should be doing to facilitate learning) 1. Distribute copies of the Discovering Squares and Squares Roots activity sheet and sets of colored square tiles. Have students work individually to complete the activity sheet. 2. When students are finished, pair them up, and have partners discuss the answers. Next, divide students into small groups to discuss the activity. Then, have a whole class discussion on perfect squares, square roots, and the activity. 3. Give each student an I Have. Who Has? card. Have the student with the starter card come forward and write the problem on the board. The student who has the answer to the problem should come forward and explain the answer, writing the steps used in the solution. Continue this process until all the cards have been used. Assessment • Questions o Why do numbers have both a positive and a negative square root? o What is the difference between 169 and − 169 ? o How many perfect squares lie between 10 and 50? How do you know? • Journal/Writing Prompts o Draw a model to represent how you can determine the two consecutive whole numbers between which the 19 lies.