Sage for Undergraduates 2nd Edition by Gregory V Bard ISBN 9781470461553 1470461552

What is Sage?
As the open-source and free competitor to expensive software such as Maple TM , Mathematica , Magma, and Matlab , the computer algebra system Sage offers anyone with access to a web browser the ability to use cutting-edge mathematical software and to display the results for others, often with stunning graphics. I’m sure that you will find Sage far easier to use than a graphing calculator, and vastly more powerful .Sage is built on the programming language Python. Therefore, if you learn Sage, then you’ll learn a lot of Python along the way. Being able to program in Python is an extremely useful skill for undergraduate students (and graduate students) in a wide array of academic disciplines including physics, economics, biology, statistics, geology, chemistry, finance, astronomy, mathematics, and every form of engineering. You’ll see examples from each of those subjects in this book, plus you’ll learn a lot about programming and coding. For your convenience, every block of code that is longer than four lines has been included in an archive on the book’s website: http://www.sage-for-undergraduates.org/ so that you won’t have to retype long blocks of code.If you’d like to dive in and start right away, which is definitely the most fun way to learn, then skip to Chapter 1, starting on page 1, at this time.I’ll now explain why a second edition was necessary and then describe the various chapters and appendices of this book.xi THE PREFACE: HOW TO USE THIS BOOK xv Appendix B. There’s also a command-line interface that can be useful for the batch processing of large data sets.Note: Generally, users will want to switch to either a local installation or CoCalc.com (formerly known as SageMathCloud) after completing Chapter 1, before tackling the projects in Chapter 2 or the advanced work of Chapters 4-6. I think the information in Chapter 3 can be learned with SageMathCell, with CoCalc.com, or with a local installation, with no disadvantage to the user.• The first edition’s Appendix C, a list of resources on the internet relating to Sage, became completely out of date surprisingly fast. That’s because URLs change frequently, and sometimes entire websites are taken down. This information will now be stored on the book’s website: http://www.sage-for-undergraduates.org/ so that URLs can be updated whenever they change. This is a much more practical way of storing this useful information. This appendix has been jokingly renamed “Appendix W” because it is essentially a list of websites.• The new Appendix C contains directions for converting Sage code that was written in the Python 2 era (versions of Sage before Version 9) for use in the Python 3 era (versions after and including Version 9). Any readers who do not have that task set before them can safely skip Appendix C.• The first edition’s Appendices D, E, and F have been removed to save space.• Finally, you will find the acknowledgments for the second edition, the acknowledgments for the first edition, the bibliography, and the index.
Let’s Dive
In! Right Now! Without any hesitation whatsoever, open up a web browser at this very moment. Type in the following URL: https://sagecell.sagemath.org/ You have now connected to the “Sage Cell Server.” Now you shall see a large window (called a cell) and a big button marked “Evaluate.” In that cell, type the code solve( x^2 -16 == 0, x ) and then click “Evaluate.” You should receive back the answer [x == -4, x == 4] xvi THE PREFACE: HOW TO USE THIS BOOK which is clearly the correct set of two solutions to the equation x 2 -16 = 0. You have now solved your first math problem with Sage. Congratulations! Welcome to Sage! You are now ready to begin reading Chapter 1.
A Note for Instructors:
The contents of each section of each chapter in this second edition is a strict superset of the contents of the same section of the same chapter of the first edition. So, if you’re accustomed to asking students to read Sections 1.1 to 1.4 (for example), then all concepts found there in the first edition will be taught, as well as some more concepts in most cases. (The only exception is that Section 5.8 of the first edition is now Section 5.10.) It was not possible to extend this correspondence to individual subsections. While there are no plans for a third edition (or a second printing of the second edition) at this time, it is nonetheless wise for instructors to use section numbers, e.g., Sections 1.1 to 1.4, instead of page numbers, e.g., pages 1-33, when creating assignments.
If You Notice Any Mistakes:
I have several paid proofreaders who assisted me in preparing this book and I am deeply grateful for their help. All of the code and mathematics has been rigorously checked as a primary requirement for a successful mathematical textbook. As a secondary (but still somewhat important) task, the grammar and spelling have also been checked, with either a 6-eyes or an 8eyes standard of proofreading (that is to say, myself and 2-3 knowledgable proofreaders).Nonetheless, mistakes inevitably occur, and I welcome any reports of typos. If you find any errors, then please feel free to email me, so that the mistakes can be noted for correction in the future. Please email me at [email protected] and place “Typo in second edition of Sage for Undergraduates” in the subject line. Please include a screenshot of the mistake, and the section number (such as 4.26.5, for example), as well as a brief description of what was incorrect.Finally, note that my ORCID (Open Researcher and Contributor ID) is https://orcid.org/0000-0001-9460-6086.Chapter 1
Welcome to Sage!
Mathematics is something that we do, not something that we watch. Therefore, the best way to learn any branch of mathematics is to actually get started and give things a try. The same is true for learning Sage! Just connect to the Sage “Cell Server,” the simplest of several interfaces to Sage, and dive right on in! Experiment with the examples on the next few pages, trying them out as you read. https://sagecell.sagemath.org/
WELCOME TO SAGE!
2 An irrelevant historical aside: Because rewriting software is a painful and timeconsuming process, many important pieces of scientific software remain written in FOR-TRAN even in the year that this footnote was written (2020), and likewise the same is true of business software in COBOL. Each new programming language incorporates features of the previous generation of languages, to make it easier to learn, and accordingly one sees some truly ancient seeds of decades-old languages once in a while. 1.1. USING SAGE AS A CALCULATOR 1.2. USING SAGE WITH COMMON FUNCTIONS 9 You can do things like N(pi, digits=200) to get a decimal expansion of pi, with 200 significant digits. The golden ratio φ = 1 + √ 5 2 ≈ 1.618033988 . . . that comes up in geometry, architecture, and the Fibonacci numbers, and which was a rather trendy 4 topic of conversation at the turn of the century, is also built into Sage. You can get 200 digits of it with N(golden_ratio, digits=200)
Controlling Excessive Precision
Back on page 5 in Subsection 1.1.4, we had expressed frustration that certain dollar amounts were given with ten decimal places. Even the second-to-last digits represented “nano-dollars” or billionths of a dollar, in that example. Indeed, throughout the USA it is a standard practice to report all 5 financial amounts below one million dollars to the penny, if possible, but essentially never more precise than that. Now, we can fix the excessive precision by using the N() function. Here is the code to do that: print(‘First month:’, N(76_543.21*(1 + 0.03/12)^1, digits=7) ) print(‘Second month:’, N(76_543.21*(1 + 0.03/12)^2, digits=7) ) print(‘Third month:’, N(76_543.21*(1 + 0.03/12)^3, digits=7) ) print(‘Fourth month:’, N(76_543.21*(1 + 0.03/12)^4, digits=7) ) However, I have to sometimes fiddle with the number of digits. For example, if the dollar amounts went into the hundreds of thousands, I’d have to use digits=8 to get Sage to show the answer to the nearest penny, which is the standard practice in finance. For example, N( 123_456.987654321, digits = 7 ) results in 123457.0 and I need to set digits=8 to get that second decimal place. 4 Primarily due to prominence in the fictional but fascinating mystery novel The Da Vinci Code, by Dan Brown, published in 2003 by Doubleday [13].5 The one exception (in the USA) where additional precision is used happens to be gasoline prices, which are reported to the tenth of a penny, or equivalently a thousandth of a dollar. This is weird because there is no coin in the USA smaller than the penny, since the abolition of the half-penny coin in 1857.