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Read more about The Essence of Mathematics Through Elementary Problems

The Essence of Mathematics Through Elementary Problems

Alexandre Borovik, University of Manchester

Tony Gardiner, University of Birmingham

It is increasingly clear that the shapes of reality – whether of the natural world, or of the built environment – are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself ‘tick’, and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside.


 


The Essence of Mathematics consists of a sequence of 270 problems – with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline.

 

The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works.

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Read more about First Semester in Numerical Analysis with Julia

First Semester in Numerical Analysis with Julia

Giray Ökten, Florida State University

First Semester in Numerical Analysis with Julia presents the theory and methods, together with the implementation of the algorithms using the Julia programming language (version 1.1.0). The book covers computer arithmetic, root-finding, numerical quadrature and differentiation, and approximation theory. The reader is expected to have studied calculus and linear algebra. Some familiarity with a programming language is beneficial, but not required. The programming language Julia will be introduced in the book. The simplicity of Julia allows bypassing the pseudocode and writing a computer code directly after the description of a method while minimizing the distraction the presentation of a computer code might cause to the flow of the main narrative.

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Read more about Fundamentals of Matrix Algebra

Fundamentals of Matrix Algebra

Gregory Hartman, Virginia Military Institute

A college (or advanced high school) level text dealing with the basic principles of matrix and linear algebra. It covers solving systems of linear equations, matrix arithmetic, the determinant, eigenvalues, and linear transformations. Numerous examples are given within the easy to read text. This third edition corrects several errors in the text and updates the font faces.

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Read more about APEX PreCalculus

APEX PreCalculus

Amy Givler Chapman, Virginia Military Institute

Meagan Herald, Virginia Military Institute

Jessica Libertini, Virginia Military Institute

This text was written as a prequel to the APEXCalculus series, a three–volume series on Calculus. This text is not intended to fully prepare students with all of the mathematical knowledge they need to tackle Calculus, rather it is designed to review mathematical concepts that are often stumbling blocks in the Calculus sequence. It starts basic and builds to more complex topics. This text is written so that each section and topic largely stands on its own, making it a good resource for students in Calculus who are struggling with the supporting mathemathics found in Calculus courses. The topics were chosen based on experience; several instructors in the Applied Mathemathics Department at the Virginia Military Institute (VMI) compiled a list of topics that Calculus students commonly struggle with, giving the focus of this text. This allows for a more focused approach; at first glance one of the obvious differences from a standard Pre-Calculus text is its size.

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Read more about Yet Another Introductory Number Theory Textbook (Cryptology Emphasis Version)

Yet Another Introductory Number Theory Textbook (Cryptology Emphasis Version)

Jonathan A. Poritz

This version of YAINTT has a particular emphasis on connections to cryptology. The cryptologic material appears in Chapter 4 and §§5.5 and 5.6, arising naturally (I hope) out of the ambient number theory. The main cryptologic applications – being the RSA cryptosystem, Diffie-Hellman key exchange, and the ElGamal cryptosystem – come out so naturally from considerations of Euler’s Theorem, primitive roots, and indices that it renders quite ironic G.H. Hardy’s assertion [Har05] of the purity and eternal inapplicability of number theory. Note, however, that once we broach the subject of these cryptologic algorithms, we take the time to make careful definitions for many cryptological concepts and to develop some related ideas of cryptology which have much more tenuous connections to the topic of number theory. This material therefore has something of a different flavor from the rest of the text – as is true of all scholarly work in cryptology (indeed, perhaps in all of computer science), which is clearly a discipline with a different culture from that of “pure”mathematics. Obviously, these sections could be skipped by an uninterested reader, or remixed away by an instructor for her own particular class approach.

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Read more about Lies, Damned Lies, or Statistics: How to Tell the Truth with Statistics

Lies, Damned Lies, or Statistics: How to Tell the Truth with Statistics

Jonathan A. Poritz

This is a first draft of a free (as in speech, not as in beer, [Sta02]) (although it is free as in beer as well) textbook for a one-semester, undergraduate statistics course. It was used for Math 156 at Colorado State University–Pueblo in the spring semester of 2017.

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Read more about Open Resources for Community College Algebra

Open Resources for Community College Algebra

Portland Community College Faculty

Open Resources for Community College Algebra (ORCCA) is an open-source, openly-licensed textbook package (eBook, print, and online homework) for basic and intermediate algebra. At Portland Community College, Part 1 is used in MTH 60, Part 2 is used in MTH 65, and Part 3 is used in MTH 95.

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Read more about Business Math: A Step-by-Step Handbook

Business Math: A Step-by-Step Handbook

Jean-Paul Olivier

Business math is the study of mathematics required by the field of business. Business professionals will work with taxes, gross earnings, product prices, and currency exchange; they will be offered loans, lines of credit, mortgages, leases, savings bonds, and other financial tools. This textbook covers all of these topics and how these financial tools can maximize their earnings and minimize their costs. It also discusses how to execute smart monetary decisions both personally and for their business.

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Read more about Trigonometry

Trigonometry

Richard W. Beveridge

The precursors to what we study today as Trigonometry had their origin in ancient Mesopotamia, Greece and India. These cultures used the concepts of angles and lengths as an aid to understanding the movements of the heavenly bodies in the night sky. Ancient trigonometry typically used angles and triangles that were embedded in circles so that many of the calculations used were based on the lengths of chords within a circle. The relationships between the lengths of the chords and other lines drawn within a circle and the measure of the corresponding central angle represent the foundation of trigonometry - the relationship between angles and distances.

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Read more about College Algebra & Trigonometry

College Algebra & Trigonometry

Richard W. Beveridge

This College Algebra text will cover a combination of classical algebra and analytic geometry, with an introduction to the transcendental exponential and logarithmic functions. If mathematics is the language of science, then algebra is the grammar of that language. Like grammar, algebra provides a structure to mathematical notation, in addition to its uses in problem solving and its ability to change the appearance of an expression without changing the value.

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