Carl LundNM, Albuquerque, US,

October 7, 2015

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Introduction to mathbook

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This is a website for a book about the foundations of basic mathematics. Since it is designed to look at basic mathematics from a different prospective, it may be considered a supplement to high school mathematics.

The book begins with a discussion of everyday mathematics. It starts with an examination of the reasons that mathematics was developed, and an examination of what is fundamental about its structure. Thus it starts with a discussion of the natural numbers (1, 2, 3, … etc) and arithmetic, and shows how the system of natural numbers has been expanded to include the integers and fractions. The second half of the book covers generalizations of these basic systems, leading to discussions of the real and complex numbers. The book concludes with a discussion of the common exponential, logarithmic and trigonometic mathematical functions.

This approach has the advantage of making more advanced mathematics accessible to the reader as soon as they have an understanding of basic algebra. Thus it is readable by people in high school, as well as persons looking for an introduction to a more general development of mathematics.

It is also an exposure to a much more rigorous kind of thinking than one usually sees in a first exposure to mathematics. This is why the presentation is considered a supplement to other approaches. The procedure is to explicitly identify the basic principles, and use formal logic to develop the applications. The material proceeds from the basic assumptions (axioms) to the proof of theorems (conjectures of truths that follow from the assumptions) via the laws of logic. The laws of logic like the Law of the Excluded Middle, for example (which says that a well-formed mathematical question is either true or false), are usually not intuitive when first encountered. And yet, when understood, are far more convincing than the kind of argument one often encounters elsewhere. It is similar to the increased understanding one gets in grammar when one learns about diagramming sentences. (Or, if you prefer, one appreciates the power of diagramming sentences when one sees the power of mathematical logic!)

To be accessible to as wide an audience as possible, the book is completely self-contained. All the proofs are given in explicit detail, and, as much as possible, the individual sections of the book are self-contained. That is, there are a minimal number of references to earlier results when proving theorems, even at the expense of repeating some of the earlier arguments.

For a discussion of the intended audience of this book, click on the Intended Audience link.

For a discussion of the approach to the material in this book, click on the Approach link.

This book does not give discussions of the applications. It is concerned with the overall underlying foundations of basic mathematics. The Content link gives examples of the material that is covered.

For the background of the author please see the About the Author link.

The book is a PDF file, which can be downloaded to your own computer. A PDF reader is needed to view the file, and print it if desired. A common, free PDF viewer is Acrobat Reader

The book, while less than a megabyte in size, can take a few minutes to download if you don't have a fast internet connection. In general, it is easier to download the file to your computer first, and then view it. You will then also have the file to look at even when you are not connected to the internet.

Since it is relatively easy to update an electronic version of a manuscript, we anticipate changes relatively often compared to editions of a printed manuscript. We have therefore added references to changes from various earlier versions following the title page. So if you no longer have the latest version, you can easily find the changes in the current version to see if you would like to update your copy.

You can see the book at the The Book link.

mathbook@comcast.net

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