Mathematics - the Practical, the Logical, and the Beautiful

Benjamin Baumslag and Frank Levin have written a book Mathematics - the Practical, the Logical, and the Beautiful which is freely available on the web at

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We reproduce the authors' Preface to Mathematics - the Practical, the Logical, and the Beautiful:-


This book is intended for:

1. Readers who liked mathematics at school but never studied it further.

2. Young people with mathematical talent.

3. Teachers who are looking for inspirational material for school.

Each reader will study different parts of the book in different ways. Not many are likely to read every page, but then, not every visitor to an art museum looks at every painting. You do what you find enjoyable and find the time for; you may even think of coming several times. Chapters can be read in almost any order. Nor is it necessary to read the whole of a chapter. Many sections indicate optional material. In particular, the solved problems are optional. The more difficult calculations are often put into the solved problems, and even if one does not read these, one still gets an understandable account.

The aim of the book is expressed by its title, namely to give examples of mathematics which is practical and useful in everyday life, examples of beautiful mathematics, and to illustrate the logical arguments used in mathematics, i.e. proof. Practical topics include approximation and the use of the powers of 10 notation. Then there is percentages, and estimating various quantities with simple calculations (Chapter 4), some knowledge of graphs, some probability and statistics. There is also a chapter on units like Watts and horsepower.

Proof is demonstrated for instance by proving Pythagoras's theorem, and using numbers to derive Euclidean Geometry.

There are some beautiful classical results such as examples of a finite Geometry and a Projective Geometry, the existence of an infinite number of primes, and the irrationality of the square root of 2. Fermat's little and last theorems and the estimation of the number of primes less than a given number N, are discussed. Finally, we return to counting but this time counting infinite sets, and have the striking results of Cantor such as there are as many points on a line of length 1 as on a line of length 2.

Much could be added, but we have chosen to be brief in order to present a more easily comprehendible book. There is much of value and interest anyway.

We have tried as far as possible to provide mathematics, which the reader can verify for himself or herself, and not have to rely on our authority.

The book begins with topics that would normally be discussed in school, and ends with topics, which would normally appear in a university course on mathematics. The careful choice of material and presentation provides an account which is understandable by those who have studied secondary school mathematics. Because we give a self-contained account, the reader who has forgotten the school mathematics will be reminded of some of the details. What is required of the reader is a flexible mind, curiosity, and the sort of patience and determination that is required to play bridge or chess or solve crossword or sudoku puzzles.

The examples are from different countries, England or Sweden or the U.S.A. But these are only examples which are meant to illustrate various methods, and the reader will with their help apply these techniques to their own interests and needs.

The material in this book is not original to us.

There are many brilliant ideas in mathematics. If we can introduce you to some of them it will be an honour and a privilege.

Last Updated January 2015