It is impressive how much you can learn and prove about irrational numbers with very little math background. This book was written sixty years ago for high-school students, and it doesn’t assume anything beyond high-school algebra. The main tools used are the Rational Root Theorem and Dirichlet’s Pigeon-hole Principle (both of which are explained thoroughly in the book). There is no calculus and no infinite processes, although it does sidestep this limitation in the transcendental number section by using infinite decimals. In general the book keeps things simple by dealing with specific examples of irrational numbers rather than general classes.

This is a proofs book. It doesn’t assume the reader already knows how to prove things, so it gives helpful hints along the way. The approach is “old school”: you learn proofs by proving things, not by taking a Transition to Proofs course. As the book states on p. 7, “If the nature of proof cannot be described or formulated in detail, how can anyone learn it? It is learned, to use an oversimplified analogy, in the same manner as a child learns to identify colors, namely, by observing someone else identify green things, blue things, etc., and then by imitating what he has observed.” There are sections about proof techniques interspersed with the exposition, for example Section 2.3 on “The Many Ways of Stating and Proving Propositions”.

Roughly the first third of the book (Chapters 1–3) is devoted to a careful study of the different types of numbers (integer, prime, rational, real), ending with a few examples of the simpler irrational numbers such as \( \sqrt{2} \), \( \sqrt{3} \), and \( \sqrt{2}+\sqrt{3} \). The proofs of these are ad hoc. Then we begin to prove irrationality more systematically, developing the Rational Root Theorem in Chapter 4, and using it there and in Chapter 5 to prove the irrationality of many specific radical expressions, logarithms, and values of trigonometric functions. Chapter 5 also covers the three classic ruler-and-compass construction problems: trisecting an angle, doubling a cube, and squaring the circle. It proves that the first two are impossible. Chapters 6 and 7 are more difficult. Chapter 6 deals with Diophantine approximation and the presentation is more abstract than what has gone before; it deals with the degree to which an irrational number can be approximated by rational numbers. It quotes but does not prove Hurwitz’s theorem that for any irrational number \( \lambda \) there are infinitely many rational numbers \( m/n \) such that

\( -\frac{1}{\sqrt{5}n^{2}} < \lambda - \frac{m}{n} < \frac{1}{\sqrt{5}n^{2}} \)

Chapter 7 proves that transcendental numbers exist, by showing that Liouville’s number

\( 10^{-1!} + 10^{-2!} + 10^{-3!} + 10^{-4!} + 10^{-5!} + \ldots \)

is transcendental. This is done very carefully in the context of decimal expansions rather than infinite series, so that we don’t have to introduce the concept of convergence.

There are four appendices on related topics: the infinitude of prime numbers, unique factorization, Cantor’s proof (by cardinality) that transcendental numbers exist, and a proof that the trigonometric functions of a rational number of degrees \( \theta \) in the range \( 0^{\circ} < \theta < 90^{\circ} \) is irrational, except for the few obvious cases \( \cos 60^{\circ} = 1/2 \), \( \sin 30^{\circ} = 1/2 \), and \( \tan 45^{\circ} = 1 \). (Chapter 5 proved this for several specific angles.)

There are numerous exercises at the end of each section, and these are well-chosen. Most are drill in applying the results of the section to further examples, and a few are to prove things. Answers and hints to selected problems are in the back of the book. This is a very different book from Niven’s earlier Carus Monograph,

Irrational Numbers. That book is aimed at college students and professional mathematicians. Its prerequisites are higher, but not extremely high: calculus and some algebraic number theory and field theory. It also starts at the beginning, and covers many of the same results, although usually from a more advanced standpoint and with different proofs. It covers some additional topics, such as normal numbers, the generalized Lindemann theorem about the linear independence of powers of \( e \) (this includes the transcendence of \( e \) and \( \pi \)), and the Gelfond–Schneider theorem.