A Course of Pure Mathematics: Third Edition

Cambridge Mathematical Library

by G.H. Hardy

Originally published in 1908, this classic calculus text transformed university teaching and remains a must-read for all students of introductory mathematical analysis. Clear, rigorous explanations of the mathematics of analytical number theory and calculus cover single-variable calculus, sequences, number series, and properties of cos, sin, and log. Meticulous expositions detail the fundamental ideas underlying differential and integral calculus, the properties of infinite series, and the notion of limit.
An expert in the fields of analysis and number theory, author G. H. Hardy taught for decades at both Cambridge and Oxford. A Course of Pure Mathematics is suitable for college and high school students and teachers of calculus as well as fans of pure math. Each chapter includes demanding problem sets that allow students to apply the principles directly, and four helpful Appendixes supplement the text.

Genres: Mathematics, Reference, Science, Nonfiction, Textbooks, Technical, Philosophy

464 pages, Paperback

First published January 1, 1908

About the author

Godfrey Harold Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis.

Non-mathematicians usually know him for A Mathematician's Apology, his essay from 1940 on the aesthetics of mathematics. The apology is often considered one of the best insights into the mind of a working mathematician written for the layman.

His relationship as mentor, from 1914 onwards, of the Indian mathematician Srinivasa Ramanujan has become celebrated. Hardy almost immediately recognized Ramanujan's extraordinary albeit untutored brilliance, and Hardy and Ramanujan became close collaborators. In an interview by Paul Erdős, when Hardy was asked what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan. He called their collaboration "the one romantic incident in my life."

Reviews

[Charles · Rating 5 out of 5]

Although the sequence of the presentation of the fundamentals of mathematics has changed over the last century, the substance has not. There is no greater evidence of this fact than this classic work by Hardy, which could be used without alteration or additional explanation as a text in modern college mathematics courses. Hardy was rightfully known as a bit of an eccentric, yet he was a brilliant pure mathematician and he will always be held in the highest regard for his actions in aiding the Indian prodigy Ramanujan. Less well known but still extremely significant is his expository writing; there are few who wrote as clearly as he did.
This book was extremely influential in the teaching of mathematics over the last century. The primary subject matter is:

*) Real variables
*) Functions of real variables
*) Complex numbers
*) Limits of functions of a positive integral variable
*) Limits of functions of a continuous variable
*) Derivatives and integrals
*) Theorems on the differential and integral calculus
*) Convergence of infinite series and infinite integrals
*) Logarithmic, exponential and circular functions of a real variable
*) General theory of the logarithmic, exponential and circular functions

There are few proofs, but an enormous number of examples. The mathematical influence of G. H. Hardy over mathematical education was and remains strong, as can be seen by reading this masterpiece.

Published in Journal of Recreational Mathematics, reprinted with permission and this review appears on Amazon

[Yasiru · Rating 5 out of 5]

While more contemporary introductions to real analysis like Rudin's strive for elegance and perhaps seem tryingly terse while tying the subject neatly together, Hardy manages to convey an excitement about the subject that may be invaluable motivation for a beginner.

There is a historical precedent for this attitude, since during Hardy's time English mathematics was almost wholly concerned with the applied side (continuing the tradition of Newton and many other fine applied mathematicians over the years). Hardy's aim was to spark interest in analysis where it had not before taken ground.

The organisation of the material and what is emphasised may not always be as things would be done today, but Hardy is ever rigorous, and the clarity of arguments together with the wealth of challenging problems make for an engaging style. This an ideal read either on its own, or as a complement to a prescribed course text on analysis.

[Mike · Rating 1 out of 5]

I decided to read through this book to see if it was worth studying in detail. As in a few pages a day and writing out mostly everything by hand to really learn the material. What I have found was a lot of the things covered in this book are in any modern calculus book and because of the old notation used in equations and the amount of skips in the proofs, etc. I was constantly going back to my calculus book to fill in what Hardy left out. So I said to myself I might as well be reading my calculus book than Hardy's book. The techniques on integration seem dated. If you want a book with similar integral problems look at the PDF of N. Piskunov Differential and Integral Calculus. He gives examples and steps that are very easy to follow. I learned that one of the integration techniques is called Euler substitution. Google it for yourself. Sure this book my have things others do not, but you can compare 10 modern math books and they each will have something not in the other. The problems require a skill that reading the book will not give you. Many are modern day Putnam equivalent. They are part of the Mathematical Tripos and students had to spend 3 years to prepare for the test to graduate. To do well you had to hire coaches, and their only job was to prepare you for the exam. So the problems are probably not the best for self study, unless you've had training in math competitions. I have the 10th edition 1952 and I've only been reading it for a short time but the cover is becoming frail and rough in my hands. It was perfectly smooth and I only read it in my room.

[Matt · Rating 5 out of 5]

Embarrassingly, the problems remain extremely difficult for me. I long for an answer and study guide.

[Ian Carmichael · Rating 5 out of 5]

Beautiful writing, and typesetting.

[William Schram · Rating 5 out of 5]

A Course of Pure Mathematics by G. H. Hardy is meant to be an introduction to math that has no direct applications. Although people with interest in such things as engineering can gain something from this book, they are not the primary intended audience. Since it is a “Course” in Pure Mathematics, this book contains plenty of examples and exercises. However, since it is also for a student, it doesn’t show you the answers to the exercises, or at least I have not found them.

Since this book is before calculators it also has a lot of interesting things about it, like the Root Extraction formula and other stuff like that. As the blurb on the back mentions, this book mostly covers mathematics that utilizes the notion of the limit. So Calculus topics are covered in here. The table of contents goes further in depth into what you might find if you just want to skim it.

All in all, this book was quite well done, with this edition being a reprint of the tenth edition.

[Lee · Rating 3 out of 5]

Got half way

[Oliver Ford · Rating 5 out of 5]

Excellent read, would recommend to any A level Further Mathematics student, I should have finished it then. Great A level/STEP/pre-College prep.

[Jorge Vieira S Junior]

Nihil