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How to Think Like a Mathematician: A Companion to Undergraduate Mathematics

How to Think Like a Mathematician: A Companion to Undergraduate Mathematics [Kindle Edition]

Kevin Houston
3.9 out of 5 stars  See all reviews (17 customer reviews)

Print List Price: £21.99
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Product Description


"In this book, Houston has created a primer on the fundamental abstract ideas of mathematics; the primary emphasis is on demonstrating the many principles and tactics used in proofs. The material is explained in ways that are comprehensible, which will be a great help for people who seem to hit the wall regarding what to do when confronted with the creation of a proof... In this book, Houston takes a systematic and gentle approach to explaining the ideas of mathematics and how tactics of reasoning can be combined with those ideas to generate what would be considered a convincing proof."
Charles Ashbacher, Journal of Recreational Mathematics

"The author provides concise, crisp explanations, including definitions, examples, tips, remarks, warnings, and idea-reinforcing questions. Houston expresses thoughts clearly and concisely, and includes succinct remarks to make points, clarify arguments, and reveal subleties."
W.R. Lee, Choice Magazine

Product Description

Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.

Product details

  • Format: Kindle Edition
  • File Size: 2478 KB
  • Print Length: 279 pages
  • Page Numbers Source ISBN: 0521895464
  • Simultaneous Device Usage: Up to 4 simultaneous devices, per publisher limits
  • Publisher: Cambridge University Press; 1 edition (6 Jun 2013)
  • Sold by: Amazon Media EU S.à r.l.
  • Language: English
  • ASIN: B00AKE1V4K
  • Text-to-Speech: Enabled
  • X-Ray:
  • Average Customer Review: 3.9 out of 5 stars  See all reviews (17 customer reviews)
  • Amazon Bestsellers Rank: #39,738 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
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Customer Reviews

Most Helpful Customer Reviews
20 of 21 people found the following review helpful
5.0 out of 5 stars Don't let the title put you away 12 Dec 2010
If you are, like I am, trying to get ready for university Maths by yourself, this is the book for you, easy to read, and the content is exactly what you need just after having taken a look back at the A level Maths (you will need at least to have another comprehensive book of A level maths at hand otherwise).

I thought twice about purchasing this book because the title makes it look like one of those "learn how to speak chinese in 15 days with no need to study". But this is not the case. It teaches you the basics of mathematical thinking so you are able to aproach a proof or a theorem with confidence, and above all, how to solve problems and study by yourself.

Also, I earn my living doing computer programming, and found that the book helped me quite a lot with my job as well. I now find myself thinking more logically and methodogically about any problem I come across, instead of rushing to type code and test.

Although it seems like a one day read when you receive it, it took me some 3 months to finnish it with about an hour a day. You have to do some research and be active with the learning (as the book will advice you to do), so it is not a book for a flight read.

I purchased "A Concise Introduction to Pure Mathematics" at the same time I bought this, but found it too hard to follow at first. But after finnished How to Think... the Concise Intro... fits nicely and I find it far far easier to read now.

By the way, I am in Spain and planning to study here, but the book is as good for UK as for any other country as far as I can see.
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27 of 29 people found the following review helpful
As a teacher of A level Maths and Further Maths, I found this book very helpful. I think every prospective maths student should read it, and do all the exercises, before starting their undergraduate mathematics course. It introduces in a rigorous but very readable style the vast majority of the topics that will be covered in the first year at university, and links them all together in a way that cultivates the more formal approach that university level mathematics requires. The author Dr Kevin Houston has been a maths lecturer at Leeds University for more than 15 years, and the book is based on his personal experience of helping to develop students' capacity for mathematical thinking and expression. There are over 300 exercises. Although the answers are not included, Dr Houston is planning to make them available on the internet over the next few months.

Conclusion--thoroughly recommended!
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14 of 16 people found the following review helpful
2.0 out of 5 stars Without solutions, the book is incomplete! 20 Aug 2012
By Apollo
Format:Paperback|Verified Purchase
I decided that I must write a review after working through about a third of the book. The book is well-written and pitched at about the right level for first/second year undergraduates - however, the absence of solutions is intensely irritating.

After completing the exercises for the chapters concerning logical implications (ch 7,8 & 9), I find myself thinking that I probably got most/all(?) of the answers correct - but I'll never be sure. This is SO unsatisfactory! Just to add to your sense of unease, these chapters stress how easy it is to be fooled by interpreting/representing natural [English] language. Furthermore, the examples don't shed light on all the exercises.

I don't mind investing effort in thinking through the exercises, but, at the end of the day, I WANT TO KNOW IF I REALLY DO GET IT!!!

VERDICT: Don't buy it yet. When the book is complete, but it.
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2 of 2 people found the following review helpful
Format:Paperback|Verified Purchase
The text book is attractively laid out and the author communicates very clearly. A lot of thought has gone in to produce a book which is easy to read and understand. There are a lot of worked examples.

However, there is a major flaw with this book. There are no solutions to the exercises. If you are studying on your own then there is no way of checking your solutions and hence your understanding.

If you are an independent student you may want to consider very seriously whether this is the right book for you.
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2 of 2 people found the following review helpful
5.0 out of 5 stars Mathematically enlightening 11 Sep 2011
By Philip
Format:Paperback|Verified Purchase
Really good book, does exactly what it says on the tin. Filled with lots of info and exercises that get you thinking about mathematics in a different way. I found the sections on how to write and solve problems like a mathematician particularly useful. Well worth buying if you want to enhance your understanding of the world of mathematics. I started the book having done half a physics degree; you could probably just about get away with reading it if you have a good GCSE understanding but it's probably best if you learn some A-level maths first.
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1 of 1 people found the following review helpful
1.0 out of 5 stars DONT BUY ON KINDLE 1 Jun 2014
Format:Kindle Edition|Verified Purchase
Many mathematical symbols do not show on kindle app they just appear as a box.. worse than useless. Do not buy on kindles, luckily I have a paper version as well which is excellent. Shame I cant give to reviews.
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11 of 14 people found the following review helpful
5.0 out of 5 stars Very well written! 20 Mar 2010
There is a difference between Alevel maths and degree level. With this book you with find the transformation a lot easier. Read this over the summer holidays and you will have more time to get involved with all that freshers fun :P and hopefully enjoy getting those high grade. A great investment!
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3 of 4 people found the following review helpful
Format:Paperback|Verified Purchase
* Physical

The book has a good font size printing in B&W text.

* Target Audience

This volume is clearly designed using feedback from students to help in the areas where perceived difficulties are common and repeatedly encountered. I feel this book's syllabus covers such a wide area it is capable of addressing new or refreshing forgotten knowledge gaps of many student's, beyond its initial design. Its so beautifully and carefully composed that I would be surprised if you were unable to find nothing totally new or refreshed by this book, almost independent of the level you may be at the moment.

* Personal areas

The parts that I particularly enjoyed were sections iii, iv, v, or to those without the book 'Definitions, Theorems and Proofs', 'Techniques of Proof', and 'Mathematics that all good mathematicians need'. As this is the stuff that I had understood, but things had become bit rusty. Its has a clear ability to remind the reader that scores with me. Its such a well - designed, helpful and monumentally clear exposition of these vital skills that I read it in 3 days and read for hours totally delighted by the way is written and the pitch its put to help its target audience.


Unfortunately, a number of pages from my copy became out of its binding, which is a pity. However, I stapled them together, and with regards of the authors skills I still consider the book as a recommended read and very enjoyable to go over topics in a fresh and entertaining manner.
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Most Recent Customer Reviews
5.0 out of 5 stars Five Stars
Brilliant resource for daughter getting up to speed after gap year before term starts
Published 3 months ago by Lmpd
5.0 out of 5 stars Interesting intro to University level Maths
Accessible, easy to read. pitched at about the right level. The author clearly has a lot of experience and communicates this well.
Published 15 months ago by Richard Hallam
3.0 out of 5 stars Fairly good
Fairly good, although after 3 or 4 months of studying uni maths it seemed a bit basic. Shame there were so many typos!!
Published 15 months ago by Chloe
4.0 out of 5 stars Excellent book for beginning University level maths
This book won't teach you how to do any specific maths techniques, but it will help you to learn how to "do maths". Read more
Published 22 months ago by Browned off
5.0 out of 5 stars Excellent book for the Undergraduate
A perfect companion for the learned Maths reader. It succeeds in helping lay down the essential foundations and mindset that are required for higher level maths. Read more
Published 22 months ago by Sam
5.0 out of 5 stars An amazing read and a brilliant Author!!
An amazing read. Not only has it really helped with my studies (First year at Uni) doing Maths but it's also really interesting. The author is great. Read more
Published 23 months ago by Kenza Hmaimou
3.0 out of 5 stars Horrible quality - Decent content
I would just like to start by stating my main cause of concern, the quality of the book itself is atrocious, the pages literally come right off!..and this is when it was brand new. Read more
Published 23 months ago by S. Karimi
3.0 out of 5 stars Expected more not just list exercises with no solutions or other...
How can a Book list exercises for the reader and not provide solutions so anyone who attempts to solve these exercises get a feedback to see If they are correct or not ? . Read more
Published on 19 Nov 2011 by Michalis
2.0 out of 5 stars Beware if you are self studying
Good book but the Kindle version, at least, does not include solutions to the exercises. You can get partial solutions from the web but not to all of the exercises, which limits... Read more
Published on 3 Oct 2011 by Baggy
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