List them here! Make sure you mention the subject and give a small review (i.e. tell us why it's so exceptional).

One of my favorites is Knuth's Concrete Mathematics : A foundation for computer science. Don't be fooled by the title, it's just meant to lure computer scientists into doing math.
This book is an excellent collection of brilliantly presented ideas. It's very accessible yet full of priceless gems unknown even to the fairly experimented mathematician. It's a delightful read, filled with absurd jokes in the margins and cracking full of beautiful identities of all kinds and various ways to derive them. (The formula for is derived in probably 10 different ways throughout the book and it becomes somewhat of a running gag - the authors promise they won't prove it again but they always end up doing so).

Yep, it's good, that one.

I'm currently doing George F. Simmons' "Differential Equations with Applications and Historical Notes". It's good stuff, he doesn't hang about, but he does give answers at the back (but not the full working). Recommended.

Whitelaw's "An Introduction to Abstract Algebra" is also very good, doesn't go too fast, again with full answers in the back.

That's a couple. There are loads more on my shelves.

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I found both of these very interesting. A lay person's insight into some pretty hairy topics.

The Code Book

FLT -The Book

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I'm not sure if you could call Simon Singh "a lay person" - he does have a PhD in particle physics. Although I suppose it depends on your intrepretation of "lay".

My favourite text book is Robinson's "A Course in the Theory of Groups". It took me a while to find in the the library though - I thought it was by Robertson, and there are a number of different ways of fitting "Course" "Group" and "Theory" into a sentence, all of which seemed to be a title of a book in the library!

I enjoyed reading "The Music of the Primes" by Marcus du Sautoy, although I still have no idea what the Riemann Hypothesis is actually asking!

My next "intellectual" read is one of Stephen Hawking's tomes - I've somehow managed to aquire "A Brief History of Time","Black Holes and Baby Universes", and "Standing on the Shoulders of Giants" without reading any of them. Does anyone have any ideas which I should start with?

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It's asking if all the zeroes of the Zeta function have their real part equal to 1/2.


As for the books... no idea, they're almost all in French

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Well, yeah. But what does that actually mean?

We had to do a talk as part of our degree - I think it was so the uni could tick a box. It was unassessed though, and this led to some very poor talks, including one by one of my friends who had not preparing for his at all, so he stuttered for 3 minutes about his dissertation then sat down (they were meant to last about 20 minutes...). Anyway, I went to one that was on the Riemann Hypothesis, which I thought would be interesting (I have a love/hate relationship with Analysis - I love her, she hates me). However, the lecturer who was looking after the talk stuck his hand up after 2 minutes, and the other 18 minutes was spent by the lecturer trying to get the poor student to define the question correctly. From what I could gather, something trixy happens with the line where the zeros are - the line is sort of artificially moved to where you want it to be, else where is a problem...what this problem is, I cannot remember. But it was probably division by zero. It usually is.

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You might want to add non-trivial there.

About books, here is my favourites;

Jean Bertoin - Levy Processes
Andreas Kyprianou - Introductory Lectures on Fluctuations of Levy Processes with Applications
Rogers & Williams - Diffusions, Markov Processes and Martingales
Ken Iti Sato - Levy Processes and Infinitely Divisible Distributions
Revuz & Yor - Continuous Martingales and Brownian Motion

If you want a nice book on analysis I would recommend Fomin and Kolmogorov - Introductory Real Analysis, which you can find very very cheap on Amazon.

The F&K book on real analysis is *far* from "introductory" - unless you count measure theory as "introductory". When I started it, it was way over my head. I might be ready to have another hack at it.

Also found: the big fat "Modern Algebra" by Seth Warner. Although it uses either archaic or non-standard notation, it *explains* everything, and by a third of the book he's justified the behaviour of the natural, real and rational number systems. Then he was off into ideal theory and lost me. Exercises can be fiendish though and there's no answers given.

On the same subject is the slim book "Abstract Algebra" by Allan Clark who builds up to Galois theory in an attracttively dense presentation. Again, no answers to exercises of varied levels of difficulty (you never know what you're going to get when you start). Most of it can be found elsewhere, though, so you can find your way around if you search.

The Schaum "Mathematical Handbook of Formulas and tables" is invaluable.

I've also got a great pile of so-so books on mathematical logic at the top of my heap at the moment which I can't move for.

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"No matter how fast or how far you run, you're still in the space where you are." -- Russell

Opps, was meant to infer the reader was lay.

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It is an introductory book in the sense that it starts from set theory, covers a bit of functional analysis, metric spaces and topology, then finishes with measure theory. I learnt all my measure theory off books, I found F&K a bit weird in that section. I much prefer the book by Roger & Williams. Very nice first chapter on Brownian motion and Levy processes and the measure theory bit is how I like it.

Indra's Pearls, by David Mumford,Caroline Series, & David Wright.

I don't think there is any book which looks so like nothing more than a "coffee table" book, assumes so little in the way of existing knowlege from the reader, but contains more real mathematics, albeit of an experimental and practial rather than rigorous kind.

Modern Geometries, by Michael Henle.
It's short and full of misprints, but equips the reader with most of what's required to be able to solve coordinate geometry problems in almost any geometry you care to mention, as well as dipping a toe into topics such as graph theory and discrete symmetry groups. Its also got a lot of interesting material on the connections between mathematics and art and literature: it's certainly the only maths book I have which along with the typical proofs and calculations has extensive quotations from writers such as Tom Stoppard and Dostoyevsky, and questions like "Analyse the use of perspective in the following works of Art..." or "Survey literary attitudes towards mathematics, particularly toward geometry (non-Euclidean and higher dimensional)"

"Fractals for the Classroom" (part 2) (I've not found part 1 yet) by Peitgen,Jurgens,Saupe.
This is another excellent experimental mathematics type book that covers a lot of ground. In particular it has a very good introductions to Julia sets and to discrete chaos (period doubling cascades, Feigenbaum constant etc).

My Favourite: generatingfunctionology by Herbert Wilf.

The exposition is crystal clear, and it's full of insight and interesting techniques. It shows how you can use generating functions for a number of things, for example: evaluating sums, obtaining asymptotic formulas for strange sequences.
I think that if you bought Concrete Mathematics, you will quite likely enjoy this book very much. - Ah!, and it smells good too ! -

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I'm looking into all these books, but I think they're a little too advanced for me. I've got Calculus by Spivak and a different Calculus book by Apostol...I hear they are rigorous.

I've also got The Calculus Lifesaver by Adriann Banner, which has proved invaluable to me.

I've been ordering old books also because I feel they have not dumbed down the subject and I was wondering if anyone has any book recommendations that they enjoyed if they're a bit older or have a bit more experience sifting through math textbooks...

I've got Principles of Mathematics by Allendoerfer and Oakley. I got it for cheap too, but the quality of content is great.

Not to worry about dumbing down, I haven't seen evidence of that in modern texts. Old ones are interesting for historical reasons, but I've found that newer ones tend to be more readable. I haven't yet found an undergrad-level maths text that I've been fully able to master.

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"No matter how fast or how far you run, you're still in the space where you are." -- Russell