# What are some good proofs to read? [closed]

I have just started my second year in a maths degrees and I am interested in reading mathematical proofs, I find the proofs to everything I do in class fascinating so I'm looking for some proofs to some more difficult problems that I can try and wrap my head around. I obviously can't jump into something like an attempt at proving Fermat's last theorem, I'm just looking for something to try and challenge myself to make sense of.

## closed as primarily opinion-based by Namaste, Aloizio Macedo♦, Matthew Conroy, Austin Mohr, user21820Apr 1 '17 at 3:54

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

• I nice, geometric flavour, medium-level book (these comments are constrained to my taste, of course) full of proofs and exercises to practice is Apostol's mathematical analysis. – Will M. Mar 31 '17 at 23:41
• An important piece of information to include would be what courses you've taken and even a sketch of what major theorems you know how to prove already. – Stella Biderman Mar 31 '17 at 23:47
• You could just read a textbook on whatever proof-based subject seems interesting to you, like number theory or linear algebra or analysis, etc. Calculus by Spivak is always worth checking out. – littleO Mar 31 '17 at 23:48
• I've done a first year maths course that included linear algebra and calculus, so I'm not too well versed in most areas but I am definitely willing to do some research to be able to understand what I want to read. I'll check out the books you have all given, thanks a lot. – Simon Goodwin Mar 31 '17 at 23:54
• I like the question and it annoys me that it is put on hold because if we only limit ourselves to the class of things for which a non-subjective answer exists we are actually disregarding that mathematics is a discipline of human beings that are individuals. Many opinion-based questions about mathematics are extremely valuable. – exchange Apr 1 '17 at 4:09

The way to understand a theorem, and what a theorem does, is to weaken the hypotheses and find examples that fit the weakened conditions but no longer quite fit the theorem conclusion. In brief, examples. See if I can find the Kendig book quote...

• I love doing this. A Halmos quote comes to mind: "A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Kaj Hansen Apr 1 '17 at 1:16
• Erdos' probabilistic method is really cool in that it uses ideas from probability in a very unique and clever way. There's some background reading you'll need to do, but not much. At the risk of self-promotion, while we're in the Ramsey theory area, check out the proof for $R(3,3) = 6$. A neat little idea, easy to digest even for a (precocious) elementary or middle school student, and yet a satisfying nugget of creativity involved.

• Another favorite of mine (due to its application to a potential magic trick for a skilled card-shuffler) is figuring out how many "perfect" shuffles are required on a deck of $n$ cards to return the stack to the original ordering (way fewer than you might think for some $n$). A little group theory knowledge is required here, though.

• There are some accessible proofs of some neat facts that involve little more than a clever application of the pidgeonhole principle. Googling would almost certainly turn something up in this regard. Ah, yes. Here's one: Consider the points of $\mathbb{Z} \times \mathbb{Z}$ in the plane. Color each either red or blue. Prove that we can find a rectangle with monochromatic vertices.

• The compass-and-straightedge construction of a pentagon: more of a proof-by-demonstration, but thought-provoking nonetheless.

• Set theory: prove that there are infinite sets of different cardinality (this one is really important). In particular, $|\mathbb{N}| < | \mathbb{R} |$.

• More set theory: prove that, when $S$ is an infinite set, $|S| = |S \times S|$.

• Even more set theory (the easiest of the three): Prove that a set $S$ is infinite $\iff$ there exists a subset $T \subset S$ such that $|S| = |T|$. As a side note on this one: it was a magical, eye-opening experience for me seeing for the first time that the cardinalities of the Cantor set and of $\mathbb{R}$ are equal.

• The existence of asymmetric cryptosystems. This was another shock-and-awe, "I cannot believe this is possible" moment for me. The premise is that two strangers who've never met can call each other on the telephone and, using some mathematical trickery, manage to exchange secrets between each other even if the telephones are wiretapped by an opposing party. Read about Diffie-Hellman in particular; it's fairly simple but requires some basic group theory.

I'll append more to this post as things come to mind.

• Interesting video, I'll look into the others as well – Simon Goodwin Apr 1 '17 at 0:17
• I don't think you first link works. – Lisa Apr 1 '17 at 0:17
• The perfect shuffle is covered in the fun book "Magical Mathematics" by Diaconis and Graham. It also exposes the reader to a wealth of other interesting mathematical constructs. – yberman Apr 1 '17 at 0:20
• Oops, not sure how that happened. Thanks for the heads up! – Kaj Hansen Apr 1 '17 at 0:20

Based on what you've said I'm inferring you've had minimal experience reading and (more importantly) writing your own proofs. Although reading proofs of specific theorems can be enlightening, I think that a more systemic approach to learning about proof based mathematics is preferred to reading random proofs.

How to Prove It: A Structured Approach by Daniel J. Velleman is a bit of an eclectic book, but a good introduction to proofs. It might be the best first book to read, especially if you have minimal experience writing your own proofs.

Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand, Polimeni, and Zhang a good grounding in proofs, taking time to developing a variety of proof techniques and set theory. Understanding both of these things (at least a little) is very important in mathematics.

Calculus by Spivak is probably the best calculus book if you haven't done proof based calculus. In my mind this book is just much better than other similar books.

An Introduction to the Theory of Numbers by Hardy and Wright would be a pretty good guide to an introduction to number theory. You might have to consult online references or other books for some context, but it's one of the best number theory books out there. Number Theory is an often ignored but fundamentally important subject, both to modern mathematics and to the history of mathematics. I very much enjoyed watching my friends taking number theory their third or fourth year gawk at the myriad connections to other fields.

• +1 Hardy and Wright. I remember feeling late to the party when I realized that quotient groups are a natural generalization of modular arithmetic. – yberman Apr 1 '17 at 0:22

I would recommend something in elementary number theory (which covers areas like primes and divisibility). It is a very accessible field, and many of the early theorems in it are amenable to testing by hand.

Divisibility tests, such as the process of "casting out nines" for example is taking advantage of modular arithmetic, part of number theory. E.g. $8451$ is divisible by $9$ since $8+4+5+1=18$, and $18$ is divisible by $9$.

I personally like "The Theory of Numbers" by Niven and Zuckerman, but there are probably newer texts that also well suited for a beginner.

You can try the wonderful book 'Proofs from THE BOOK' by Martin Aigner and Gunter Ziegler.

This book is a collection of mathematical proofs where you have more than one proof for a theorem. This book was dedicated to Paul Erdös!

If you want to get a glimpse as to how the greatest minds in mathematics worked, try the book 'Journey through genius' by William Dunham.

A few books about problem solving in mathematics may help you understand how to prove 1)The Art of Mathematics by Bollabas

2)Berkeley Problems in Mathematics by Silva

3)1001 problems in classical number theory by Konnick