(Part 2) Best products from r/mathematics

Here's an easy read that I liked: Concepts of Modern Mathematics by Ian Stewart. It gives a pretty broad overview. And you can't beat the price of those Dover paperbacks.

You may also be interested in a more thorough exploration of the history of the subject. Try History of Mathematics by Carl Boyer.

Honestly, two great books for you to use are:

1. Engineering Mathematics by Stroud
   
2. Advanced Engineering Mathematics by Stroud
   
   It takes a 'cookbook' approach, that is, learning the techniques and application, of the most important/relevant engineering mathematics. So no deeper, pure mathematics, learning proofs approach.
   
   Both are available for 'free' on the internet . . . at the libgen place and elsewhere ;)
   
   I used them to refresh a lot of my mathematics for a graduate programme in finance.
   Absolutely loved them!
   
   

Journey Through Genius: The Great Theorems of Mathematics by William Dunham. It's currently US$10.14 in paperback on Amazon.com. It includes plenty of interesting mathematics, as well biographical profiles of a number of mathematicians. It's also definitely suitable for someone your brother's age and with his current mathematical background.

Honestly, if she has passion for math to the extent that she wants to learn calculus over the summer, she'll find the classroom pace annoyingly slow. AP Calculus can be taught in 2 months, but they stretch it into 8 months.

I always recommend Stewart's calculus book,
http://www.amazon.com/Calculus-6th-Edition-Stewarts-Series/dp/0495011606/ref=sr_1_1?ie=UTF8&qid=1372050088&sr=8-1&keywords=stewart+calculus+6th+edition
It's a college-level textbook, but it starts where a high school student should be comfortable. Only the first 7-8 chapters apply to AP Calculus.

The book that really hooked me on math (I was an undergraduate math major) was G. H. Hardy's, "A Mathematician's Apology". You can find free versions online, because over 50 years have passed since publication. But the free versions I saw don't contain the introduction by C.P. Snow that the book has. So you might consider getting the book, either out of the library or from Amazon.

Two other recommendations would be:

• James R. Newman's 4-volume, "The World of Mathematics"
  
  
• E. T. Bell's, "Men of Mathematics"
  
  All three of those kept me duly inspired before and during my undergraduate years.

Daniel J Velleman's How to Prove It : A Structured Approach


This book is a pretty dang good intro to proofs, I highly reccommend it. This is the first edition, so you'll be able to find a used copy for super cheap.

The books Engineering Mathematics and Advanced Engineering Mathematics are fantastic self-teaching texts which will take you from basic arithmetic to complex analysis in a series of frame-by-frame "programmes". They will reinforce what prior knowledge you do have (allowing you to brush up on important algebra, for example), while gently and swiftly bringing you through trig, pre-calc, and then calculus. All of that is in the first volume (which also has matrix algebra, probability, and statistics). You may not need the second volume, but it's just as good for the same reasons. I've found that it's basically impossible to be confused by these books. They make sure you learn.

I highly recommend Cornerstones of Decidability. It is presented in an accessible way but does not sacrifice rigour in the process. Rozenberg and Salomaa are not only very accomplished theoretical computer scientists, they are outstanding teachers as well.

I'm a huge origami fan. Great books for teachers:

https://smile.amazon.com/Origamics-Mathematical-Explorations-Through-Folding/dp/9812834907/ref=mp_s_a_1_2?ie=UTF8&qid=1538344551&sr=8-2&pi=AC_SX236_SY340_FMwebp_QL65&keywords=origami+explorations+math&dpPl=1&dpID=41HUKVEXbXL&ref=plSrch

https://smile.amazon.com/Project-Origami-Activities-Exploring-Mathematics/dp/1466567910/ref=mp_s_a_1_9?ie=UTF8&qid=1538344609&sr=8-9&pi=AC_SX236_SY340_FMwebp_QL65&keywords=math+origami&dpPl=1&dpID=515Ywuc4QYL&ref=plSrch

I know of a few:

Critical Mass: How One Thing Leads to Another, by Philip Ball. This one is absolutely amazing. It covers how large group of small objects/individuals (each one only obeying simple rules based on what their neighbours are doing) can behave in large scale ways, eg landslides, traffic congestion, schooling fish, swarming birds, pedestrian flow. For me, it was eye opening because I had never realised that many of the random occurrences are actually still governed by some predictable rules.

Prime Obsession, by John Derbyshire. It is about a famous conjecture that has remain unresolved, and is surprisingly difficult to crack. This book covers some of the history, the fundamentals of the mathematical ideas, some anecdotal stories about some of the mathematicians who have worked on this problem. It is designed to be pretty accessible, so you do not need to be an expert in this area to be able to enjoy this book.

I love Russell's Principles of Mathematics for its exposition of the history of mathematical development, and found it really useful for putting some of the hairier concepts into context. Have you read it?

I really enjoyed going through Abstract Algebra: An Introduction by Hungerford. There is an option to begin with Group Theory or begin with Ring Theory, if that's important to you. I think Hungerford's book is a very good introduction with plenty of exercises.

John Bird has great books running the gamut from simple arithmetic to differential equations.

You can download them on gen.lib.rus.ec or buy from Amazon. If you want to stick to high school material then the book to go for is Basic Engineering Mathematics and if you want an "all-in-one" then it's Understanding Engineering Mathematics.

If you learn everything in the latter, you will most likely only need to supplement your knowledge of probability and discrete mathematics during your bachelor studies and learn how to do mathematics at a more conceptual level and work with proofs. But what you would need to supplement will be covered during your studies and would not likely be assumed as prior knowledge in courses unless the courses stipulate certain undergrad courses in probability and discrete math or math for CS as prerequisites.

Any suggestion for interesting proofs to remember?

I really love (one of) the proofs to Fermat's Little Theorem. I might actually be able to reproduce it.

There is of course Proofs from the book.

My go to book for anything graph theory related is the intro book by West.

Great book for undergrad / first year grad students. Goes into detail on numerous topics and if I can recall, you can find a bit of good application there. I know computer science replies on applications of graph theory quite a bit, so you may be able to delve further into that.

http://www.amazon.com/Engineering-Mathematics-K-A-Stroud/dp/0831134704/ref=pd_sim_sbs_b_1?ie=UTF8&refRID=05GTV6C48KY4RJSHCK3W