Reddit mentions: The best number systems books

It sounds like you might perhaps want a background in Number Theory and/or Basic Logic and/or Set Theory. The thing about math is that there is a lot...

My advice for a text that might serve you well is N.L. Biggs' Discrete Mathematics (http://www.amazon.com/Discrete-Mathematics-Norman-L-Biggs/dp/0198507178). If you are at all interested in computer science, this is also a great book for that because it introduces some of the mathematical rigor behind it. Some people have a smidgen of difficulty with this text because it doesn't give some names to proofs/algorithms that maybe you've heard whispered (e.g. Dijkstra's shortest path and Prim's minimal spanning tree). A text that I tend to think is on par with Biggs', but many think is vastly superior (I love both, but for different reasons) that covers some (most) of the same topics is Eccles' An Introduction to Mathematical Reasoning (http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/ref=pd_sim_b_4?ie=UTF8&refRID=1BB6VKRP59S2420M132F). This book has a wonderful focus on building from the ground up and emphasizes clearly worded and mathematically rigorous proofs.

You seem genuinely interested in mathematics, but I do want to warn you about some more ahem esoteric (read: improperly worded, perhaps?) problems that ask such things as why 1 is greater than 0. The mathematics here is largely armchair - lacking any fundamental logic. There would be no issue with redefining a set of bases such that "" is greater than "1". However, if you want to have rationale of the concept of things being greater than another, that's more like number theory. You can learn the 10 axioms of natural numbers and then build from there.

Both of the books I mentioned will cover stuff like this. For example, they both (unless I'm not remembering correctly) delve into Euclid's proof of infinite primes, something which may interest you.

Briefly (and not so rigorously), assume that the number of primes, p1, p2, p3, ..., pN, is finite. Then there exists a number P which is the product of these primes. Based on the axioms of natural numbers, since all primes p1,p2,...,pN are natural numbers P is a natural number and so is P+1. Consider S = P+1. If S is prime than our list is incomplete, assume S isn't prime. Then some number in our list, say pI, divides S because any natural number can be written as the product of primes. pI must also divide P because P equals the sum of all primes. Therefore if pI divides S and pI divides P, then pI divides S-P = 1. That's a contradiction because no prime evenly divides 1.

Stuff like this is super cool, super simple, and super beautiful and you absolutely can learn it. These two books would be a great place to start.

There are essentially "two types" of math: that for mathematicians and everyone else. When you see the sequence Calculus(1, 2, 3) -> Linear Algebra -> DiffEq (in that order) thrown around, you can be sure they are talking about non-rigorous, non-proof based kind that's good for nothing, imo of course. Calculus in this sequence is Analysis with all its important bits chopped off, so that everyone not into math can get that outta way quick and concentrate on where their passion lies. The same goes for Linear Algebra. LA in the sequence above is absolutely butchered so that non-math majors can pass and move on. Besides, you don't take LA or Calculus or other math subjects just once as a math major and move on: you take a rigorous/proof-based intro as an undergrad, then more advanced kind as a grad student etc.

To illustrate my point:

Linear Algebra:

1. Here's Linear Algebra described in the sequence above: I'll just leave it blank because I hate pointing fingers.
   
   
2. Here's a more serious intro to Linear Algebra:
   
   Linear Algebra Through Geometry by Banchoff and Wermer
   
   3. Here's more rigorous/abstract Linear Algebra for undergrads:
   
   Linear Algebra Done Right by Axler
   
   4. Here's more advanced grad level Linear Algebra:
   
   Advanced Linear Algebra by Steven Roman
   
   -----------------------------------------------------------
   
   Calculus:
   
   
3. Here's non-serious Calculus described in the sequence above: I won't name names, but I assume a lot of people are familiar with these expensive door-stops from their freshman year.
   
   
4. Here's an intro to proper, rigorous Calculus:
   
   Calulus by Spivak
   
   3. Full-blown undergrad level Analysis(proof-based):
   
   Analysis by Rudin
   
   4. More advanced Calculus for advance undergrads and grad students:
   
   Advanced Calculus by Sternberg and Loomis
   
   The same holds true for just about any subject in math. Btw, I am not saying you should study these books. The point and truth is you can start learning math right now, right this moment instead of reading lame and useless books designed to extract money out of students. Besides, there are so many more math subjects that are so much more interesting than the tired old Calculus: combinatorics, number theory, probability etc. Each of those have intros you can get started with right this moment.
   
   Here's how you start studying real math NOW:
   
   Learning to Reason: An Introduction to Logic, Sets, and Relations by Rodgers. Essentially, this book is about the language that you need to be able to understand mathematicians, read and write proofs. It's not terribly comprehensive, but the amount of info it packs beats the usual first two years of math undergrad 1000x over. Books like this should be taught in high school. For alternatives, look into
   
   Discrete Math by Susanna Epp
   
   How To prove It by Velleman
   
   Intro To Category Theory by Lawvere and Schnauel
   
   There are TONS great, quality books out there, you just need to get yourself a liitle familiar with what real math looks like, so that you can explore further on your own instead of reading garbage and never getting even one step closer to mathematics.
   
   If you want to consolidate your knowledge you get from books like those of Rodgers and Velleman and take it many, many steps further:
   
   Basic Language of Math by Schaffer. It's a much more advanced book than those listed above, but contains all the basic tools of math you'll need.
   
   I'd like to say soooooooooo much more, but I am sue you're bored by now, so I'll stop here.
   
   Good Luck, buddyroo.

Not much, the nice thing for upper math courses is they do a good job of building up from bare bones. If you have some linear algebra and a multivariable calc course you should be good. The big requirement is however mathematical maturity. You should be able to read, understand, and write proof.

A very basic intro to proofs course is usually taught to first year math students, this covers set notations, logic, and some basic proof techniques. A common reference is "How to prove it: a structured approach", I learned from Intro to mathematical thinking. The latter isn't as liked, it does seem to cover some material that I think should be taught early. A lot of classical number theory and algebra, for example fundamental theorem of arithmetic, and Fermat's little (not last) theorem are proven. Try to find a reference for that stuff if you can.

It's really important to do a proof based linear algebra class. It helps build the maturity I mentioned and will make life easier with topology. But even more importantly teaching linear algebra in a more abstract way is important for a physics undergrad as it can serve as a foundation for functional analysis, the theory upon which quantum mechanics is built. And in general it is good to stop thinking of vectors as arrows in R^n as soon as possible. A great reference is Axler's LADR.

Again not strictly required, but it helps build maturity and it serves as a good motivation for many of the concepts introduced in a topology class. You will see the practical side of compact sets (namely they are closed and bounded sets in R^(n)), and prove that using the abstract definition (which is the preferred one in topology). You will also prove some facts about continuous functions which will motivate the definition of continuity used in topology, and generally seeing proofs about open sets will show you why open sets are important and why you may wish to look at spaces described only by their open sets (as you will in topology). The reference for real analysis is typically Rudin, but that can be a little tough (I'm sorry, I can't remember the easier book at the moment)

Edit: I will remove this as it doesn't meet the requirements for an /r/askscience question, we usually answer questions about the science rather than learning references. If you feel my answer wasn't comprehensive enough feel free to ask on /r/math or /r/learnmath

Here are some great books that I believe you may find helpful :)

• Elements of Algebra by Leonhard Euler
  
• Foundations of Mathematics by Ian Stewart
  
• Geometry Revisited by H. S. M. Coxeter
  
• Excursions in Calculus by Robert M. Young
  
• New Horizons in Geometry by Tom M. Apostol and Mamikon A. Mnatsakanian
  
• The Calculus Gallery: Masterpieces From Newton to Lebesgue by William Dunham
  
• A Primer of Real Functions by Ralph P. Boas
  
• Understanding Analysis by Stephen Abbot
  
• Chaos: Making a New Science by James Gleick
  
• Logic: An Introduction to Elementary Logic by Wilfred Hodges
  
  and last but definitely not least:
  
  
• A Mathematical Gift I, II, III by Kenji Ueno et al.
  
  Later on:
  
  
• Conceptual Mathematics: A First Introduction to Categories by F. William Lawvere
  
• Twelve Landmarks of Twentieth-Century Analysis by D. Choimet & H. Queffélec
  
• Theory of Complex Functions by Reinhold Remmert
  
• Algebraic Combinatorics. Walks, Trees, Tableaux & More by Richard P. Stanley

Well, if you want something light and accessible and suitable for the layperson, I'm quite fond of Jan Gullberg's Mathematics from the Birth of Numbers. It goes over basically everything you would typically learn in primary and secondary school, and it presents everything with historical background. But it doesn't go into tremendous detail on each topic, and it doesn't provide the most rigorous development. It's more of a high-level overview.

But if you really want to learn some mathematics, on a deep and serious level, be prepared to read and study a lot. It's a rewarding journey, and we can give you book recommendations for specific topics, but it does take a lot of discipline and a lot of time. If you want to go that route, I would recommend starting with an intro to proofs book. I like Peter J. Eccles's An Introduction to Mathematical Reasoning, but there are many other popular books along the same line. And you can supplement it with a book on the history of mathematics (or just read Gullberg alongside the more serious texts).

There's a pretty good outline of the stuff here. The third really follows from the previous two, fairly directly. The real connection between this product and the Euler-Mascheroni constant is then basically the fact that Merten's Constant is equal to (Euler-Mascheroni) + (Extra term involving primes). This is the hard part of Merten's Theorem and many modern treatments of the theorem just show that such a constant exists, but don't relate it to the Euler-Mascheroni constant (see this). This article, however, does do Mertens some justice. Though, This book does the same and presents it nicely, using more modern considerations (but I can't link a whole book to you).

It depends a bit on what your college offers and what you think you would benefit from.

On the math side an intro to proofs course would be helpful because it teaches you a lot of math and reasoning that undergirds a lot of cs. A book like this is good for self study: https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188

If your school has a good philosophy dept, then check out the classes on critical reasoning and logic too. Herrick's logic book is not bad for a first course.

take the most advance math courses you can. do undergraduate research...summer programs, independent studies. make sure to write a math research paper. it doesn't have to be published, but a published paper would look great. give a talk about your research at an undergraduate math conference. go to many math conferences. many schools require the math subject test gre, which is difficult and requires a fair amount of study outside of coursework.

that being said, since you are still a beginner, be warned that upper level math is very different than high school math. after a certain point, computations are no longer of use and all math is theoretical and abstract. you will be focusing on "proofs" and generally these are much more logic based and theoretical than any math you do before university. any proofs you did proofs in a highschool geometry class are also not relevant. to get a better idea, look at an elementary proof-writing book. for example http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188/ref=sr_1_2?s=books&ie=UTF8&qid=1320289226&sr=1-2#reader_0521597188

more specifically, once you are enrolled in a phd program, you will have to take at least 2 years of coursework. you will also need to pass one or two sets of "qualifying exams", the number and style of testing is based on the university. these test you on your basic knowledge of math, and also on the subject of your research. to obtain a phd you have to do NEW mathematical research and then write a dissertation about it. the research part of the phd can take 2-4 years on average.

Paul Nahin has published many good historical math books that don't skimp on the mathematical underpinnings. I particularly enjoyed An Imaginary Tale: http://www.amazon.com/An-Imaginary-Tale-Princeton-Science/dp/0691146004

Regarding Spivaks: I'm also working on it, and found that my proof technique was lacking. An Introduction to Mathematical Reasoning (Eccles) was helpful for me: http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188

Classes started back yesterday and I'm already working on the first chapter in my Number Theory book. Does anybody have any experience with The Whole Truth About Whole Numbers?

I'm also in the second semester of Real Analysis and we began working toward the Riemann Integral yesterday. Mathematics truly amazes me, and I'm always left in wonder at these guys who figured this stuff out.

I'm currently in my first year of undergraduate Maths and our course uses the book 'An Introduction to Mathematical Reasoning: Numbers, Sets, and Functions' by Peter J Eccles. It's such a helpful book aimed at introducing first year university students to pure mathematics, the book has definitely helped me feel confident in my pure module.

It states propositions and theorems and proves them and gives problems for you to solve or prove with the solutions at the back.

Learn to write proofs. Learn how to write mathematically. It's not just about knowing how to solve calculus problems.

Pick up this book. Very likely, it's math you already know - but that's not what it's about. It's about learning to show your work in a clean, elegant, and concise manner.

This was the class as it was last quarter (Spring 2012), they used this textbook. I live off-campus and only go to the UW to drop off homework and almost never talk to anyone, so you almost definitely don't know me, but perhaps we walked right by each other one time without ever knowing it. EXCITING!

It depends on where they are and what the purpose is. If you are trying to discourage them (and there might be valid reasons to do that), I'd say try measure theory.

Maybe use the Bartle book.

That would give them a taste for how abstract things can get and also drive home the point tiny books can require a lot o work.

On the other hand, if you want to do something that will help them, they An Introduction to Mathematical Reasoning.

It won't break the bank and, despite a few small typos, covers a lot material fairly gently.

Apologies if you felt my tone was lacking, however I did mean what I said. The best discussion of this subject is to be had in a first year algebra book: specifically take a look at http://www.amazon.com/Introduction-Mathematical-Thinking-Algebra-Systems/dp/0131848682.
My own knowledge base is insufficient to explain the process and reasoning.

Start with a book like this:

http://www.amazon.com/books/dp/0521597188

or this:

http://www.amazon.com/How-Prove-It-Structured-Approach/dp/0521675995

or the one teuthid recommended. When you're doing self-study, it's doubly important to be able to read and follow most of the material.

An Introduction to Mathematical Reason - Peter Eccles. Very good book.

http://www.amazon.com/Introduction-Mathematical-Reasoning-Peter-Eccles/dp/0521597188

Pick up a math reasoning and proofs book. Eccles Introduction to Math Reasoning is good.

There is actually a book called An Introduction to Mathematical Reasoning.

I used this book in my first mathematical reasoning class at my university.

If you are still an undergrad and your school offers a "how to prove stuff and how to think about abstract maths" course take it anyway. No matter how far along you have come.

An example text for such a course is this one:

https://www.amazon.com/Introduction-Mathematical-Reasoning-Numbers-Functions/dp/0521597188

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As for Linear Algebra (the most useful part of all higher mathematics for sure (R/math: if you disagree, fight me on this one...i'll win) ) I will tell you i learned a LOT from these two texts:

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https://www.amazon.com/Linear-Algebra-Introduction-Mathematics-Undergraduate/dp/0387940995

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https://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mathematics/dp/3319110799/ref=pd_lpo_sbs_14_img_0?_encoding=UTF8&psc=1&refRID=APH3PQE76V9YXKWWGCR9

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