Reddit mentions of Elementary Number Theory: Second Edition (Dover Books on Mathematics)

You are in a very special position right now where many interesing fields of mathematics are suddenly accessible to you. There are many directions you could head. If your experience is limited to calculus, some of these may look very strange indeed, and perhaps that is enticing. That was certainly the case for me.

Here are a few subject areas in which you may be interested. I'll link you to Dover books on the topics, which are always cheap and generally good.

• The Nature and Power of Mathematics, Donald M. Davis. This book seems to be a survey of some history of mathematics and various modern topics. Check out the table of contents to get an idea. You'll notice a few of the subjects in the list below. It seems like this would be a good buy if you want to taste a few different subjects to see what pleases your palate.
  
  
• Introduction to Graph Theory, Richard J. Trudeau. Check out the Wikipedia entry on graph theory and the one defining graphs to get an idea what the field is about and some history. The reviews on Amazon for this book lead me to believe it would be a perfect match for an interested high school student.
  
  
• Game Theory: A Nontechnical Introduction, Morton D. Davis. Game theory is a very interesting field with broad applications--check out the wiki. This book seems to be written at a level where you would find it very accessible. The actual field uses some heavy math but this seems to give a good introduction.
  
  
• An Introduction to Information Theory, John R. Pierce. This is a light-on-the-maths introduction to a relatively young field of mathematics/computer science which concerns itself with the problems of storing and communicating data. Check out the wiki for some background.
  
  
• Lady Luck: The Theory of Probability, Warren Weaver. This book seems to be a good introduction to probability and covers a lot of important ideas, especially in the later chapters. Seems to be a good match to a high school level.
  
  
• Elementary Number Theory, Underwood Dudley. Number theory is a rich field concerned with properties of numbers. Check out its Wikipedia entry. I own this book and am reading through it like a novel--I love it! The exposition is so clear and thorough you'd think you were sitting in a lecture with a great professor, and the exercises are incredible. The author asks questions in such a way that, after answering them, you can't help but generalize your answers to larger problems. This book really teaches you to think mathematically.
  
  
• A Book of Abstract Algebra, Charles C. Pinter. Abstract algebra formalizes and generalizes the basic rules you know about algebra: commutativity, associativity, inverses of numbers, the distributive law, etc. It turns out that considering these concepts from an abstract standpoint leads to complex structures with very interesting properties. The field is HUGE and seems to bleed into every other field of mathematics in one way or another, revealing its power. I also own this book and it is similarly awesome. The exposition sets you up to expect the definitions before they are given, so the material really does proceed naturally.
  
  
• Introduction to Analysis, Maxwell Rosenlicht. Analysis is essentially the foundations and expansion of calculus. It is an amazing subject which no math student should ignore. Its study generally requires a great deal of time and effort; some students would benefit more from a guided class than from self-study.
  
  
• Principles of Statistics, M. G. Bulmer. In a few words, statistics is the marriage between probability and analysis (calculus). The wiki article explains the context and interpretation of the subject but doesn't seem to give much information on what the math involved is like. This book seems like it would be best read after you are familiar with probability, say from Weaver's book linked above.
  
  
• I have to second sellphone's recommendation of Naive Set Theory by Paul Halmos. It's one of my favorite math books and gives an amazing introduction to the field. It's short and to the point--almost a haiku on the subject.
  
  
• Continued Fractions, A. Ya. Khinchin. Take a look at the wiki for continued fractions. The book is definitely terse at times but it is rewarding; Khinchin is a master of the subject. One review states that, "although the book is rich with insight and information, Khinchin stays one nautical mile ahead of the reader at all times." Another review recommends Carl D. Olds' book on the subject as a better introduction.
  
  Basically, don't limit yourself to the track you see before you. Explore and enjoy.

Several good books have already been mentioned in this thread, but some good books are hard to get into as a beginner.

I recommend Elementary Number Theory by Underwood Dudley as a good starting point for a beginner, as well as something like Apostol or Ireland-Rosen if you want more details.

I think it makes sense to start with something like Dudley to get an overall framework, and then rely on more detailed books to flesh out the details of whatever topics you're interested in more.

In particular, I think Dudley's book has an approach to Chebyshev's theorem (i.e. there is always a prime between n and 2n) that's great for beginners, even if someone with a bit more experience can streamline that proof a little.

Elementary Number Theory, 2nd edition by Dudley, it's a fantastic book on number theory, deals with all sorts of old theorems by Euclid and Euler. Written in a really easy-to-comprehend way, and if you're really stuck on an exercise the internet can help.

Sure, there are lots of cool websites that don't ask for crazy prerequisites. One which I share with all of my friends who are starting out in math is the Fun Facts site, hosted by Harvey Mudd College.

As far as learning specific materials, you can try Khan Academy for what are perhaps some of the more elementary topics (it goes up to differential equations and linear algebra). If you want to learn more about number systems and algebra I think that either picking up a good, cheap book on number theory, or even checking out the University of Reddit's Group Theory course (presented by Math Doctor Bob) are both very strong options. Otherwise, you can check out YouTube for other lecture series that people are more and more frequently putting up.

Set Theory:

Naive Set Theory

Number Theory:

Elementary Number Theory

Introduction to Analytic Number Theory

A Classical Introduction to Modern Number Theory

Topology:

Topology

Introduction to Topological Manifolds

There's a couple options. You could pick up a basic elementary number theory book, which will have basically no prerequisites, so you'll be totally fine going into it. For instance Silverman has an elementary number theory book that I've heard great things about. I haven't read most of it myself, but I've read other things Silverman has written and they were really good.

There's a couple other books you might consider. Hardy and Wright wrote the classic text on it, which I've heard still holds up. I learned my first number theory from a book by Underwood Dudley which is by far the easiest introduction to number theory I've seen.

Another route you might take is that, since you have some background in calculus, you could learn a little basic analytic number theory. Much of this will still be out of your reach because you haven't taken a formal analysis class yet, but there's a book by Apostol whose first few chapters really only require knowledge of calculus.

If you decide you want to learn more number theory at that point, you're going to want to make sure you learn some basic algebra and analysis, but these are good places to start.

Also pick a book on Number Theory, pretty useful in computer science. I would recommend: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?ie=UTF8&qid=1500074415&sr=8-1&keywords=elementary+number+theory+dudley

I would highly recommend spending some time learning number theory first. Much of crypto relies on understanding a fair amount of number theory in order to understand what and why stuff works.

The book antiantiall linked is fantastic (I have a copy), however if you don't have a strong foundation in number theory will likely be a bit over your head.

Here is the textbook that was used in my number theory course. It isn't necessarily the best out there, but is cheap and does a good job covering the basics.

Number theory is pretty cool. I enjoyed Dudley's book for a number of reasons.

I like Underwood Dudley's book, and it's now available cheap from Dover: https://www.amazon.com/Elementary-Number-Theory-Second-Mathematics/dp/048646931X/ref=sr_1_1?s=books&ie=UTF8&qid=1468275037&sr=1-1&keywords=dudley+number+theory