Reddit reviews Abstract Algebra: An Introduction
We found 5 Reddit comments about Abstract Algebra: An Introduction. Here are the top ones, ranked by their Reddit score.
Used Book in Good Condition
I was in the same position as you in high school (and am finishing my math major this semester). Calculus is not "math" in the sense you're referring to it, which is pure mathematics, without application, just theory and logic. Calculus, as it is taught in high school, is taught as a tool, not as a theory. It is boring, tedious, and has no aesthetic appeal because it is largely taught as rote memorization.
Don't let this bad experience kill your enthusiasm. I'm not sure what specifically to recommend to you to perk your enthusiasm, but what I did in high school was just click around Wikipedia entries. A lot of them are written in layman enough terms to give you a glimpse and you inspire your interest. For example, I remember being intrigued by the Fibonacci series and how, regardless of the starting terms, the ratio between the (n-1)th and nth terms approaches the golden ratio; maybe look at the proof of that to get an idea of what math is beyond high school calculus. I remember the Riemann hypothesis was something that intrigued me, as well as Fermat's Last Theorem, which was finally proved in the 90s by Andrew Wiles (~350 years after Fermat suggested the theorem). (Note: you won't be able to understand the math behind either, but, again, you can get a glimpse of what math is and find a direction you'd like to work in).
Another thing that I wish someone had told me when I was in your position is that there is a lot of legwork to do before you start reaching the level of mathematics that is truly aesthetically appealing. Mathematics, being purely based on logic, requires very stringent fundamental definitions and techniques to be developed first, and early. Take a look at axiomatic set theory as an example of this. Axiomatic set theory may bore you, or it may become one of your interests. The concept and definition of a set is the foundation for mathematics, but even something that seems as simple as this (at first glance) is difficult to do. Take a look at Russell's paradox. Incidentally, that is another subject that captured my interest before college. (Another is Godel's incompleteness theorem, again, beyond your or my understanding at the moment, but so interesting!)
In brief, accept that math is taught terribly in high school, grunt through the semester, and try to read farther ahead, on your own time, to kindle further interest.
As an undergrad, I don't believe I yet have the hindsight to recommend good books for an aspiring math major (there are plenty of more knowledgeable and experienced Redditors who could do that for you), but here is a list of topics that are required for my undergrad math degree, with links to the books that my school uses:
And a couple electives:
And a couple books I invested in that are more advanced than the undergrad level, which I am working through and enjoy:
Lastly, if you don't want to spend hundreds of dollars on books that you might not end up using in college, take a look at Dover publications (just search "Dover" on Amazon). They tend to publish good books in paperback for very cheap ($5-$20, sometimes up to $40 but not often) that I read on my own time while trying to bear high school calculus. They are still on my shelf and still get use.
Whenever I explain algebra (rings or groups) I start with the issue of defining what a "number" is. This leads us to the ideas of addition, multiplication, etc. The basic idea that I try to convey is that a "number" is a very general concept and that instead of trying to define a "number," we instead define how certain kinds of "numbers" behave and interact. This gives us an algebra. I give examples of groups and rings to point out the similarities between seemingly different algebraic structures.
A lot of people have pointed out that groups can be understood in the context of symmetries. My first course in algebra actually took a different route. We were taught groups after we were taught rings. It isn't traditional, but Hungerford does a pretty good job and it appeals to an early math student's intuition.
ps: I use the term "number" (with quotes) very loosely. I know that to many people the word number is synonymous with integer but to a layman a "number" is really just something with which we "do math."
I think Dummit and Foote would be too difficult. It's almost always used for a second course in algebra, not a first. I'd try http://www.amazon.com/Abstract-Algebra-Introduction-Thomas-Hungerford/dp/0030105595 first, or maybe Artin's "Algebra."
You can try Hungerford's Introduction to Abstract Algebra or Schaum's Outline of Modern Abstract Algebra. The former is very clearly written and great for self-study, but provides a thorough introductory course that may be much more intensive than you are interested in. The latter covers (in a less rigorous fashion) the principles of linear algebra, number systems and modern algebra. It is exceptionally easy to read and understand, but is not much of a textbook, and lacks most of the depth of the first text. To be honest, it is more of a guide to the foundations of algebra.
If you are intent on studying abstract algebra intensively, I would recommend the first book. However, it appears to me that the second text would definitely be more up your alley, as a stronger foundation in algebra will be invaluable when your student is ready to pursue this subject in a more comprehensive way.
Tacking onto what /u/ReneXvv said, if you would like a more historical (and in some ways, intuitive approach) Hungerfod's Algebra: An Introduction. Again, this format is slightly different (he begins with rings and then moves onto groups, which is opposite the usual approach), but it's well-designed to get your feet wet into the topic. From there you could graduate onto the more "standard" texts, like Hungerford, Dummit & Foote, or my new favorite, Rotman (who is also slightly nonstandard in his approach and builds nicely into category theory).