I am looking for a math textbook. I'm 15 and I'd like to complete algebra 2 geometry and perhaps something about probability/ number theory or trigonometry would be nice too. Later I wanna do precalculus and calculus. The important thing is I want the textbook to truly be challenging and going well beyond what's usually taught in high school (honestly the harder it is, the better). The main focus should be problem solving although they don't have to be aimed straight at competetive math. Also it doesn't have to be 1 textbook covering it all (not that there is one probably).
You will find textbooks for all of these at artofproblemsolving.com. They formed in order to help students like you.
I have used their number theory book, while tutoring a young student (who was great at math), and I loved it.
=== Edited (9/28/19) to add:
Another good resource is Henri Picciotto's mathed.page. You'll find free pdf's there for at least two of his books: Geometry Labs and Algebra: Themes, Tools Concepts.
To supplement Sue's great suggestion:
Another advantage of the artofproblemsolving.com is that it provides a whole lot of support for students like you, whether or not you purchase or use the textbooks available there.
You'll find, e.g., see AoPS Community, which provide a number of online forums in which all students can participate, ask questions, e.g., about homework problems in various topics; and you can even help other students by providing solutions to their questions. There are also opportunities to engage socially with other bright students through many of these formats, learn about post-highschool opportunities, etc.. In addition, there are links to contests, and tools to search the site and ask for help.
Also, see AoPS Resources, which provides "a wide variety of free resources for avid problem-solvers", including - Math Training and Tools, including Videos - News and Information - Just for Fun
Also, you might be interested in participating (free of cost) in AOPS Math Jams.
Math Jams are free online discussions hosted by Art of Problem Solving for a variety of purposes, including:
- Informational sessions about Art of Problem Solving classes.
- Review of the AMC and AIME competitions.
- Introductions to new topics in math and other areas of study.
- Special events, such as our annual discussions with admissions officers and professors from outstanding universities such as MIT, Carnegie Mellon, Stanford, and Duke.
Much of the AoPS curriculum, including for Intermediate Algebra (Alg.II), Intro to Number Theory, PreCalculus, and Calculus, while they offer new paperback hard-copies for sale, they also provide online textbooks and solution, all of which are available for relatively reasonable prices (far less than the prices of college texts) and are highly competitive, if not cheaper, than highly reputable non-AoPS texts for the topics in which you are interested.
Finally, in the AoPS Bookstore, you can click on Other Products available, and if you scroll down to towards the bottom, you'll find Other Books with a few lower priced high quality alternatives, but not in all the subjects you list.
Two books offered in the Other Books list, Intro to Inequalities and Intro to Number Theory, are largely taken from the book Introductions to Number Theory and Inequalities,, also produced by the UKMT.
You might want to review some texts, in pdf format, in elementary number theory, and a few with more advanced (largely undergraduate) texts in elementary number theory. You might find one or more to be helpful to you, with no cost to you.
Similarly, with respect to Geometry: This depends on whether you've you've covered any geometry before now. For the basics in high school geometry, you might like High School Geometry, Wikibooks. A nice supplement to this free text would be Geometry by Construction: Object Creation and Problem-solving in Euclidean and Non-Euclidean Geometries, which sells for $25 on Amazon.com.
A possible (freely available textbook in pdf format) you might use to address Algebra II AND Trigonomtry can be found here. It does presume any potential student has mastered basic geometry (e.g., as covered in the first pdf text that I list for geometry). To learn analytic geometry (addressed in this text), and to learn basic trigonometry, basic geometry is really a prerequisite, or corequesite.
Another Algebra II text, used in an Honors Algebra II course, with each chapter of the text available freely for download in pdf format, can be found here. You'll find additional supplementary material to download at linked at the bottom of page at the bottom of the Algebra II course page first linked above.. This text has a trigonometry unit, as well as a unit in probability and statistics.