I cannot express strongly enough how worthwhile an endeavor it is to give Sheldon Axler's Algebra and Trigonometry
its due. While I initially presumed this would be a simple review of material, I have so far been pleasantly surprised by the density and diversity of the exercise sets. If one is self-motivated, one will appreciate how many of the exercises really force one to think like an ancient scholar, like Euclid would.
This has altered my approach, and I will proceed with mindfulness ... seeing myself not so much as a former student living an uneventful life, but as one who aspires simply to be a learned citizen of the world.
I know that you [Holden] are more attracted to Number Theory and the more esoteric branches of mathematics, but I have never seen an approach to Algebra and Trigonometry like Sheldon Axler's. The exercises are a treasure trove, where one can momentarily put aside the realization that it would be better never to have been born, and get lost in analytic geometry, embracing this material as one of the ancient scholars would have, or even as an escaped slave who has come to lead a scholarly life outside of academia. The Internet has made such scenarios possible. (The intersection of the modern and ancient worlds?) This technology is revolutionizing the way we educate ourselves.
What is also cool about this particular textbook is that many of the solutions are worked out in detail at the end of each section, eliminating the need to track down a solution manual.
This book is all about the exercise sets, which would compel you to sharpen your pencil and stretch your mind. I thought this book would be used just for reference, but it is turning out to demand my full attention.
For me, this is where patience and humility and intellectual honesty come into play.
My goal is to stretch my mind and force myself to think carefully as opposed to racing through to get it over with. Most of what I know of algebra, trigonometry, and analytic geometry I associate with calculus. I have not allowed myself to approach these subjects as ends in themselves since 1993 when I was preparing to go to community college to take calculus in 1994, ten years after graduating high school. I want to revisit the fundamentals to get a better appreciation for how much algebra (analytic geometry) and trigonometry is the foundation of the more advanced subjects. Many things that I associate with calculus are really algebraic or trigonometric. I can never consider my study of the fundamentals as complete.
Now I suddenly found myself grappling with problems in Differential Equations and Multivariable Calculus, thinking I would just pick up where I left off in 2002.
While I could work through the exercises, I was not calm, and even kind of frantic. I want to redevelop strategies for problem solving.
I think that Sheldon Axler's textbook will help me rebuild my preliminary foundations.
Also, there is color pdf for Dale Hoffman's Contemporary Calculus
, freely available on line. That series of texts are sold in separate volumes, actually less expensive freshly printed at lulu DOT com than used from Amazon. For instance, CC IV is $18 to print at lulu, but shockingly like $40 used through Amazon. No, it's not reachable through Library Genesis (yet). The author (Hoffman) encourages self-motivated students to make use of these online resources rather than investing in the texts.
But, the font is small so you might be better off using the online pdf in color (also since you can't be lugging books around while traveling around for your employer).