What are the classic or great books for Algebra, Geometry, and Trigonometry that are similar to what Spivak, Courant, and Apostol are for Calculus?

Question

There are classic textbooks for Calculus like Spivak, Courant, Apostol, etc that do a fantastic job at explaining the fundamental concepts and theory along with great problem sets. My question is the following:

Are there any classic textbooks for Algebra, Geometry, and Trigonometry that are equivalent to how Spivak, Courant, and Apostol are for Calculus?

When I say "Algebra" I mean elementary algebra not abstract algebra. I would like a book that explains concepts like polynomials, factoring, sequence and series, inequalities, absolute value, and other interesting algebraic structures with non trivial problem sets. Something like the algebra in the math olympiad.

For geometry, Introduction to Geometry by Coxeter fits the bill, I think. — MathematicsStudent1122, May 15 '16 at 22:11
What do you mean by elementary algebra? Do you mean solving for X in equations like 2X+5=9? Do you mean introductory linear algebra textbooks? — Colin McLarty, May 15 '16 at 22:13
The former. By elementary algebra I mean what would be taught in high school algebra or college algebra, but not watered down. Things like polynomials, intermediate value theorem, inequalities, absolute value, cramers rules, matrix algebra, division algorithm, etc with lots of their properties and problems that aren't completely trivial. Maybe similar to the algebra in the math olympiad. Not linear algebra though. — Happy, May 15 '16 at 22:18
Look at these algebra books and these trig books and these geometry books. — Dave L. Renfro, May 16 '16 at 21:47

score 0 · Answer 1 · answered Apr 07 '23 at 08:21

I think what you mean is algebra, trig, and geometry books with rigor as the calc text you mentioned. I know some, for algebra and trig I recommend Modern Algebra and Trigonometry by Vance. This book provides the elementary concepts of algebra and trig but with rigor and full proofs on all the theorems. He even added concepts from abstract algebra and analysis such as field, order, completness. The problem is hard. You are asked to prove a lot of things and don’t worry, he explain the necessary methods of proof. He even introduced limits.

For geometry, it is Geometry by Moise and Downs. It is a good book with proofs that includes analytic geometry. Nothing much to say because geometry isn’t my priority.

What are the classic or great books for Algebra, Geometry, and Trigonometry that are similar to what Spivak, Courant, and Apostol are for Calculus?

1 Answers1