High school math book for non-idiots

[eof]

I have tried to find a book that would cover high school math together with some tricky problems. I have a cousin who hasn't studied math in about 10 years, but he needs to do the GRE. He wanted to start reviewing in a way that would actually teach the math instead of just showing tricks that you need to memorize. The other issue is that he studied math in another language, so he doesn't know the english terminology. He's going to do the GRE in the spring, so there's really no hurry.

Euclidean geometry is especially a weak spot, because it's almost completely neglected in the curricula he followed in school. He did place in the top 5% nationally in math at the time he graduated high school, so he's completely capable of learning.

I know there are these review books for math grads i.e. "All the math you need for grad school" etc. so I was wondering if there's something similar for the high school level i.e. something concise but still complete.

 

[emyt]

I always give "Principles of Mathematics" By Allendoefer as a recommendation for a solid book covering "pre-calculus" (and even some calculus and algebra) material. It has a good focus on proofs, unlike other books that would cover the same material

 

[rasmhop]

I liked Basic Mathematics by Serge Lang. Harder than most high school books, but good if you're really trying to understand the math and have been exposed to it before. It's probably not the most effective if the ultimate goal is simply to score well on the GRE, but if your goal is also to understand the math presented in high school, then it's great. It might deviate slightly from the topics usually covered in high school, but overall I think it's pretty broad.

 

[brainy kevin]

It's excellent that you want to learn the material and not just the tricks, but I don't believe all of high school mathematics can be in one book, even by excellent authors like Lang. These are some brilliant books that really teach you the material. Feel free to pick and choose the ones that look more interesting.

Algebra:

Algebra, Gelfand. A book on algebra, nice coverage and good problems.

An Introduction to Inequalities, Beckenbach and Bellman. It talks about mathematical inequalites of all sorts.

Foundations of Analysis, Landau. A hard, rewarding book. It goes through and proves the commutative law, the associative law, and so on.Geometry:The Elements, Euclid, Green Lion Press. I'm not joking. It has an amazing coverage of more geometry than you could ever possibly imagine, much less need. There aren't any problems however, but if you only get one book, get this one. The Green Lion Press edition is the way to go, it's cheap and in one volume, as opposed to Dover, which is in three.

Geometry Revisited, Coxeter
Advanced Euclidean Geometry, Johnson
College Geometry, Altshiller-Court
These are books on "advanced" geometry. They all have problems, teach you some modern developments in geometry, and you can buy all three for less than an average textbook.Trigonometry:

Trigonometry, Gelfand. A very nice treatment of trigonometry. You'll need some trig before reading the advanced geometry books.Precalculus:

You can get your precalculus from a calculus book like Spivak. I'm not aware of any good precalculus textbooks.

Well, that's about all the math you'd need from high school, and then some, but if you want some more recommendations you should read J. Bogaarts excellent Amazon listmania list of geometry books. Also, here are some books of a number-theoretic flavor that I've glanced at and liked. They are all cheap.

Continued Fractions, Khinchin
Infinite Sequences and Series, Knopp
Theory and Application of Infinite Series, Knopp
Elementary Number Theory, Landau

That's all I've got. I hope my list helped a little.

 

[n!kofeyn]

I second the recommendation of the Gelfand books. Israel Gelfand, a world-renowned Russian mathematician who actually passed away recently, wrote these books for http://gcpm.rutgers.edu/former_description.html", originally in Russia and then elsewhere. They are very brief, but good. They may be too easy for your cousin though, considering he was rather good at math, and these books won't contain as many problem sets as other more standardized texts will have.

Although, I might recommend going through them (probably starting with the function/graphs or algebra book), and then once he's done, just start going through one of those GRE prep books. Don't spend too much time trying to learn everything the right way, as his final goal is to do well on the GRE. It's just one of those tests that is a necessary evil.

All the Gelfand books, in order of intended study, are:
https://www.amazon.com/dp/0817635335/?tag=pfamazon01-20
https://www.amazon.com/dp/0486425649/?tag=pfamazon01-20
https://www.amazon.com/dp/0817636773/?tag=pfamazon01-20
https://www.amazon.com/dp/0817639144/?tag=pfamazon01-20