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Could you recommend any high school mathematics books that are rigorous and present high school mathematics in a higher level.I just realized that I only memorized a lot of things that I was taught in school and want to revise and understand those concepts deeply.

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Have you seen Mathematics Made Difficult by Carl Linderholm? – MJD Feb 20 '13 at 20:12
Different high schools within the same county, state and country have sometimes pretty different requirements for mathematics (this doesn't happen so badly in countries like France or Israel, where they have the same national final exams for all the high schools in the country), let alone in different countries. – DonAntonio Feb 20 '13 at 20:13
Look over the books I mentioned in my answer to this StackExchange Question. Note that the two links embedded in my answer take you to many fine items that are freely available on the internet. – Dave L. Renfro Feb 20 '13 at 20:23
up vote 3 down vote accepted

You might want to visit the Art of Problem Solving's Website to explore for potential resources. It is geared toward high-caliber high-school students (and without doubt, helpful to those beyond highschool, as well). There are classes available to enroll in, recommended curriculum with textbooks available for purchase, and a whole ton of free resources accessible on-line at the website, including an interactive community, and some resources available in pdf.

There's a range of material available, and at the "textbook" link, you can customize a search for the level of curriculum more appropriate to your needs. The site also provides resources for topics beyond the algebra-geometry-precalc-calc courses: e.g., number theory.

Personally, I took a long break between highschool and university study, and "jumped in" right where I left off. It took a little extra effort at first, because I had to review some old material which I needed to know in order to use it with the new material.

But that's how one learns in general: as we need it, we sometimes need to review it, in order to use it, but that's true of math graduate students and professors alike. Were it not for having to teach courses in college algebra, trig, calculus, many would be challenged to recall, on the spot, some of the most elementary of identities, etc. we learn -> we know a little; we relearn -> we know a little again, and we begin to understand. It's an ongoing process.

But take heart, relearning what you've covered earlier is vastly easier than learning it for the first time ever.

  • If you encounter an area you need to "review", a nice resource is the Khan Academy where you can find virtually any topic or technique in math, at virtually any level - from arithmetic to algebra to calculus and differential equations - broken down into 10-minute video tutorials, and select all and only those areas you need to review.
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I can understand you want to "shore up" what you already know, but most of the time the solution to this problem is to go ahead and learn newer more advanced things. It is better to be exposed to more things.

You will have plenty of opportunity to review the old stuff as you go along, and you will probably get a better perspective on it as you see it in more advanced settings.

A good reason to avoid reading material written at a highschool level is that a lot of it is organized for the awful curricula in place at high schools. That means it is almost by definition pedagogically bad. (Don't get me wrong, I realize there are exceptions to my complaint, but most of the books are bad.)

Case in point: highschools think "rigor" is teaching two-column proofs in geometry, when actually that is a fantastically boring and overly formal way of doing things that mathematicians would not dream of doing in daily life.

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@user1978522 May I ask what topics particularly interest you? – rschwieb Feb 20 '13 at 20:18

Try Basic mathematics by Serge Lang.

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Here is a great book for you. It was written by one of the outstanding mathematicians, expressly for students in your situation:

It is intended that you can understand it and open your eyes to the world of math. Good Luck.

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The Forester books.

That's algebra 1, but there's others that cover trig, calculus, etc.

The concepts are presented very well and have interesting applications/challenge problems. I recommend these.

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Concrete Mathematics by Ronald Graham, Donald Knuth, and Oren Patashnik would be my suggestion that covers a wide spectrum of topics with some rigor and at a high level. There was a portion used for my advanced graduate course in Asymptotic Enumeration but that may be more near the end of the book than the earlier stuff as Asymptotics is Chapter 9.

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