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Re: Relearning the Basics
Honestly, I know of no book covering high school material in a good manner. Everyone I know that's crusing have learned their stuff not from a textbook but from some or other awesome math teacher. (hence why i am planning on writing a comprehensive precollege level text at some point)
Wish you luck.
Wish you luck.
Re: Relearning the Basics
The only thing I really know that really starts with definition of numbers and works up from there is Real Analysis (Rudin does this more or less). Of course, if you haven't taken Calculus, this would definitely be overkill and probably not extremely successful (assuming you're a normal person, if you're some sort of prodigy, I guess you might be capable). Also, when you say start at the beginning, you might want to first look into Set Theory and Logic. These are more fundamental than numbers, at some sort of level, and are definitely key in defining what numbers actually are.
To properly give you advice, it might be helpful if we knew exactly what you want to cover, and what your goals are.
To properly give you advice, it might be helpful if we knew exactly what you want to cover, and what your goals are.

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Re: Relearning the Basics
Are you looking for more formalism and rigor, or just trying to make sure you didn't miss anything important? I can't offer suggestions if it's the latter, but if you want more rigor, the <a href="http://en.wikipedia.org/wiki/Peano_axioms">Peano axioms</a> are an interesting example of a barebones definition of the natural numbers. (And from there, of course, you can define the integers, the rationals, and so on.) For really extreme formalism, <a href="http://us.metamath.org/mpegif/mmset.html">Metamath</a> is an interesting site that builds up math from logic and set theory, and they have some book recommendations (which I can't comment on). Or you could read Principia Mathematica.
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Re: Relearning the Basics
I'm pretty sure when you mean "start at the beginning" you don't mean you want to learn everything totally formalized (I may be wrong about this) but just want to refresh your memory about what you learned.
For this, I recommend going to your local bookstore and picking up a book on math for grade 3ish standardized tests. Work through the problems, then go to the next level. You can probably do more than you think. The only exceptions might be some of the theorems in geometry, but those are fairly easy to prove if you forgot.
If you do want to learn math starting from the basics, like Peano arithmetic and ZFC, the internet has many good resources for these. I can't think of any good books though.
For this, I recommend going to your local bookstore and picking up a book on math for grade 3ish standardized tests. Work through the problems, then go to the next level. You can probably do more than you think. The only exceptions might be some of the theorems in geometry, but those are fairly easy to prove if you forgot.
If you do want to learn math starting from the basics, like Peano arithmetic and ZFC, the internet has many good resources for these. I can't think of any good books though.
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Re: Relearning the Basics
In that case, I would recommend buying a good precalculus textbook and going through it, making sure you can do the exercises. If you have a good understanding of limits, trigonometry, graphs, and specific techniques like partial fractions, you should be more prepared than most other people in class.
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Re: Relearning the Basics
Blarnay wrote:I have this feeling if I start from the beginning and ensure that I have every topic down set than the new topics that I get to in Calculus will be easy (since math is cumlative). At the same time I think it would be a good time to rethink how I work with numbers and break down any bad habits or find ways to optimize processes that before I might have been slow with so I think the pages about number theory should help.
Just a note. Number theory is useless for calculus for all intent and purposes. Calculus I is about real numbers, continuous function, and limits, while number theory is about positive integers, which are not continuous. Unless you mean number theory in a different sense than we do.
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Re: Relearning the Basics
achan1058 wrote:Blarnay wrote:I have this feeling if I start from the beginning and ensure that I have every topic down set than the new topics that I get to in Calculus will be easy (since math is cumlative). At the same time I think it would be a good time to rethink how I work with numbers and break down any bad habits or find ways to optimize processes that before I might have been slow with so I think the pages about number theory should help.
Just a note. Number theory is useless for calculus for all intent and purposes. Calculus I is about real numbers, continuous function, and limits, while number theory is about positive integers, which are not continuous. Unless you mean number theory in a different sense than we do.
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Re: Relearning the Basics
The correct answer is look up state standards for mathematics from elementary to high school, then go through grade by grade finding things that review each concept. Pick a state, any state.
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Re: Relearning the Basics
auteur52 wrote:The only thing I really know that really starts with definition of numbers and works up from there is Real Analysis
This book might be good if that is what you are looking for:Taylor and Mann 0471025666
Re: Relearning the Basics
A textbook that teaches calculus in a rigorous way often times builts up the number line in the first chapter. Maybe you're looking for a book like Apostol? This fits your requirement of building things up from the basics and goes to calculus and beyond (ie, into to linear algebra). Or maybe you want a book about just the number system, the most appropriate one is numbers: Rational and irrational by niven. This is a good choice if you don't want to a difficult calculus book right away (which is a bad idea if you don't understand everything below it perfectly).
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