# A book on basic math that explains how math really works

## Main Question or Discussion Point

Please, recommend me a book (or books) on basic math/pre-algebra that really explains how math works from the inside out. What I mean by that is a book that goes really deeply into exploring math giving you true understanding of what's going on behind the scenes so to speak. For example, why the long division method even works, why when dividing one fraction by another fraction is supposed to be carried out the way it is, why dividing by zero doesn't really make much sense, why a negative number multiplied by another negative number gives a positive number and stuff like that. So a book that gives the basis for basic math and arithmetics. Not like the books which only explain how to superficially perform math operations and algorithms. I looked into the for dummies series, but they are all like that, don't provide you with that kind of explanation.

Thank you.

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I don't really think that such a book exists. Don't get me wrong, there are a lot of books that explain exactly what you want. A lot of books explain how the long division works, how multiplication of fractions works, why multiplication of negative numbers result in a positive number. The problem is that such a book is usually quite advanced. The subject that treats such questions is abstract algebra, and it's a college-level math course.

Furthermore, if you want everything to be explained, then you will need to deal with proofs. So before reading abstract algebra, you should read up on mathematical proofs.

If you wish to try out abstract algebra, then I can recommend the book "A first course in Algebra" by Fraleigh. Everybody who is familiar with proofs will be able to tackle this book. But it will still be a difficult read.
If you want to do proofs and sets first, then Velleman's "how to prove it" is a good read.

I realize that I have not been helpful at all. As I understood your questions, you wanted easy books that explain everything. I have given you hard books. The problem is that there are no easy books that explain it all (as far as I'm aware). I think your best bet is posting the questions you have on the forums here. That way you won't have to read an entire book.

Algebra by Gelfand is well motivated and fairly rigorous (at least for high school algebra standards). Lectures on Elementary Mathematics by Lagrange also seems to be like something you are looking for (you may find it online here).

Most arithmetic algorithmic (e.g. long/synthetic division, multiplication with multi-digit numbers, &c.), however, are things that you should easily be able to justify on your own if you know a bit of algebra.

If you want to find out how mathematics is done, I'd recommend watching Vi Hart's Youtube channel and reading Hilbert's Geometry and Imagination.

Good luck!

Mathematics: Form and Function by MacLane

Mathematics: Form and Function by MacLane
MacLane? Really? That may be too advanced, don't you think? The OP asks for a book on basic math/pre-algebra!

MacLane? Really? That may be too advanced, don't you think? The OP asks for a book on basic math/pre-algebra!
Well, I've read just the beginning, but it seems any person could understand a lot of this book. Look at wiki entry. I guess at least first part of the book is readable by anyone. It definitely is "behind the scenes" book. But yeah, this is perhaps too much philosophy, it seems the OP is looking for something more practical.

I think it to a large degree is a matter of psychological satisfaction, which can vary between individuals; some people ask the question "Why?" more often than others. Eventually, you'll hit a wall where you feel satisfied with the mathematical theory "behind" things. Personally, I'm using "Basic Mathematics" by Serge Lang as complementary studying in high school right now, and I think it's enough for me. However, it's probably not for you because it assumes some things without proof, like distributivity, associativity and commutativity. I, on the other hand, think it's OK to do so since I think I have a sufficient intuitive understanding why those properties must be true.

I would suggest that you look into the above mentioned books and see if something fits you. Good luck!