Self-contained undergrad math resources for someone with extremely weak foundations in math?

Question

I've long been interested in various math related subjects (technology, philosophy, sciences, computer science, languages, etc.) without really invested time to actually any learn any of them. I probably sucks at them, as I usually fail my math and sciences when I was in secondary/high-school (I didn't pay attention in class, now I'm regret about it). And I don't really know what $a^{2} + b^{2} = c^{2}$ really means until quite recently.

Now I am more motivated and decided to spend some time to sit down and learn about them. I plan to start from math because it is more or less a foundation, or provides crucial intuition for all these subjects.

It's probably reasonable to start from secondary-school-level math though I found this learning path not interesting enough to keep me motivated. Hence I wish to get right into college-level math, but with a self-contained material that doesn't require much basic math, especially the formulas (e.g. $x=\frac{-b\pm\sqrt{4ac+b^{2}}}{2a}$, $\sin\theta=\cos(90^{\circ}-\theta)$, which I have no idea what they are). It's more preferable if these basic math can be introduce with a college-level manner (e.g. Being constructed/proved from lower level concepts). After some Googling I've found this book (Comprehensive Mathematics for Computer Scientists that provides general introduction to college-level math with, hopefully, reasonable dificulty for me.

The book basically build up a comprehensive portion of computer science related math ("including sets, numbers, graphs, algebra, logic, grammars, machines, linear geometry, calculus, ODEs, and special themes such as neural networks, Fourier theory, wavelets, numerical issues, statistics, categories, and manifolds" [quoted from Amazon]) from propositional logic and axiomatic set theory, without going deep into the details of these constructions (e.g. "it discusses graph theory, but does not mention the graph coloring problem or the shortest path problem" [quoted from an Amazon review]).

So my questions are:

1. Are this book suitable for my needs in the way that it:
   • Reintroduce basic math so that one can reinforce his/her foundation.
   • Self-contained to the extent that minimal to none previous mathematical background is required.
   • Provides a good coverage of introductory college-level math which serves as a sound foundation for further study of more advanced undergraduate subjects.
2. What level of math (e.g. first term of pure math major, first year of computer science math) can someone obtains given that he/she absorbed a reasonable portion of this book?
3. How many hours should someone with poor to no math foundation expect to get through it?
4. What other books, series, or lectures do you suggest? And
   • What level of math can be archived going through it?
   • How long will it takes?

Answer 1

It seems that you are looking for mathematics that is not too technical. In this regard, I do recommend the textbook 'Comprehensive Mathematics for Computer Scientists' because it contains the type of theory that forms the foundations of mathematics - i.e. sets and logic - without going into numerical analysis. Logic and set theory can be intuitive and fun and is definitely useful for application in other subjects such as computer science and philosophy.

On the other hand, theoretical mathematics that is abstracted from numbers, such as linear algebra (chapters 20 -25), can be difficult to master if you are not dedicated, as there are many complicated concepts to understand. For a student with no math foundation, many of the chapters in this textbook will be fast-paced.

If you are looking to "reintroduce basic maths", then this book is not suitable. A better textbook would be one that is aimed at high-school students or a first-year introductory course. The level of math that you will be at once you have mastered the book will be about the end of a second-year pure maths course at university. You can expect to require 5 hours per week for a year $\approx$ 250 hours in total.

An alternate solution, which is a "self-contained" book and requiring "minimal to no previous mathematical background" is to find a descriptive coffee-table style book on mathematics, such as 50 mathematical ideas you really need to know by Tony Crilly (London: Quercus). In my opinion, the more description that the book gives and the fewer formulae, the easier it is to understand what is going on. This book can easily be read by dipping into and out of different chapters, which each take about half an hour to read and understand (there are 50 chapters).

Answer 2

One of the books that i would say would be very good is 'Discrete mathematics by Rosen', it covers a lot of basics like graph theory, set theory, combinatorics, propositional logic,number theory, it covers most of the topics in a very self contained way, regarding the time needed to complete could vary a lot considering some of the more demanding parts are discussed very briefly and sometimes you would like to more read of it than what is given. $$\quad$$ Regarding online lectures one of the sites you can visit is MIT open courseware, they have also their own notes for most of the topics which are excellent to read, i don't know what might be very good for the calculus part but you can refer to MIT open courseware for that also, calculus by Michael spivak covers most of the calculus that you will need initially, but it takes considerable time to read it thoroughly. And also don't forget to have a look at those books again that you had to study at school, as for me I found that although they may not be very mathematically very rigorous but does explains the topics by providing some very nice examples.

Answer 3

If you're looking to get into pure mathematics, I would like to throw in Introduction to Mathematical Reasoning by Eccles. This book will admittedly not give you much in terms of basic mathematics from high school, but it will equip you with the necessary tools to read and write mathematics seriously. It is pretty accessible, and covers topics in mathematics that you won't have been exposed to otherwise, like mathematical induction, prime numbers, diophantine equations, and an elementary consideration of concepts like "infinity."

The utility in such a treatment is that it will allow you to forego the kind of "naive" learning that you do in high school. Instead of being drawn pictures and having to memorize formulas, this text will give you the tools to ask questions as to why a particular process or formula works, and what's really at play when you apply it.

This book is definitely college level, as I used it in my freshman class as an introduction to rigorous mathematics, and will help you to learn about all the other fields of mathematics.

If/when you feel good about these concepts, a personal favorite for studying algebra and the kind of material related to the formulas you've written above is the problem book Polynomials by Bardeaux. I like this book because it guides the reader through the concepts with questions of varying difficulty. It requires no prior knowledge to pick up, except how to manipulate algebraic equations (i.e how to "solve for some term"). Because of this, I like the book, as over the course of the text, many important theorems in other branches of math are proven for their polynomial cases, and it is highly illustrative of the concepts at play. From these two, it is possible for you to pick up any other introductory text in any branch of math of your choosing and begin your studies there - you should have all the necessary tools theoretically and computationally.