Maths
This friendly self-help workbook covers mathematics essential to first-year undergraduate scientists and engineers. In the second edition of this highly successful textbook the author has completely revised the existing text and added a totally new chapter on vectors. Mathematics underpins all science and engineering degrees, and this may cause problems for students whose understanding of the subject is weak. In this book Jenny Olive uses her extensive experience of teaching and helping students by giving a clear and confident presentation of the core mathematics needed by students starting science or engineering courses. The book contains almost 800 exercises, with detailed solutions given in the back to allow students who get stuck to see exactly where they have gone wrong. Topics covered include trigonometry and hyperbolic functions, sequences and series (with detailed coverage of binomial series), differentiation and integration, complex numbers, and vectors.
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2009–2024years
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- List of good books, sorted by difficulty:
- Maths: A Student's Survival Guide (ISBN-13 978-0521017077)
- Review Text in Preliminary Mathematics - Dressler (ISBN-13 978-0877202035)
- Fearon's Pre-Algebra (ISBN-13 978-0835934534)
- Introductory Algebra for College Students - Blitzer (ISBN-13 978-0134178059)
- Geometry - Jacobs ( 2nd ed, ISBN-13 978-0716717454)
- Intermediate Algebra for College Students - Blitzer (ISBN-13 978-0134178943 )
- College Algebra - Blitzer (ISBN-13 978-0321782281)
- Precalculus - Blitzer (ISBN-13 978-0321559845)
- Precalculus - Stewart (ISBN-13 978-1305071759)
- Thomas' Calculus: Early Transcendentals (ISBN-13 978-0134439020)
- Calculus - Stewart (ISBN-13 978-1285740621)
The main goal of learning is to understand the ideas and concepts at hand as “deeply” as possible. Understanding is a mental process we go through to see how a new idea is related to previous ideas and knowledge. By “deeply” we mean to grasp as much of the ideas and relations between them as possible. A good metaphor for this is picturing knowledge as a web of ideas where everything is somehow related to everything else, and the more dense the web is, the stronger it becomes. This means that there might be no “perfect” state of understanding, and otherwise it is an on-going process. You could learn a subject and think you understand it completely, then after learning other subjects, you come back to the first subject to observe that now you understand it deeper. Here we can use a famous quote from the mathematician John V. Neumann: “Young man, in mathematics you don't understand things. You just get used to them”, which I think really means that getting “used to” some subject in Mathematics might be the first step in the journey of its understanding! Understanding is the journey itself and not the final destination.
Solve as many exercises as you can to challenge your understanding and problem-solving skills. Exercises can sometimes reveal weaknesses in your understanding. Unfortunately, there is no mathematical instruction manual for problem-solving, it is rather an essential skill that requires practice and develops over time. However, it could be greatly impacted by your level of understanding of the subject. The processes of learning and problem-solving are interrelated and no one of them is dispensable in the favor of the other. There are also general techniques that could be helpful in most cases which are found in some books on problem-solving (which are included in the roadmap).
Teach what you have learned to someone else or at least imagine that you are explaining what you learned to someone in the best possible way (which is also known as the Feynman Technique). This forces you to elaborately rethink what you have learned which could help you discover any weaknesses in your understanding.
Learning how and when to take notes is not easy. You don't want to waste your time copying the entire book. Most modern books have nice ways to display important information such as definitions and theorems, so it's a waste of time to write these down since you can always return to them quickly. What you should do is take notes of how you understood a difficult concept (that took you a relatively long time to understand) or anything that you would like to keep for yourself which is not included in the book, or to rewrite something in the book with your own words. Notes are subjective and they should be a backup memory that extends your own memory.
Read critically. Books are written by people and they are not perfect. Don't take everything for granted. Think for yourself, and always ask yourself how would you write whatever you are reading. If you found out a better way to explain a concept, then write it down and keep it as a note.
Cross-reference. Don't read linearly. Instead, have multiple textbooks, and “dig deep” into concepts. If you learn about something new (say, linear combinations) — look them up in two textbooks. Watch a video about them. Read the Wikipedia page. Then write down in your notes what a linear combination is.
Learning is a social activity, so maybe enroll in a community college course or find a local study group. I find it's especially important to have someone to discuss things with when learning math. I also recommend finding good public spaces to work in—libraries and coffee shops are timeless math spaces.
Pay graduate students at your local university to tutor you.
Take walks, they're essential for learning math.
Khan Academy is not enough. It has broad enough coverage, I think, but not enough diversity of exercises. College Algebra basically is a combination of Algebra 1, Algebra 2, relevant Geometry, and a touch of Pre-Calculus. College Algebra, however, is more difficult than High School Algebra 1 and 2. I would tend to agree that you should start with either Introductory Algebra for College Students by Blitzer or, if your foundations are solid enough (meaning something like at or above High School Algebra 2 level), Intermediate Algebra by Blitzer. Basically, Introductory Algebra by Blitzer is like Pre-Algebra, Algebra 1, and Algebra 2 all rolled into one. It's meant for people that don't have a good foundation from High School. I would just add, if it is still too hard (which I doubt it will be for you, based on your comment), then I would go back and do Fearon's Pre-Algebra (maybe the best non-rigorous Math textbook I've ever seen). Intermediate Algebra is like College Algebra but more simple. College Algebra is basically like High School Algebra 1 and 2 on steroids plus some Pre-Calculus. The things that are really special about Blitzer is that he keeps math fun, he writes in a more engaging way than most, he gives super clear—and numerous—examples, his books have tons of exercises, and there are answers to tons of the exercises in the back of the book (I can't remember if it's all the odds, or what). By the time you go through Introductory, Intermediate, and College Algebra, you will have a more solid foundation in Algebra than many, if not most, students. If you plan to move on to Calculus, you'll need it. There's a saying that Calculus class is where students go to fail Algebra, because it's easy to pass Algebra classes without a solid foundation in it, but that foundation is necessary for Calculus. Blitzer has a Pre-Calculus book, too, if you want to proceed to Calculus. It's basically like College Algebra on steroids with relevant Trigonometry. Don't get the ones that say “Essentials”, though. Those are basically the same as the standard version but with the more advanced stuff cut out.
- I am currently learning maths independently. I'm using the book, Maths: A Student's Survival Guide by Jenny Olive. I'm towards the end of the first chapter and feeling confident with basic algebra now! I picked it up after seeing it recommended here.[1]
The book explains a topic concisely and then gives exercises. Importantly, the exercises don't assume previous knowledge and you can solve them by applying previous explanations. Highly recommended!
- The book that made things click for me was Maths: A Student's Survival Guide by Jenny Olive. Give it a shot, use a calculator for everything. IMO in the age of computers let computers do the calculations and just learn to have intuitions.
- suggestion for you. "Maths - A Student Survival Guide" by Jenny Olive: https://www.amazon.com/Maths-Students-Survival-Self-Help-Eng.... She basically starts out with the simplest algebra (fractions) and gradually works up to topics in 1st semester Calculus. And she starts each chapter with a short quiz to test yourself and skip ahead if already know the material. This book is great for what you are describing, if I'm understanding you. I picked it up when I was preparing to return to college after being away for many years. I supplemented it with another book I highly recommend: "Mastering Technical Mathematics" by Stan Gibilisco and Norman Crowhurst: https://www.amazon.com/Mastering-Technical-Mathematics-Third.... I found that Jenny Olive's book was well designed and preferred it's style to any math textbook I have ever used. Even so occasionally I would get bored while working thru it. That is when I flip thru the Stan Gibilisco's book, which was full of interesting looking problems and examples. When I would try to solve one of them, it would become apparent that I still need to work on the fundamental concepts that were prerequisites for solving the problem. Thus I would return to Jenny Olive's book right where I left off, re-energized by the desire to master those fundamentals that she covers so well
- I almost lol'ed when I read this, but I think I have good suggestion for you. "Maths - A Student Survival Guide" by Jenny Olive: https://www.amazon.com/Maths-Students-Survival-Self-Help-Eng... She basically starts out with the simplest algebra (fractions) and gradually works up to topics in 1st semester Calculus. And she starts each chapter with a short quiz to test yourself and skip ahead if already know the material. This book is great for what you are describing, if I'm understanding you. I picked it up when I was preparing to return to college after being away for many years. I supplemented it with another book I highly recommend: "Mastering Technical Mathematics" by Stan Gibilisco and Norman Crowhurst: https://www.amazon.com/Mastering-Technical-Mathematics-Third...
I found that Jenny Olive's book was well designed and preferred it's style to any math textbook I have ever used. Even so occasionally I would get bored while working thru it. That is when I flip thru the Stan Gibilisco's book, which was full of interesting looking problems and examples. When I would try to solve one of them, it would become apparent that I still need to work on the fundamental concepts that were prerequisites for solving the problem. Thus I would return to Jenny Olive's book right where I left off, re-energized by the desire to master those fundamentals that she covers so well.
- This one is great for start. It has many exercises too.
https://www.amazon.com/Maths-Students-Survival-Self-Help-Eng...
