In this post, I will share with you a guide on how to approach mathematics from scratch (with the assumption that you know how to read, write, and do basic arithmetic operations like add and subtract).
Right off the bat, if you want or need an inspiration to delve into this discipline, I highly recommend consulting some texts on philosophy (specifically those concentrating on logic and some epistemology). I highlight some of my personal recommendations below as well as videos that will serve as a complementary guide.
To understand why philosophy is relevant at all in this regard, it suffices enough to say that philosophy encompasses almost every discipline you can find and mathematics is no exception to this; one could argue that mathematics is the twin of philosophy and in certain circumstances (like definitions and theorems) borrows its rules. Nonetheless, these three books should serve a great starting ground for establishing the motivation needed for working through mathematics.
Thinking It Through: An Introduction to Contemporary Philosophy by Kwame Anthony Appiah [Chapters 1 to 4, and 8 to 9]
A concise introduction to logic by Patrick J. Hurley
Language, Truth, and Logic by A. J. Ayer
YouTube Videos:
Pre-Algebra (Heart of Algebra: Fractions, Ratios, and PEMDAS/BODMAS):
After you're done with the journey in The Library of Alexandria, find yourself comfortable enough to come out of the cave and delve into the magic of a Platonist's utopia - I'll stop....
Anyway, next step is going through all elementary mathematical operations such as addition, subtraction, multiplication, division, modulus, fractions, and ratios.... If you're still not clear on those, the next step is to delve through one of these books:
Pearson's Pre-Algebra (8th Edition or newer)
YouTube:
You can go for that or follow a YouTube lecture series by Khan Academy at this point in time. You probably won't need to follow any book for the first few Mathematics levels/topics since there's an abundance of videos on those topics.
Algebra:
Algebra and Trigonometry by James Stewart, Lothar Redlin, and Saleem Watson
College Algebra by Robert Blitzer
YouTube:
Trigonometry:
Algebra and Trigonometry by James Stewart, Lothar Redlin, and Saleem Watson
Trigonometry by Charles P. McKeague and Mark D. Turner
YouTube:
Pre-Calculus:
Precalculus: Mathematics for Calculus Axler's Precalculus: A Prelude to Calculus by James Stewart, Lothar Redlin, and Saleem Watson
Precalculus: A Prelude to Calculus by Sheldon Axler
YouTube:
[Note: I should be honest and let the reader know at this point that if you solely want to be spoonfed the ideas of mathematics and not understand how it's done, then the fastest way to do so is to just skim through each and every one of these lectures. However, if you want to be able to solve and understand them in depth, then it's ABSOLUTELY necessary that you practice and attempt problems after getting introduced to each topic or concept.]
Calculus I (single-variable: differential):
CALCULUS: EARLY TRANSCENDENTALS by James Stewart
YouTube:
Math Sorcerer (Start from Lecture 1.2)
MIT OpenCourseware (from start till Lecture 21)
Calculus II (single-variable: integral):
Same textbook as above.
MIT OpenCourseware (Lecture 22 till the end)
Calculus III (multi-variable):
Same textbook as single-variable.
After this point, you can pick any mathematics topic you want to concentrate on but advanced topics like analysis and abstract algebra will require a rigorous course on proofs. I will emphasize the importance of that after the next 3 areas are covered.
Ordinary Differential Equations:
Differential Equations w/ Boundary Value Problems by Dennis G. Zill
Elementary Differential Equations by Rainville and Bedient
YouTube:
Linear Algebra:
Linear Algebra and its Applications by David C Lay, Judi McDonald, and Steven R Lay
Introduction to Linear Algebra by Gilbert Strang
YouTube:
After this, if you want to delve into the sciences or engineering of any kind without looking too deep into a course but want to familiarize yourself with the methods, then I recommend this:
Advanced Mathematics for Science & Engineering:
A First Course in Partial Differential Equations: with Complex Variables and Transform Methods by Hans F. Weinberger
along with this YouTube playlist I made:
Typically, I recommend going through a Mathematical Methods textbook since it gives you an overview of all the mathematics that you'll need to apply in a STEM field but those serve more like a handbook rather than a proper textbook to introduce yourself with. A textbook like Mathematical Methods by Boa or by Riley is appropriate enough as references and if you're taking a course on Mathematical Methods, then they serve as a good complements alongside the lectures.
That being said, if you want to delve into each subject separately and more rigorously, then I recommend getting a textbook or taking a course that concentrates fully on just one of those subjects. Any course that combines multiple subjects are usually not rigorous enough and only delve into the bare minimum needed to apply for subsequent non-mathematics courses that list them as pre-requisites.
Additionally, a Probability & Statistics course with some complementary data analysis is also necessary:
Probability and Statistics for Engineering and the Sciences, by J. Devore.
After this, if you're further interested in mathematics, I recommend going over to Part 2 of this series of post(s), where I talk more about proof-based courses and/or topics as well as some applied mathematics and statistics courses.
