How may I teach myself pure mathematics "from scratch"? I'm in my 19's and I keen on becoming a mathematician yet currently I can't fund going to university. I am to work as a customer service representative in order to linger. I harness mornings to study maths.
How may I teach myself pure mathematics "from scratch"?
I reckon I don't have the skills to start this major either: Syllabus (best books in your minds, courses,etc).
 A: Look at undergrad math curricula of some good universities - that should tell you the order of math subjects that's typically used to teach undergrad students.
Then study the subjects one by one. You can look up good books for each subject on Amazon going by customer reviews. Or you can search for a reference request question for that subject on SE.
Download the PDF of a good book if available; you can also read course notes for those subjects on college/university websites just so you know what topics are typically covered in undergrad courses.
For example, a linear algebra book will usually contain much more than what's taught in a first-year undergrad linear algebra course, so referring to course materials can help you keep track of what to cover in a first course/reading.
Hope that helps!
A: You could look for videos on Coursera and on YouTube. MIT OpenCourseWare is pretty good and free on YouTube. They will advice you the books.
A: Although I'm confident that those older than us have much more wisdom on this topic, I do believe that self-learning math at our young age in these times may be a dramatically different experience than those self-taught decades ago. As someone who is currently 18 and has dreamed about being a pure mathematician for several years, let me tell you what's worked for me.
Motivation
Despite being indescribably passionate about math myself, one of the real struggles for self-teaching is motivating myself to learn subjects that may not interest me, but are crucial fundamentals. For instance, I'm not terribly passionate about calculus specifically, but I'd like to become adept in it for the sake of it's profound implications on other fields that I am more interested in. Personally, what motivates me to take on such challenges is attempting undergrad-level problems that are interesting and that I come across either on youtube or those that I create for myself purely recreationally. I often browse lists of unsolved problems in mathematics to inspire what could be achieved. Who doesn't like to imagine discovering something new?
Resources
Of course this all depends on how you prefer to learn, here are my thoughts on different materials.
Books
I have about a dozen books on mathematics ranging from high-school to graduate level. I've never read a single one. Perhaps this is a personal or generational thing but with technology giving rise methods of learning that transcends paper, I can't seem to bring myself to maintaining interest in reading more than a few pages. That's not to say that textbooks and the like aren't useful (in fact some of greatest minds in math majorly taught themselves through textbooks), they just may not be "hands-on" enough for the purpose of endeavour.
Acadamia
Although I am lucky enough to be an undergrad student in mathematics and have had remarkable professors, I don't think you're really missing out on the potential for "better" education. Most subjects that I've had the opportunity to be taught can easily be accessed and learned prior to entering the classroom. That being said, I've heard many recomendations for free courses like MIT OpenCourseWare. Also, I'm more than confident if you live near a University or College, some instructors will happily let you sit in on their lectures for free.
In my opinion, perhaps the greatest resource in Academia are the people. The legendary mathematian Ramanujan famously spent his early years "exhaust[ing] the mathematical knowledge of two college students". I've never had the opportunity for a math "mentor" of sorts, but I'm confident that they would be a delight for intellectual satisfaction.
YouTube
A ridiculous amount of what I know when it comes to mathematics comes from YouTube. I suppose I'm a visual learner in that sense. I'm sure you're aware of most (if not all) of these chanels, but here's some that I follow regularly:

*

*Matt Parker's Stand-up Maths

*Burkard Polster's Mathologer

*PBS Infinite Series

*Brady Haran's Numberphile

*Grant Sanderson's 3Blue1Brown

*Jens Fehlau's Flammable Maths
Usually I find myself watching these videos, scribbling along on a piece of paper, then browsing some related articles.
Wikipedia
Many take Wikipedia for not being credible, yet I've found it to be irreplaceable when it comes to needing to quickly get the grasp of a concept or to grab a few equations to play with. Say you want to learn about Abstract Algebra. You could hop on Wikipedia's article and read until some foreign terminology like Homomorphism pops up, then simply go to that article and rinse and repeat. Within an hour either you've conceptually mastered the jist of algebra, or gone down a fascinating rabbit hole. Either path is great!
Tactiles
I think a lot of people have a hard time grasping mathematics as most of the time spent in the field is during thought, as it's not exactly tactile. Trying to be as "hands-on" as you can is likely your best bet. Whether that be playing with software such as Desmos or WolframAlpha, learning a programming language, filling notebooks with geometry & symbols, or making toothpick models of graphs; they're all safe spaces for personal creativity.
Overcoming cognitive dissonance
Above all, my greatest struggle in teaching myself mathematics is my own ego. I failed a calculus class three times because I thought I was beyond what the course had to offer. I wasn't. I may have been already comfortable with all that was taught, but I really needed to be there anyway for the sake of learning the skills of patience and work-ethic. In summary, you may come across subjects where you've learned every lesson, but that only means there's a deeper lesson beneath the surface. That is the magic of mathematics.
I hope some of my experiences can be insightful. I wish you only the best; never stop learning!
