Self Study Map of Mathematics

When choosing what and when to self study, it's helpful to have a bird's eye picture of all math courses. In this video, I give a sample of what such a map might look like. So many courses are for need of space not included, for example statistics.

^ if the link above does not work, here's an Internet Archive version from June 2021:

[SteamShinobi]

THIS IS THE VIDEO I HAVE BEEN LOOKING FOR.

I couldn't stand math in high-school, but I love branching into every subject. So I got my degree going in liberal-arts, linguistics to be exact, and then I used the knowledge I gained there to go back and re-learn every other subject I could think of. The only issue was I knew most of the sciences very well in terms of where to go from point a to whichever destination, but math, I am illiterate af. I last touched polynomials, and I have had no idea where to go since then to keep studying math so that I can move beyond grasping the theory of sciences, and actually procede to learn formula, understanding equations better, having better overal mathematical literacy, and then hopefully also brancing into applied mathematics via the combination of theory & practice.

I may just be a liberal arts student, but I am so grateful this video has provided the guidance I need to now move past a plateau that has kept me confusing myself and running in circles, hopefully I can actually come out of this as someone who is literate in math, a dream of mine, and I am super grateful.

[adrianarreguin]

I just started college at 27 and have to re-learn my high school mathematics(Forgot a ton of stuff)! This gave me a road map of where to begin! Thank you for the in depth explanation

[corbinwilson]

I remember watching this video at the beginning of my math journey. I now have an associates in math, I’ve taken calc 1-3, Linear Algebra and ODE. Currently I’m studying proofs so I can move onto even more mathematics like analysis and abstract algebra

[Mynhassty]

You should make a video giving book recomendations for every subject in this self-study map!

[Phylaetra]

So - 'linear algebra' is something you will see in algebra II, then a sophomore/junior level course in college, then _again_ as a senior/early graduate level course. You learn new things each time.

I would _strongly_ recommend some linear algebra before differential equations. It's not necessary, but it is _super_ helpful.

Once you start getting into 'proofs' courses, you need a tutor who can at the very least point out your errors in the proofs. Errors can be very subtle, and you simply cannot learn it on your own because you cannot know when you have made an error. You also want to work on your writing skills. A senior-level Real Analysis proof may be a couple of pages long and include a significant amount of prose writing (that goes for pretty much any of the non-computational courses).

The higher levels of mathematics are largely about _proving_ things, and it is really a social activity. Try to find at least one other person to work on more advanced things with and bounce ideas off of each other. Even working on the same proof, you may find different approaches (and possibly different approaches from what an 'answer guide' might provide. You should also get used to reading mathematical literature - papers, books, etc..

I will say that statistics is good to know, especially if you start looking at more advanced analysis (which gets you into a field called 'Measure Theory', tying statistics into 'regular' mathematics. Galois Theory is interesting, but not actually that important if you are more interested in various aspects of 'finite' mathematics (like number theory, combinatorics, graph theory, numerical methods etc.). Not mentioned is the extension of Real Analysis into Complex Analysis (that is, calculus involving complex numbers - that is numbers that include the square-root of -1). Differential Equations extends into Partial Differential Equations, which can then lead to Functional Analysis and Harmonic Analysis. And at some point you start to see how all these 'different' things all tie back together (like with Analytic Number Theory, which uses techniques for Real and Complex Analysis to approach problems involving (for example) the distribution of prime numbers).

Let's see - extension of Real Analysis into Topology. But then there is Algebraic Topology as well. Etc. Etc. Etc.

But - really - find a math buddy and work on things together and have at least one person who knows a _lot_ who can check your work on proofs, once you get there.

[blocksource]

I hate school. I hate the way I learn in school. I hate everything about school.

But I love learning. I'm a self-taught programmer and definitely would like to know more about math.

[zeldamage]

I'd say if you continue the branch of analysis, you end up at functional analysis (which links to differential equations), topology and differential geometry. The last one is the domain of Gregory Perelman. Different than abstract algebra, but certainly not less difficult!

[Selftaught_ai]

Thank you! Finally, a resource that get's straight to the point

[tinaaaaaaaaaa]

this is really what I was looking for. Thanks a lot dude😊

[holymotherofgodpleasenobut]

you don't know how helpful this is to me omg, I've been finding a way onto where to start, since online school ruined me with my math studies😭😭

[shortfusedynamite]

This is so nice thank you. Just went back to school after almost 6 years and took like a warpspeed level math class that I got totally lost in and have been scouring the internet for something like this.

[ayafaustino]

I am starting a pre-medicine college course next year, and we are taking some advanced math classes there so I figured that I'd go over every lesson from the beginning to fill in knowledge gaps.

Math has always been difficult for me but because of school closure, I found that I learn Math better at my own pace when I self-study. And this helps A LOT. Thank you for saving my high school academics, and now you're saving my uni grades <3

[SharkyShocker]

I have to say, as someone that ended highschool taking Algebra II and then going through Trig, Pre-Calc, Calc 1, Calc 2, and now struggling with Multivariable Calculus... it was sobering for everything previous to be referred to as "normal high school student" level.

[parammaewal]

Would love if you could do this video again and add prob/stats, and discrete math

[circushonk]

Thank you, very helpful, was looking for this

[connor]

It’s almost like my phone could hear me talking to recommend this video!!!

I returned to school at 25 to get a degree in economics, I hadn’t done any mathematics past simple algebra since high school, so the school required me to take many refresher courses. So far the math has been my favorite aspect of the degree and I want to take my understanding farther!

[vilkoos]

Excellent route to learn the math you need for classical physics and the engineering that is based on it (e.g. mechanical or electrical engineering) . . . this is probably not the best route if you want to end up in pure mathematics or computer/data science . . . then you might start at set theory and logic, linear algebra should be done as soon as possible, discrete optimization can replace calculus . . . so, before you plan a route you might want to consider where you want to end up.

[matthewzeller]

I don't know why this was recommended to me but it was really informative, thank you 👍

[travelingwithyoumyfriend]

Thank you for your video making me have a way to learn maths again.

[TaimTeravel]

Oh
This is exciting.
My grades are so bad I couldn't apply to college, but I love math and am determined to learn it. Thank you so much.