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I had some mathematics education during my high school and Electrical Engineering studies, but I never used any of them during my career as a software professional. Now I am again coming across Linear Algebra, complex numbers and Calculus while learning certain topics in computer science such as Quantum Computing and Machine Learning.

Where I did my studies, maths education was pathetic as we were taught in a way that barely helps to develop intuition and real appreciation for the subject. I had to do rote learning all of the matrix operations and formulas and differentiation rules. Same with Fourier transformation etc which I can vaguely remember. Then we repeatedly solved problems using an outrageous number of sample problems so that you can score in the exam.

That is why my maths knowledge is very poor for the level I was formally educated and I realized that in-order to be able to do good quality work in the future using mathematics as a tool, I need to reeducate myself.

After doing scores of hours of curriculum and text book research and course auditing, I compiled a list of courses and text books suitable for self study at my level.

My curriculum is below:

Semester 1:

  • Single Variable Calculus - MIT OCW and textbook by James Stewart, 8th ed
  • Discrete Mathematics - MIT OCW and corresponding textbook from them
  • Linear Algebra - MIT OCW and textbook by Gilbert Strang

Semester 2:

  • Multi Variable Calculus - MIT OCW and James Stewart, 8th ed
  • Differential Equation - edX (textbook not decided yet, probably Stewart is enough)
  • Matrix Methods in Data Analysis, Signal Processing, and Machine Learning - MIT OCW and a new textbook from Strang

Semester 3:

  • Linear/Non linear/Convex Optimization - edX
  • Introduction to Complex Analysis - coursera

Semester 4:

  • Probabilistic Systems Analysis and Applied Probability - MIT OCW and textbook by Bertsekas
  • Statistics for Applications - MIT OCW and textbook The elements of statistical learning

I made this "semester" thing based on the idea that all of these single courses are covered in a single semester in a college. There are also some rough estimation from course creators on effort needed to self study the topic - which is usually around 150 hours.

I am 2-3 months into this process and now I am realizing that I won't be able to cover a given course in 150-200 hours which I had allotted. The problem seem to be doing exercises, especially what I consider as difficult ones. As an example, Discrete Mathematics book had an innocent looking problem "Prove that $\sqrt2 ^ \sqrt2$ may be rational using Proof by cases"

This is one of the 50 or so problems given and this alone took me a few hours and a whole lot of new insights such as transcendental numbers, Gilfond't constant etc. That's great, but I would be done in...may be 4 years instead of 2 I had planned.

So my question is - where do I draw lines when it comes to exercises and how do students/professors in good Western universities manage to finish a course in a single semester? Do they just do only part of the textbook? What about problems? Attempting only some of the similar problems sound reasonable, but what if they are very diverse and interesting questions? As I am detached from formalized education for too long, it would be appreciated to get some insights into these issues.

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    $\begingroup$ Something is amiss with "Prove that ${\sqrt 2}^{\sqrt 2}$ may be rational using Proof by cases", as this is not true and thus cannot be proved. It's not true because ${\sqrt 2}^{\sqrt 2}$ is the (positive) square root of $2^{\sqrt 2},$ which is known to be transcendental (highly nontrivial, as this basically required solving Hilbert's 7th problem. (It's not difficult to show that the square root of a transcendental number is transcendental, but too much of a tangent for you to worry about.) The problem was probably to show that (continued) $\endgroup$ Mar 18 at 17:02
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    $\begingroup$ an irrational number raised to an irrational power can be rational, and a well-known nonconstructive proof (but not something a student should be expected to come up with) involves exponentiations like this. This can also be shown directly via ${\sqrt {10}}^{\log 4} = \left({10}^{\log 4}\right)^{1/2} = 2,$ but this particular problem is certainly not something you should be concerned too much with. The problem, of course, is without a teacher or someone guiding you a bit, how are you supposed to know which problems are really worth your time and which are basically math puzzles? $\endgroup$ Mar 18 at 17:11
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    $\begingroup$ See "CURIOSA" at the Wikipedia page for Constructive proof. In my opinion, the hint is not particularly helpful. Better would have been to say "Show that an irrational number to an irrational power can be rational by considering cases involving ${\sqrt 2}^{\sqrt 2}$ and $\left({\sqrt 2}^{\sqrt 2}\right)^{\sqrt 2}.$" By the way, regarding my last comment, it is not difficult to prove that $\log 4$ (base $10$ logarithm) is irrational. Write $\log 4 = p/q,$ rewrite as an exponential equation, raise to $q$ power . . . $\endgroup$ Mar 18 at 17:16
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    $\begingroup$ why I posed the question --- Much of your question is probably too specific to your situation to be appropriate here, but I upvoted it anyway (a couple of hours ago) because I think it strongly suggests a legitimate more general question that is appropriate for this site, which is something along the lines of what guidelines might there be for someone self-studying mathematics to avoid wasting too much time on issues tangential to one's desired subject. $\endgroup$ Mar 18 at 20:01
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    $\begingroup$ I believe so too - basic maths and CS education is moving online and self directed. No offence to good teachers here, but I would have profited if only I had been tutored LA by Strang, Discrete Maths by those MIT professors etc.. even though it is remote and async. But that doesn't mean that it is complete and easy.when it comes to problem sets and guidance, there is a gap to be covered. Some textbooks mark open research questns or very difficult ones with a star. But not all textbooks do that and sometimes pedagogical goals of a broad course like discrete maths is not clear to a learner $\endgroup$
    – senseiwu
    Mar 18 at 20:24

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