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I went through may topics on learning maths effectively, learning math at later years (I am 30, so I read it to get motivated)

One thing I missed is a natural flow of topics that one must learn to progress

I am a Software Engineer and terrible at maths. I don't know anything except some basic maths. I don't know why logarithms are used for example, or how to solve a particular problem using math. You get the idea? yeah, terrible

I am sick of it and get over it, so I am starting to learn maths in my free time.

I just got a copy of What is Mathematics and will start now.

But question I have is
- What are some high level topics.
- in what order I should learn those topics?

What I would like?

  • A natural progression in my math learning

If this sounds vague problem, I am sorry, let me know and I would try to fill in the missing gaps

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  • $\begingroup$ By basic maths, do you mean arithmetic? $\endgroup$
    – Don Larynx
    Sep 8, 2013 at 20:53
  • $\begingroup$ I suggest taking a look at current sample study plans to see a rough "order". Low-level topics (i.e. the basics) are mostly Analysis and Linear Algebra. Many others (Stochastic, Numerical Mathematics, ...) use results from those to establish their theories.$$$$ Some quick googling got me this result as a starting position. $\endgroup$
    – AlexR
    Sep 8, 2013 at 20:57
  • $\begingroup$ You are a software engineer? OK, did you get an Bachelor's degree in engineering? In that case you must have seen quite some math I would suspect. Or have you become an engineer through a different route? If you are terrible at math, as you indicate, and you want to refresh yourself, I would pick a Precalculus book and go through that first. You can also take this course at your local community college. It doesn't take you the seams out of your wallet either. Precalculus is kind of the basic someone should understand to embark on any higher level math. $\endgroup$
    – imranfat
    Sep 8, 2013 at 21:01

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Short and simple: You should study whatever catches your interest.

For example, if you would like to solve real-world problems such as how fast salt is flowing in per kg/liter in a container compared to how fast it is flowing out, you would use a Bernoulli equation.

Otherwise, if you'd like to think about the metaphysics of an object, you would be more interested in studying pure mathematics. For example, if I wanted to know whether the set $\Bbb R$ can be counted, that is something a pure mathematician would answer.

In the former case, one would need a good grasp of algebra (to manipulate certain expressions) and calculus (to understand differential (equations)). Geometry may be useful if the equations should be represented on $\Bbb R^2$. The equations one sees in calculus all use $\Bbb R$ as their domain.

However in the latter, one would have to use the real numbers consistently and have a good grasp of applications of real numbers. Then and only then one can understand how important it is to ask "Can the real numbers be counted"? or some similar question.

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