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I tried to get into Spivak's Calculus only to find that I've never been taught the type of Math presented there.

First chapters talk about the properties of numbers, then mathematical induction, assertions, etc. In other words, things I've never heard of.

While I did take some calculus courses related to derivates, integrals and differential equations, most of it was very methodical as in ''use this equation to get the answer''.

How does one transition from simple math to the more rigorous and complicate type of math presented in these type of books?

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Based on your needs you can choose a proper book. Usually, there are two kind of books here. Those that emphasize on the theory and those that emphasize on the application. Often, engineering students are not that much into the theory and they use books which care more about application.

Books like Spivak have an emphasis on the theory.

If you want to learn the theory, study a little math history and master applications, I would suggest to use the Apostol's calculus.

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    $\begingroup$ Very well then, will check Apostols book. Thank you very much. $\endgroup$
    – Borodin
    Commented Jun 20, 2016 at 21:33

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