Help Guide my Fundamental Understanding, getting to the core

[GeekPioneer]

Ok so this is kinda wishy washy but I don't know how else to explain this other than I feel like I am missing a critical piece of information as far as fundamental "math" goes. I am highly intelligent and and am a Mechanical Engineering Major who gets all A's, but I feel like I am missing a "foundation." I've been watching these http://www.youtube.com/watch?v=dW8Cy6WrO94&feature=list_related&playnext=1&list=SP55C7C83781CF4316 videos which is a lectures series on the history of math/algebra/calculus and It is helping somewhat. I know this is a very hard question to answer and I know myself better than any of you of course. However I am curious to know what some of guys think about my desire to have a "ground up" understanding, I was very uninterested on High School and have just recently discovered my geeky/intellectual nature at 26 years old and my life has been completely reinvigorated with a desire to learn anything and everything about anything especially physics and calculus/math/number theory. Sorry to rant but I am very curious to see you guys thoughts on going back to fundamentals! Any book recommendations or comments are highly appreciated!

 

[Antientrophy]

I think you posted in the wrong section.

 

[GeekPioneer]

Well...were should it go?

 

[Matterwave]

What's your request? I didn't quite catch it from the paragraph you posted. You want resources for learning the history of math/science?

 

[GeekPioneer]

Yeah its pretty hard to describe but I want resources for building from the "ground up" a complete and concise understanding of the fundamental principles of math/number theory/algebra/calculus, which we all know is the foundation on which physics rests. Maybe its just that almost at the end of my freshman year and I'm wanting to much to soon. But and example would be that I really have no concept of what the trig functions are and where they came from (sine, cos, or cosh perhaps). I just know how to do calculations with them and that they relate angles and lengths of triangles. What I would really like to know is how the trig functions where developed and any other contextual information that would help me understand what these functions are at the very core of there being?

 

[Cb1984]

If you really want a "ground up" approach, you can always start with reading Euclid's Elements ( openlibrary.org has a few examples you can download, and on http://aleph0.clarku.edu/~djoyce/java/elements/elements.html you have decent annotated web version) ;)

 

[GeekPioneer]

Ive never heard of this, but "ground up" is exactly what I want! Anybody know a "ground up" book in relation to physics.

 

[James_Harford]

Try the three-volume set "The Feynman Lectures on Physics". It is an extraordinary work, and will give you an eyefull of what this lively Nobel prizewinner thinks (thought) about the "foundations on which physics rests", among other things.

 

[GeekPioneer]

oooo great, will do

 

[James Leighe]

Physicists rarely worry too much about mathematical rigor ill warn you, so look for texts on mathematics or particularly rigorous treatments of physics.

Wikipedia is nice if you don't want to spend thousands on books lol. If nothing else it will help point you in the right direction so you can get the right books.

As for sine cosine etc, it's ratios of side lengths of a right triangle (and their inverses). Euler's formula gives some more deep insight into trig but not a whole lot since it's mostly used for complex angles, it's basically pretty simple.

Differentiation is just taking a limit of a difference function, and for integration the simple treatment is taking a limit of a riemann sum but there are other ways to think about it.

Edit:
You should be able to perform the calculations for taking a derivative and an integral manually using the difference function/Riemann sum to have a real understanding of calculus. But I imagine they teach that in class no?