Mathematics resource advice for theoretical thinker?

Post Reply New Topic

Hi, I need some advice on finding a math textbook or resource for selfstudy.

I did fairly bad in mathematics in HS partly because I had undiagnosed sleep disorder and partly because my foundation was bad... but the thing is
I think I could be good at it. I've actually found that I understand mathematical consepts pretty well as long as that is what it is explained as.. Consepts that are logical.
But I have yet to find a book that will explain math to me in this manner on the HS or subHS level. It seems to be very "do this then this" without actually explaining the logic.
What happens then is that my eyes glaze over because it sounds like just more "because I say so" BS - even though I know it is wellfounded. I just can't focus unless I'm being given theoretical and well structured information.
I can't wait to start with math at uni because I know it will be more theorethical and thourough. But I need my HS courses first...
What I need is a book that treats basic mathematics (from scratch preferably) in this theorethical manner... I don't care if I have to learn much more than the curriculum demands or spend more time on it as long as I learn the theroetical foundations.

Sigh.. it's so ironic that my brain can only learn complex stuff and not easy stuff And nobody will believe me when I tell them so either!

That's how HS math and below works. To some extent, you have to muddle through that tedious junk just so that you become fluent in dealing with basic algebra, number operations, etc.


Depends what you do. Unless you're a math major, you are not going to get anything at uni that is more theoretical than the math you saw in high school.


OK, you have several options, but fair warning: learning math this way (especially if all you want to do is take a derivative) is going to be a lot more difficult and time-consuming than just memorizing the algorithms they give you.

THIS BOOK is a solid introduction to algebra and a little bit of number theory from a relatively rigorous base.

On a more advanced level, Contemporary Abstract Algebra by Gallian is a wonderful book, and covers abstract algebra extremely well.

To learn calculus the rigorous way, you could try Elementary Classical Analysis. That book is pretty dense, but if you can get through it you will understand the why of calculus.

You can probably grab previous editions of each of those books cheap. They don't change too much from edition to edition, especially the Gallian one.

The derivations for the basic calculus principles aren't too difficult, though they will take some time.

I have found Wikipedia and Wolfram Mathworld to be very helpful, though they are rather advanced.

If you want to play around with more elaborate numbers or check your work, Wolfram Alpha is extremely helpful.

As for the usefulness of math, it is used a lot in many engineering fields. I am studying chemical engineering and most of my professors dearly love differential equations. You probably won't be forced to gain a theoretical understanding unless you major in math, but having one can still help a lot if you do something else.

Yes, I know, And I've tried. I'm just really bad at muddeling. I'm a perfectionist to my own detriment.



I'm interested in physics. I'm under the impression it would be well worth it to have good theoretical basis in math... though I could be wrong.



Thanks for some excellent suggestions!

Yes, it most certainly is worth it, but most undergraduate physics programs will not force you to take any theoretical math classes. For the first bit of a physics degree (classical mechanics) you will need to learn linear algebra and differential equations, both of which are fairly tedious muddle-through classes. It is possible to take a more theoretical approach to linear algebra, but then it's harder to get to the applications. With differential equations, I would actually recommend staying away from the deep theoretical stuff. The rigorous math behind differential equations is more painful than topology (and actually requires advanced topology in many places).

From what I find (not just in math but in everything) is that if you don't understand a subject after repeatedly learning it, self-studying won't help. Is there some math tutoring center at your school? Or some general tutoring center that can help with math? Even some disability programs (like the one at UConn) will offer tutoring specifically tailored for students with disabilities (probably not AS Specifically - even UConn, with its dedicates AS Program doesn't have that). Even if they don't have one of their own, your disabilities office should know where to direct you to get help. But keeping on studying it on your own when you don't get it at all won't help. It may even make it worse if you wind up misunderstanding something and studying the incorrect thing.

Something I've done when teaching myself arithmetic and algebra is just to test the properties to get an intuitive handle on why they work, such as with the distributive property:

If you have an addition of numbers each being multiplied by the same number, such as

n*a + n*b

then it's equal to the sum of those numbers multiplied by n also,

n * (a + b)

Or, in the usual textbook table form:

n * (a + b) = n*a + n*b

Whenever I came across a rule like this, I'd try it out to test it and see how it worked. Lets say n = 2, a = 3, and b = 5, fill in both sides of the equation and test it:

n * (a + b) = n*a + n*b

2 * (3 + 5) = 2*3 + 2*5

2 * 8 = 6 * 10

16 = 16

Now, this is a far cry from understanding exactly how the property works, but it helped me to have an intuitive understanding and confidence to proceed further to learn more rather than get stuck in the mud. Learning chemistry later on helped me to remember and appreciate algebra more though, since there's a fair bit of application with gas laws and equations regarding enthalpy and such - but having an application with a concrete basis can help in general.