When should the real understanding begin?

[kenewbie]

I'm an adult, never went to high school. I've recently started to study math on my own, and so I've had to start with pre-calc.

I'm wondering if I am supposed to be able to see the connections between things at this point, or if that comes later.

I'm talking about first year stuff, basic trig and algebra. I mean, I understand basic geometric concepts and that the sine is the relationship between the opposing catheti and the hypothenuse, stuff like that. But if the world went up in flames and all our knowledge with it, I would be hard pressed writing down geometry for future generations, starting at basic properties and working my way up to sine tables.

Same goes with algebra, I can do the work just fine, but I don't feel like I really understand what the root of a polynomial is, or the relationships between all the different approaches.

So, should I just keep doing this stuff over and over (and perhaps get a different textbook to get more than one perspective) or does this "overall understanding of how everything is deduced" come at a later point?

k

 

[atyy]

Yes, you should keep doing this stuff over and over (the ability to solve real problems is the most important thing - it ensures you don't have just a vague or erroneous understanding). Reading lots of books for different perspectives is also necessary. I usually need about 10 or more books to learn a new subject, so a good library is really helpful, unless you're a millionaire. Each book will explain some things well and other things not so well - some may even have errors on certain points (this is true even of the justly celebrated Feynman lectures). Understanding always takes time. I still learn incredibly nice and simple things about elementary geometry even though it's been many years since I studied the subject well enough to get through an undergraduate physics programme. On the other hand, it is often difficult to learn something unless you understand how it fits into the big picture. So when I read a book, I often skip over parts I don't understand, read later parts or other books to get a different perspective, then come back to where I was stuck.

As for the roots of a polynomial, perhaps this may help:

Suppose you have an equation:

x²+15=8x [Equation 1]

To find the possible values of x, what we do is first we subtract 8x from both sides of the equation:

x²+15-8x=8x-8x

which becomes

x²+15-8x=0 [Equation 2]

So the original problem of finding the value of x in Equation 1 is the same problem is finding the roots of the polynomial in Equation 2. Basically all equations you wish to solve can rewritten in a form with zero on the right hand side, so all problems are equivalent to "finding the roots" of some equation, which is why "finding the roots" is given so much emphasis.

Of course you don't always have to "find the roots" to solve your equation. For example, if your equation is:

x=5

You can, but you don't solve it by rearranging it into:

x-5=0

and finding the root of that polynomial, although that would be strictly correct.

 

[dodo]

kenewbie,
I'll paint a personal analogy, that may or may not be shared by other people, but which I believe will be illustrative.

Maths at the level you describe are of necessity a simplification; you will receive influences from more complex subjects as you advance. Here is the personal analogy: those influences will come either from the "top" or from the "bottom".

From "above" you'll reach the most probable enlightenment point when you learn calculus, and subjects spawning from it. How to solve problems related to curves (like polynomials, but not restricted to those), or where do the numbers in logarithm or sine tables come from, is best viewed under the light of calculus.

From "below" you'll get an understanding of structure. By now you might have been exposed to terms like "commutative" or "associative", which might appear as overdescriptive and futile. You may be used to do common algebra, manipulating and cancelling terms, without a real base of why is this valid. You might be intuitively noticing symmetries and relationships between the operations you perform, without any explanation for them. These elements are a glimpse of the structure below maths, of how the entire building is constructed. If you head toward a practical approach to maths, you may get some of this when studying linear algebra; some careers, mathematics of course but also, to some degree, physics and IT, will get some abstract algebra fundamentals.

The proper names for these "above" and "below" things are, respectively, Applied and Pure mathematics. The dividing line is blurry; one example that comes to mind is linear algebra as being directly applicable to solve systems of equations. Vectors and matrices are citizens of the Pure side, while appearing ubiquitously in applications.

I hope this is of some help. If it is of some consolation, the intrigue never ends: there appears to be always one more subject you'll need to know.