====== Math and Intuition ======

Hi Gerald,

Keep in mind, I'm not a professor, I do not have a degree in
mathematics, and I would be considered simply a student.

> 1. Where could I find Algebraic Theorems?

The best places would be College Algebra Textbooks and perhaps
Wikipedia and Mathworld web sites. I personally refer to textbooks
myself first.

Beyond this, nothing in math just "exists"-- procedures in mathematics
have been used since God knows when, and it's only because folks like
Pythagoras, Euler, Newton, and others that we have any formal
statements and rules about what works and what doesn't in math. It's
only within the last few hundred years (if I recall correctly) that
people have really been trying to answer questions like "What is
Algebra?" via rings, groups, and such.

> 2. When solving problems, how much should one remember about either
> Geo./Alg. theorems/identites in that
>
> no one can remember everything and what should be remembered about them as
> in,

I think a challenge is trying to find the "intuition" in any subject.
I would say finding the intuition of Algebra is probably easier than
Calculus if only because we tend to use Algebra more than Calculus on
a day to day basis.

I don't think that anyone should expect mastery of a subject after one
course of it, it takes independent study and interest.

> what is your approach?

When I wrote the seven step guide, I had not had a course in formal
logic. Since then though, I have, and I believe the best way to learn
any math is to identify the core facts and implications, and simply
base all your operations on them-- much like simplifying a logic
statement. Of course I still think it's important to find the
intuition as well.

For instance, if you re-examine differentiation in calculus, you might
notice that there is a core rule you can remember without having to
remember others-- logarithmic differentiation.

With the laws of logarithms, you can turn a differentiation problem,
for example:

<m>dx[ {(x + 3)^5}{(x + 5)^3} = y]</m>

  * Take the log of both sides:

<m>dx[ ln[{(x + 3)^5}{(x + 5)^3}] = ln y]</m>

  * Simplify by laws of logarithm:

<m>dx[ 5 ln(x + 3) + 3 ln(x + 5) = ln y]</m>

  * By laws of differentiation:

<m>5 dx[ln(x + 3)] + 3 dx[ln(x + 5)] = dx[ln y]</m>

  * Then by logarithmic differentiation:

<m>5 {1/(x+3)} + 3 {1/(x+5)} = {y prime} / {y}</m> (Note this requires knowledge of the power rule but not much more.)

  * Simplify/factor:

<m>2(4x + 17)/{(x+3)(x+5)} = {y prime} / {y}</m>

  * Identity of y:

<m>2(4x + 17)/{(x+3)(x+5)} = {y prime} / {(x + 3)^5 (x + 5)^3}</m>

  * Finally, multiply both sides by <m>y/1</m>, simplify, and your final answer is:
<m>2(4x + 17)(x + 3)^4 (x + 5)^2 = {y prime}</m>

What you might notice is that you avoid product rule, quotient rule,
among other rules you might otherwise be required to remember...

And with logarithmic differentiation, you can derive product rule,
quotient rule, even the trigonometric rules... thus in my opinion it
is more fundamental than other rules. I would imagine there would be a
similar situation with integration (probably involving <m>e^x</m>), but I
haven't researched that yet.

 --- //[[tjgrant@tatewake.com|Terence J. Grant]] 01/17/2007 21:53//