Challenges for the mathematically challenged?

[johnb]

I must heartily concur with Allen about the value of MIT's Open Courseware.
It is a collection of free (Open Source) books and guides and quizzes and tests.

If you are diligent, you could use them to go so far as to put yourself through the complete series of college engineering mathematics. Or just pick and choose the parts that interest you most and skip/skim the things you find most difficult and least rewarding.

I used these to help get my daughter through a calculus class where her instructor was a difficult person: a brilliant mathematician but a miserable excuse for a teacher/mentor.

I've also gone through them myself to maintain my skills as well as just for fun. (Even though I'm a software engineer, I find I only need calculus about once every 4-5 years or so. Or discrete math, or statistics, i.e. just long enough NOT using it to have to brush up every single time.)

Here's one man's opinion (mine) on the difficulty vs. fun of each topic Allen listed.

Absolutely necessary groundwork:

• Pre-algebra
• Algebra
• Geometry
  

Necessary for most other topics below:

• Algebra II and Trigonometry
  

Now for my very strong opinions:

Calculus I, II, III -- difficult for most people. Very useful for solving problems where 1 or more "something's" are happening continuously. Try learning it a bit: you'll either be fascinated by it or else completely put off. Physics beyond the high school level uses this heavily.

Differential Equations -- "very fancy calculus" :-)

Linear Algebra (college matrix algebra) -- much easier than calculus. Very useful for computer graphics, or for solving "sets" of problems. For example, you might have a problem where you have 3 things you need to spend your money on, and finding the "cheapest" or the "best" solution for any one of those means you can't afford cheapest or best on the other 2. What's the optimal mix of spending for all three things?

Complex Analysis -- useful for electronics. Not too hard if you found Trig easy. Kind of cool because it lets you solve things you couldn't otherwise.

Boolean Logic -- very easy, and if you work with computers you'll be surprised at how much of it you already know.

Discrete math -- lots of Sigma signs here. More difficult than matrix math or boolean logic. A toss-up whether it's harder than complex analysis or vice versa. It's the opposite of calculus: all values are though of as discrete chunks not continual smooth curves. Just like calculus, however, you'll either love this stuff and use it a lot, or hate it and decide to skip. It's very good for proving that nobody can improve on some particular algorithm.

Number Theory -- absolutely fascinating! Are there MORE irrational numbers (numbers that cannot be expressed as a fraction, like the square root of 2) than rational numbers? Is there more than one kind of infinity? Is there any true difference between an ordinal (this is the 11th thing in the list) and a cardinal (there are 14 things in the list)? You'll start thinking very concretely about some very abstract things!

Abstract Algebra -- haven't really dipped my toes in this yet.

Topology -- cool beans. You can prove a coffee cup is topologically identical to a doughnut. I have barely scratched the surface of this, though, because the proofs become very high-concept as you start wading through.

Combinatorics / Graph Theory -- another difficult subject that you'll either find a great use for and love it, or immediately hate it.


Now I'll bet that a lot of people vehemently disagree with my very subjective opinions above. Someone here may have even found easy all the things I found difficult, yet struggled with the things I found easy. Then there are those natural mathematicians who just absorb all this with ease. Also those brave souls who recognize the value and power of mathematics and are NOT naturally inclined, but chose to fight through it anyway!


Hope this helps!!