My Math Page

General Thoughts

Guess-and-check is not mathematics, it's a primordial form of brute force. The reason you're learning mathematics is to refine your thinking processes so that your mind will be a razor rather than a club (think: would you want to shave with a club?).

Imagination and creativity are keys to the kingdom. If you choose to be otherwise, it's okay, but you won't enjoy yourself as much as you would have if you'd taken the chance on something different.

Mathematics is a language. Its symbology consists of numbers, variables, and other squiggly-looking things (some of which are more boxy than squiggly). Its assumptions are axioms, its truth set is theorems, and its way of getting at new truth is to extrapolate the current body of truth to new territory by impartial, unemotional, objective, rigorous argument. It may take you a few weeks to get to the point of thinking in this fashion.

Mathematics is power. It is the power to think beyond the insanity of daily existence, and to deal with a world where everything which has been established is orderly and coherent (even chaos theory!).

Mathematics is pure thought. It has no reliance on the real world. It has no need for experimentation to determine itself. Its truth is generally expressed in a single form.

Mathematics is beautiful. It contains the elements of all that is in art: number, form, perspective, resolution, and interaction. Discrete math is pointillism; calculus is occasionally cubism. Trigonometry, in its basic sense, is rather minimalistic, yet sometimes approaches the byzantine complexity of romanticism. Algebra, perhaps is found in the stark clarity of Mondrianism. Geometry is sculpture. Each of these branches, however, are interrelated in tremendously deep ways - and so you find that after the landscape becomes more and more complex, it also begins to simplify because of these profound linkages between subjects.

It is not mere truth which matters; this is only knowledge. It is application of that truth which makes a difference in your grade and in your heart and mind; this is wisdom (or at least savvy). Take what you know and learn how to apply it to new challenges as you meet them.

Logic

Check your assumptions. Check your assumptions at the door. Check your assumptions at the door, and throw the hatcheck ticket away.

If the statement says it's so, then you have to adhere to that. Don't take away from it, and don't assume other unstated possibilities.

If the statement says it ain't so (say so, Joe), then you have to adhere to that. Don't add to it, and don't disallow other unstated possibilities.

Algebra Problem-Solving Theory

How to solve an algebra problem:

1. Define variables representing what you want to solve using "let" statements.

2. Translate the statements in the original problem into equations in terms of your variables. Usually this will mean one equation per sentence (more than one in the case of a compound sentence).

3. Manipulate your equations (algebraically!) to arrive at an answer. Be unfailingly brutal to yourself, or someone else (your teacher, your mentor, or even your friend who corrects your quiz) will do you the favor of taking care of that responsibility for you.

4. Check your answer against the original problem!

Keep the quadratic formula close at hand. When asked to derive it, you can write the formula down, back-derive the original equation in standard form, turn your paper upside down, and put your name in the upper corner (which used to be at the bottom). Reversibility is essential!

Geometry Problem-Solving Theory

"The Toolbox":

1. Geometry is like owning a toolbox. You need to have tools to solve problems.

2. Geometry is like using a toolbox. You need to know what tools are in your box and how to use each one of them.

3. Geometry is like using a toolbox effectively. You need to know what you're trying to accomplish, and you need to know which tools will get the job done.

4. Geometry is like maintaining a toolbox. If you don't polish your tools now and then, they'll get rusty and you'll forget how to use them - or worse yet, that you own them.

5. Geometry is like being responsible for a toolbox. Use it or lose it. Use it properly or else the tools inside will come back to bite you through improper application.

Geri points out: "Geometry and algebra are the same thing. WELL, technically they are dual things. Nevertheless, they are essentially the same thing and you can change one into the other and anything you can do in one you can do in the other." She knows of what she speaks, since she knows a whole lot more math than I.

Discrete Math Problem-Solving Theory

The universe has holes (also wholes) in it. In some sense, nothing is ever next to anything else. But of course that's true only if you can count them.

Don't settle for a half answer. Sometimes, also, don't settle for an answer with a 1/2 either.

Be imaginative and observant. Look for patterns which lead to the solution. Start with small cases and (gradually?) work up to big ones, or even infinite ones.

You don't need to solve Fermat's Last Theorem. Andrew Wiles has already done it for you. If you want a copy of his proof, it's only a couple of hundred pages long and encompasses something approach all of number theory up to the present. I understand it only up to "Let p be an odd prime" (the first sentence), after which point I'm hopelessly befuddled - I hope you do better than I!

Trigonometry Math Problem-Solving Theory

Life (at least as far as trig identities is concerned) boils down to sines, cosines, and tangents.

sin^2 + cos^2 = 1, tan^2 + 1 = sec^2, and 1 + cot^2 = csc^2.

Many problems dealing with triangles will require either or both of the Law of Sines and the Law of Cosines. Remember them perfectly!

Angle addition/subtraction formulas (from which you can derive double angle and half-angle formulas) are the beginning of a new adventure. You will be required to go beyond their mere statement and raw application - count on it.

DeMoivre's Theorem reigns supreme - cis boom bah!

Calculus Math Problem-Solving Theory

Everything starts from limits. You have to know your limits.

Derivatives are rate of change; integrals are accumulation of rate with respect to the independent variable.

This is a new way of thinking. Liebniz may have beaten Newton to the punch on this, but Newton's was the first published theory. You will learn how to solve problems which you couldn't solve before - treasure the experience!

Re: Calculators

Calculators are not good at mathematics. Calculators are great at brute force computation (see "General Thoughts" above), but they cannot do your thinking for you. If you can attach your calculator to your brain as if it were a Borg implant, that's fine - but it still won't do your thinking for you!

Calculators are great at giving numbers to ten or twelve significant digits. They, however, are not capable of providing exact answers, especially for the picky world of mathematicians where 10-1000 is enough to say a value is "not exact". You can't trust your calculator to give you the umpteenth digit - it can't, and it won't. And if there's some sort of repetition to the decimalized answer, see if it can be expressed as a fraction!

Calculators have batteries which eventually run out. You have a brain doesn't need batteries. (Just thought I'd throw this minor revelation in...)