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Practicing Math

Olly / / Math Posts

I spent my study time today on more algebra review, encompassing both reading and exercises. I also derived the quadratic formula, which was fun.

I don’t have any mathematical insights to share, but I did spend some time thinking about the ways in which doing math is like a sport or perhaps a martial art. For me right now, the most immediate similarities are the importance of practice to any kind of success and the way advanced skills build on more elementary ones. There is also the fact that watching someone do math and doing it oneself are very different experiences. (These are both something students often miss. Teachers usually make math look easy, but when the students try it themselves, they struggle. Rather than concluding that math requires practice, the students conclude that they are just not good at it.) Math is also something that has applications, but is often done for its own sake, something that has a long history through which it has been transmitted from teacher to pupil, and something that formalizes thought as a physical art might formalize movement.

Pertinent Points

Olly / / Math Posts

Today I continued to work through the exercises in the appendix of my calculus book that provides a review of some concepts from algebra.

One series of problems involved finding the intervals on which various expressions containing variables would be less than or greater than zero. One method to do this that was outlined in the textbook was to find the points at which the expression would be equal to zero, use those to divide the number line into intervals, then test a point from each interval to determine whether the expression was positive or negative on that interval. All the time I was doing this I was thinking, “this is all well and good for expressions that correspond to continuous functions, but what about other types?” (The property of continuous functions that makes this work is the Intermediate Value Theorem, I believe.)

Sure enough, near the end of the series of problems, some expressions appeared that would be discontinuous when interpreted as functions. The answer to my question is that you need to treat each point of discontinuity as a point at which to divide the number line, the same as each point where the expression would equal zero. This was never explained or even hinted at in the text, though, which puzzles me. I don’t think of this text as leaving much for the reader to discover. Maybe there is more than I remembered, though. If so, there may be value in doing all of the exercises.

Squares Again

Olly / / Math Posts

I’m not sure yet how I’m going to organize my review. So far I’ve been dabbling widely in the materials I have available. Yet I think it would be a good idea for me to mix drills in the calculation-focused subjects I may eventually want to tutor—calculus, analytic geometry, and so on—with review of some more proofs-based math. Otherwise I’m going to feel as if I’m missing half the show.

Today I was reading Chapter Zero by Carol Schumacher, one of my old textbooks, which was written as an introduction to proofs-based math. In its discussion of definitions, it says the following:

[W]e might naively define a square as “a four-sided, equilateral polygon,” but we would quickly see that such a figure need not have right angles. If we are actually only interested in right-angled figures, then we could refine our definition to be “a four-sided, equilateral polygon with four right angles.”

That immediately had me wondering whether, to know that a four-sided, equilateral polygon is a square, one must know the measures of all of its angles. I suspect that one is sufficient. I believe there are no equilateral, four-sided polygons that have one right angle and do not have four right angles. I haven’t sat down to work out a proof yet, though.

Squares as Products of Squares

Olly / / Math Posts

While watching a video about factoring, it occurred to me that, if n is a square number, then n is also the product of square numbers. This is trivially true, of course, in that n is the product of itself and 1, which is a square. If the root of n is not a prime, though, then n can be expressed as a product of squares in at least one other way. To see why this is, consider 36:

36 = (6)(6) = (3)(2)(3)(2) = (3)(3)(2)(2) = (9)(4)

Given any factorization of the root of n, n can be expressed as the product of the squares of the root’s factors.

This is all pretty obvious, but I thought it was interesting. In the future, I may consider if there is a geometric interpretation of this fact and what light it might shed.1

  1. To do. ↩︎

Take Drugs, Do Math

Olly / / Updates

The end of my time at college was traumatic, and since math was my major, it was inevitably swept up in the trauma. So much so that for the past 15 years I have had clear trauma responses whenever I tried to do math. Initially, I could not access math knowledge at all, as if it were behind a blank wall in my mind. Later, whenever I tried to think about it, I had shortness of breath, pounding heart, and shaking. Those symptoms gradually eased and haven’t troubled me for a couple of years. Math still caused me anxiety, however; I struggled to concentrate and ended up exhausted by even short periods of review.

Earlier this week, in consultation with my psychiatrist, I started using CBD-infused gummies while working on math. The results have been striking. Today I was able to study for almost two hours without ill effect. I think I may finally be in a position where I can reengage with the subject I loved so much.