Today’s study time was devoted to more algebra review, including some problems that involved factoring using polynomial division. I have a vague memory of being taught polynomial division around the turn of the millennium, but I had heard the name more recently. There is debate these days about whether, in a society where almost everyone carries a calculator almost everywhere, it is still worth the instructional time to teach students numerical long division. (Eliminating it wouldn’t be unprecedented. I was never taught to compute square roots by hand, as older generations were.) At any rate, one of the arguments in favor of keeping numerical long division in the curriculum is that, without it, students won’t be able to do polynomial division. I’m not sure that would be much of a loss, honestly. It is a fairly niche method of factoring, and I managed to study a lot more math than most people without running into any other application of it. I think the greater drawback to eliminating long division would be that, without that tool to convert one to the other, the connection between fractions and decimals would become something mysterious only the calculator understands.
Speaking of conversion between fractions and decimals, in the portion of The Joy of X that I read last night, Steven Strogatz mentions the fact that the decimal representations of rational numbers always terminate or repeat. Of course, I wondered how one could prove that. I haven’t had time to work on it yet, but I think I can see the vague outline of how it might be done (probably using an argument related to the long division algorithm). The thing I don’t have any ideas about yet is how to prove that a number represented by a repeating decimal must be rational. I may find time to work on both of these tomorrow, but I may decide on more algebra practice instead. I am growing impatient to move on to a new review topic.
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