Last Post?

I’ve just got the word… the latest version of next quarter’s schedule has me taking a First Calculus course instead. I’ll go ahead and admit I’m pleased with this development. But I won’t let the opportunity go by to post this link to my “Introduction to Natural Numbers” lecture notes, posted a while back in one of my numerous other blogs but originally in my “book”. The PDF version is in flux… most of it’s available at the homepage but it needs work so I’m not linking here.

A Decent Interval

There have evidently been some changes made in the new version of the course concerning conventions about interval notation, as opposed to set-builder notation. The general feeling up until now has been that it was hopeless to try to get students to use “set-builder” (and write, for example, in place of the clumsier ). I share this feeling if we speak of, let’s say 10* students campuswide. I can get my students up to speed in a matter of weeks and have done it many times. But I’ve seen (from the hall) a lecturer at the board claiming that a certain solution set was {x | x is all real numbers}. It made me want to throw up. This is even worse than the already give-up-why-live nonsense propagated by our text (with “a real number” in place of “all real numbers” one has ill-motivated circumlocution, but not actual solecism).

However. Set theory is not all that hard to understand if you’re willing to believe, as the instructor I saw wasting everybody’s time evidently does not, that all the symbols we use actually have definite meanings. These meanings, moreover, are actually rather simple (when taken one at a time… difficulties of course arise when we combine the various symbols…).

So. Where to begin. Since it’s 103, probably with the number line. In fact, start with a line. Draw it. A segment (from side to side) with two “arrows” at its either end. None of the points have names. When they do, their names will also be the names of numbers. But first we’ll choose one “arrow” and call it the positive one; the other we’ll of course call negative. We can then think of the number line itself—the set of points on the line—as “everything between the arrows” and write . The idea here is that the symbol for “minus infinity” here has essentially the same meaning as the left-hand arrow (by convention, the positive direction is to the right… anyhow, when we’re reading English…): no matter what actual point of the line we pick, the “arrow” is always imagined to its left. Similarly, mutatis mutandis, for the right arrow and “positive” infinity (but no “plus sign” is required… and neither is the verbal plus-sign “positive”). All is by way of trying to force on your attention that the drawing of the line and the typographic code have certain rather striking resemblances.

One now goes on to mark… if only in imagination… a point called Zero and another called One (with Zero to the left of One). These are of course given their usual symbols… and in fact the symbols are the actual names (you can “zero in on” something but you can’t “0 in on it”—prose is not textmessaging—the word has various meanings depending on context, but the symbol 0, will, for us, always and only denote the number zero).

With Zero and One in place, we can display two “new” (new to this discussion) kinds of interval. These are, first, the “open” interval (0, 1), denoting the set of points between Zero and One—in the “strict” sense of “between”, and second, the “closed” interval [0,1] which includes all of those points and also Zero and One themselves.

With this interval in place, one can go on to construct the entire number system, placing copies of the Unit Interval side-by-side to create the Integers and then subdividing all the intervals in various ways to produce the Rational Numbers. One typically fudges the details of how, since the Rationals do not cover the entire line (or, anyway, don’t if we insist that the line has certain desirable properties like “limit points”), we require the Reals in order to make sense of the concept of “infinite precision”. At which point we can make sense of any interval we like: , for example.

Now we can start doing unions and intersections and begin to see precisely how these relate to the English words Or and And. The “precision” I refer to is essentially the fact that calculations have answers: just is and no other way to say what it is works anywhere near as well (in this context).

Probably there should be drawings and wavings of pencils and maybe even some laughing or cursing… one doesn’t see what’s going on without a struggle and there will be many ways to arrive at the conclusion. But one conclusion. And one upon which even Authoritarians and Anarchists can agree. Here is power.

Read The Whole Thing

This course is a remedial preparatory course designed to improve the student’s algebra and problem solving abilities. The course includes: functions; systems of equations in two variables; applications and modeling; properties of exponents; scientific notation; polynomial arithmetic, factoring and equation solving; rational expression arithmetic and simplification; and complex fraction simplification. These topics are taught using an approach that integrates algebraic, graphic and numeric methods whenever possible.

The whole syllabus might be found here with an appropriate password… I don’t know (this is an experiment; here goes).