However weird the linear interpolation of Charlotte, North Carolina’s population may be outside the range from 1970 to 1980, it seems to do nicely enough between those years. And that’s as we might expect, since we used the actual population data from the census days of 1970 and 1980 to form this interpolation. But we don’t have to make a linear interpolation. We could in principle use any function, but let’s try a simple one. This would be a quadratic polynomial, one where the variable x gets raised all the way to the second power, and one that brings back faint memories of the quadratic formula, which is one of the rare pieces of mathematics for which I have a work-related anecdote. Ask sometime if you’re interested.
[ I cannot and do not try to explain it, but yesterday was a busier-than-average day around these parts, with a surprising number of references coming from an Entertainment weekly article about the House series finale for some reason. In this context a “surprising” number is “any number other than zero” since I don’t know why anyone would go from there to here. I watched House, sometimes, sure, and liked it, but kind of drifted away when there was other stuff to do, you know? ]
That’s enough time spent establishing the heck out of the idea of a polynomial. Let’s actually put one in place. My goal back when was estimating what the population of Charlotte, North Carolina, was around 1975. I had some old Census data from 1970 and 1980 giving its population on the first of April, the earlier year, as 840,347; and the first of April, 1980, as 971,391.
[ According to the WordPress statistics, trapezoids are just the hook bringing people into here. I didn’t realize there was such a big community of people who need trapezoid information. If I did I’d have played up my search engine terms more. ]
If anyone had doubts about using polynomials as a generally good thing I hope either the doubts or the doubters are quieted now. My next couple goals are simple ones: I want to set up polynomials to interpolate what the population of Charlotte, North Carolina, was around 1975. That is, I’ll be creating at least one equation of the form where somehow the right choices of numbers for , et cetera will mean if I put the right number in for x I’ll get out of it an estimate of the population. I’ve got symbols. I need to figure what I want them to mean.
[ Curious: one of the search engine terms which brought people here yesterday was “inner obnoxious”. I can think of when I’d used the words together, eg, in a phrase like “your inner obnoxious twelve-year-old”, the person who makes any kind of attempt at instruction difficult. But who’s searching for that? I find also that “the gil blog by norm feuti” and “heavenly nostrils” brought me visitors so, good for everyone, I think. ]
So polynomials have a number of really nice properties. They’re easy to work with, which is a big one. We might work with difficult mathematical objects, but, rather as with people, we’ll only work with the difficult if they offer something worthwhile in trade, such as solving problems we otherwise can’t hope to tackle. Polynomials are nice and friendly, uncomplaining, and as mathematical objects go, quite un-difficult. Polynomials can be used to approximate any function, which is another big one, as long as we don’t take that “any function” too literally. We still have to think about it some. But here’s an advantage so big it’s almost invisible: to evaluate a polynomial we take some number x and raise it to a variety of powers, which we get by multiplying x by itself over and over again. We take each of those powers and multiply them by a corresponding number, a coefficient. We then add up the products of those coefficients with those powers of x. In all that time we’ve done something great.
I’ve done as much as I want with piecewise constant interpolations, at least for the moment. The next step that makes sense to me is to look into polynomials. They’re a powerful tool to use in interpolations, but that doesn’t stand out, because they’re powerful tools for most uses. They’re very popular mathematically, since a few polynomials can turn what was a young student’s natural interest in mathematics into a passionate lifelong loathing, with the occasional dream of being haunted by the “quadratic formula”. It’s worth taking a few paragraphs to see what polynomials are, and why they’re popular among those who get past that class.