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 built up a linear interpolation for Charlotte’s population between 1970 and 1980. In principle, I could extend this beyond those years, and project what the population was before Census Day 1970 or after the census of 1980. Generally, we’d call that — the values the interpolating polynomial takes on, outside the range of data we started with — an extrapolation rather than an interpolation, but they’re pretty closely tied together. If we understand one we’re doing pretty well understanding the other.
[ 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.
Just because there are in principle uncountably many possible equations for any line doesn’t mean we ever actually see any of them. Actually, we just about always pick one of a handful of representations. They’re just the convenient ones. I’m going to say there’s four patterns that actually get used, because I can only think of three that turn up, as long as we’re sticking to Cartesian coordinate systems and aren’t doing something weird like parametric descriptions, and I want to leave some hedge room for when I realize I overlooked the obvious. The first one — that I want to talk about, anyway, and just about the first one anyone encounters — is called the slope-intercept form, and it’s probably what someone means if they do talk about “the” equation for a line.
[ Oh, wow. Yesterday’s entry had way fewer hits than average. I also put an equation out right up front where everyone could see it. I wonder if this might be a test of Stephen Hawking’s dictum about equations and sales. Or maybe I was just boring yesterday. I’d ask, but apparently, nobody found me interesting enough yesterday to know for comparison. ]
It shouldn’t be too hard to translate the the idea “I want to know the population of Charlotte at some particular time” into a polynomial. The polynomial ought to look something like y equals some pile of numbers times x’s raised to powers, and x somehow has to do with the particular time, and y has something to do with the population. And it’s not hard to do that translating, but I want to talk about some deeper issues. It’s probably better explaining them on the simple problem, where we know what we want things to mean, than it would be explaining them for a complicated problem.
[ 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.
Polynomials turn up all over the place. There are multiple good reasons for this. For one, suppose we have any continuous function that we want to study. (“Continuous” has a technical definition, although if you imagine what we might mean by that in ordinary English — that we could draw it without having to lift pen from paper — you’ve got it, apart from freak cases designed to confuse students taking real analysis by making continuous functions that don’t look anything like something you could ever draw, which is jolly good fun until the grades are returned.) If we’re willing to accept a certain margin of error around that function, though, we can always find a polynomial that’s within that margin of error of the function we really want to study. I have read, albeit in secondary sources, that for a while in the 18th century it was thought that a mathematician could just as well define a function as “something that a polynomial can approximate”.
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.
I have a couple of other thoughts about these piecewise constant functions which I’ve been using to make interpolations. The basic idea is simple enough; we pretend the population of Charlotte was a constant number, the 840,347 it happened to be on the 1970 Census Day, and then leapt upwards at some point to the 971,391 it would have on the 1980 Census Day. Maybe it leapt up immediately after the 1970 Census; maybe immediately before the 1980; maybe at the exact middle moment between the two; maybe some other day. Are those all the options we have?
[ We didn’t break 3,100 yet, and too bad that. But over the day I did get my first readers from Turkey and the second from the United Arab Emirates that I’ve noticed. Also while my many posts about trapezoids are drawing search engine results, “frazz sequins” comes up a lot. ]
I think I’ve managed, more or less, acceptance that a piecewise constant interpolation makes the simplest way to estimate the population of Charlotte, North Carolina, when all I had to work with was the population data from the 1970 and the 1980 censuses. In 1970 the city had 840,347 people; in 1980 it had 971,391, and therefore the easiest guess to the population in 1975 would be the 1970 value, of 840,347. We suppose that on the 1st of April, 1970 — that Census Day — the population was the lower value, and then sometime before the 1st of April, 1980, it leapt up at once by the 131,044-person difference. Only … how do I know the population jumped up sometime after 1975?
[ I’d like to thank all who’ve read me or passed on links to me for getting my total hit count above 3,000. In fact, as I write this, the total seems to be 3,033, which is a pleasantly 3-ish number. I suppose that it’s ungrateful to look for 4,000 right away, but after all, I do hope to be interesting or useful, and both of those seem to correlate pretty strongly with being read. In any case, I’ll see how long it takes to reach 3,100, and be silent about that if it’s a number of days too embarrassing to mention. ]
The task I’ve set myself is finding an approximation to the population of Charlotte, North Carolina, for the year 1975. The tools I have on hand are the data that I’m fairly sure I believe for Charlotte’s population in 1970 and in 1980. I have to accept one thing or I’ll be hopelessly disappointed ever after: I’m not going to get the right answer. I’m not going to do my job badly, at least not on purpose; it’s just that — barring a remarkable stroke of luck — I won’t get Charlotte’s actual 1975 population. That’s the nature of interpolations (and extrapolations). But there are degrees of wrongness. Guessing that Charlotte had no people in it in 1975, or twenty millions of people, would be obviously ridiculously wrong. Guessing that it had somewhere between 840,347 (its 1970 Census population) and 971,391 (its 1980 Census population) seems much more plausible. So let me make my first interpolation to Charlotte’s 1975 population.
[ I’m grateful to all for the help in reading my pages here. I’ve not quite reached 3,000 hits, but it’s within sight. If you do know of people who might be interested in either what I’m doing now — and it should be clearer after today’s post — or articles I’ve written in the past, please let them know, or let me know if I could be doing better at reaching interested audiences. ]
I left off the list of places I’d lived the city of Charlotte, North Carolina. There’s justice in my doing so. We lived there only for a couple years, when I was extremely young. I have only a few memories of the place, most of them based on the popcorn machine they had in my preschool program. I don’t know what else I got out of that, but I certainly appreciated seeing popcorn pop. Also I had two brothers born then. But, mostly, I can’t say that Charlotte made much of an impression on me. I couldn’t identify any major features of it from memory, and challenged to point to it on a map I might point at Delaware instead, or wander off to find a soda. Plus, I last lived there somewhere around 1975. I can accept that the population of South Amboy, New Jersey, may not have changed very much since the mid-1970s, but not that Charlotte’s hasn’t.
[ I don’t wish to be too shameless here, but I’m closing in on 3,000 visitors to my little blog here. Can we get there? Kindly pass on a reference to people you think might be interested; if I matched my most-popular-ever day I’d reach 3,000 tonight easily. ]
I’ve lived almost my entire life in New Jersey, which has its effects on my world view; for example, it produces an extreme defensiveness about the state — really, has there been a fresh Jersey Joke since Benjamin Franklin’s quip about it being “a barrel tapped at both ends”, and they’re not even sure it wasn’t James Madison who said that instead, if anyone ever did? — and a feeling that one should refer to Bruce Springsteen as “Bruce”, as if we’d ever knowingly been in the same zip code simultaneously. Add to that not understanding what is wrong with other states that you’re forced to pump your own gas, and not being able to get a cackling laughter and a voice-over announcer wailing “Rrrrrrrrraceway Park!” out of the head, and you’ve got a first sketch of my personality. (I seem to have missed going to Action Park. My father insists he took me there; I grant he may have taken my siblings, but I don’t remember ever getting there, and the fact I have all my limbs suggests I never did go there.) But there are some other impressions that one gets from growing up in New Jersey.