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’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.