A Polynomial Of What?


[ 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 y = a_0 + a_1 \cdot x + a_2 \cdot x^2 + a_3 \cdot x^3 + \cdots + a_n \cdot x^n where somehow the right choices of numbers for a_0, a_1, a_2, 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.

The thing about that equation for y as the sum of a bunch of products of coefficients and powers of x is that there isn’t anything inherent in the x or the y or any of the coefficients that means “population” or “year” or so on. They’re just pure numbers, maybe a ‘4’ or a ‘212’ or if I feel like stretching a ‘867,924’.

In a way, that’s a strength: the polynomial makes use of pure numbers, ones that haven’t got any semantic content. As long as I start with the correct numbers and don’t make any arithmetic errors, I’ll get correct numbers back out of it. Even if I make errors, if I have some idea how big those errors I make are, I can estimate how far I can possibly be from the correct answer. And if I come up with a routine that’s very good at interpolating the population of Charlotte, I can use the same routine to interpolate the population of, say, Loudonville, New York, or maybe something more exotic yet, like how many baseball games might have been won by the Philadelphia Phillies as of 1950.

But this means the work of translating my problem into an equation, and of translating the answer back into what I want to know, remains. You know this work, as it’s the hard part of every word problem: sorting through a bunch of paragraphs and trying to figure out how any of this talk about airplanes travelling in different directions in problems that obviously used to be trains going in different directions in earlier drafts of the book can be turned into the equations that were all over the actual textbook material.

And now you remember the hard part of those word problems. If you got through to an equation, it probably wasn’t so bad. After all, the textbook and classes were all about how you solve an equation, and if you manage to follow a couple of basic steps — add (or subtract) the same number to both sides of the equation, multiply (or divide) both sides of the equation by the same number — you’ll eventually drift into the correct answer. It’s the setting up that’s so hard.

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