The Best Thing About Polynomials


[ 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.

When we figure out the value a polynomial takes for some particular x, we don’t ever have to do anything except multiply numbers together and add numbers together. Oh, technically, we might do some subtraction — if we have a negative coefficient, for example, or a negative x, we might have to work out 4 – 2, or something like that, but we can think of subtraction as just addition with negative numbers. And we might have to do divisions — suppose one of the coefficients is 1/4 — but that’s multiplication by a reciprocal, and if you don’t remember reciprocals, then, let’s just say we did all this without doing anything but addition, subtraction, multiplication, and division. All the stuff that made up mathematics back in elementary school when everybody likes mathematics, and before algebra starts putting letters into it.

Basic arithmetic, that’s nice. We learned to do it early on, and given a fresh sheet of paper and if we only have to work with whole numbers, we can still do it pretty well. But here’s the even better thing: we don’t have to do it. When’s the last time you had to do a bunch of multiplications? When’s the last time you needed to do a bunch of additions by yourself? Maybe in a store, figuring out about what the bill should be, so you can get all snooty and sarcastic if the cashier rings it up wrong. Maybe figuring out convoluted mixes of change so that you can give the cashier $7.12 to cover a $6.87 bill and see what happens. But you don’t have to do that. You’re just having a bit of fun. The cash register is better at adding number than you are, and it always will be as long as it’s plugged in and nobody’s taken a mallet to it.

And that’s what makes polynomials so great. Machines are very good at addition and subtraction, multiplication and division. If you give them the correct numbers to start from they won’t make mistakes [1], and they’ll get done incredibly quickly. They won’t get bored with doing it, either.

So through polynomials we can make an approximation of any function we might be interested in, and turn it over to a computer to do the actual calculations. We lose some precision — we’ll get a slightly wrong answer — but we can make the size of that error as small as our needs dictate.

That’s great news for us. It takes a flash of insight to realize that we can figure out the common logarithm of 7, working by hand. (Here’s one way, which requires knowing something about logarithms for the fine points; if you don’t remember logarithm fine points just skip to the close parenthesis marks and trust me that I’m saying something true and I might fill in the gaps later on. So how do we find the logarithm of 7? By the “common logarithm” I mean the logarithm base ten, and again, if you don’t know why that’s interesting please just take my word that it is. The common logarithm of 7 is one-half the logarithm of 49, because 49 is the square of 7, and the logarithm of the square of a number is twice the logarithm of the original number. 49 is just about one-half of 100. So, the logarithm of 49 must be the logarithm of 100 minus the logarithm of 2, because the logarithm of one number divided by another — such as 100 divided by 2 — is the logarithm of the first number minus the logarithm of the second. The common logarithm of 100 is exactly 2; the common logarithm of 2 is a little bit more than 0.3, which takes another similar flash of insight; think about how close, though, that two raised to the third power is to ten, or that two to the tenth power is to one thousand. So the logarithm of 49 is something a little under 1.7, and therefore the common logarithm of 7 must be almost 0.85, which is satisfyingly right — it’s actually approximately 0.8450. You maybe were impressed with the logical steps there, but more likely, your eyes glazed over the entire paragraph and if asked, you’d say this is the kind of thing you never understood.)

But it takes no imagination, no flashes of insight, nothing but an indefatigable nature to instead work out the sum — trust me that this is the sum we want, please — of:

1 - \frac{1}{2.3}\left( 0.3 + \frac12 0.3^2 + \frac13 0.3^3 + \frac14 0.3^4 + \frac15 0.3^5 + \cdots \right)

where the \cdots means we keep on adding terms on the end there, one divided by a whole number times 0.3 raised to that number, until we are sure we’ve used enough terms that we aren’t making too large an error by stopping. (Just the terms I wrote out there is pretty good already. Go ahead, try entering it into the Google search bar and see what the sum is. That 2.3 right up front should actually be a slightly different number, but I’m approximating, to make life easier.)

The collection of polynomials, then, lets us play with all kinds of interesting functions, and leave the hard work of figuring out what the polynomials are equal to, and to do wonderful things like drawing them so we can understand them, to computers that will do all the calculating for us. We really have to like tools that let us do what we’re really interested in without making us work for it.

[1] By “not making mistakes” I should admit that computers actually always make mistakes. Or at least almost always. The trouble is that we’re often interested in numbers like 1/3, while computers don’t work with fractions so much as an imitation of decimals known as floating point numbers. 1/3 has an infinite number of decimal digits. The computer doesn’t. Just as there’s a difference between 1/3 and 0.333, so there’s a difference between most interesting numbers and their floating point representations. If we’re smart, and we are smarter than the computer, we’ll avoid doing problems which turn these tiny errors into big ones.

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Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there.

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