I've just started reading Paul Nahin's "Dr. Euler's Fabulous Formula: Cures Many Mathematical Ills." I haven't read his well-received predecessor to this book, "An Imaginary Tale: The Story of "i"," but it's definitely on my list.

So far, I have not been impressed by Dr. Euler's.... I know it's an entirely crass and superficial thing to lead with, but I was immediately put off by his photograph on the dust jacket. The beard. The casual posture. The ice cream. I felt like I've known too many math types like this: incredible snobs with a practiced air of earthiness about them.

The text itself hasn't, thus far, elevated my opinion of Dr. Nahin. He starts early on by trashing the art of Jackson Pollack. Now, I'm no big fan of Pollack myself, but I certainly wouldn't publicly proclaim that his work is devoid of artistry or talent, as Nahin does.

I haven't gotten very far, but so far I've found the math itself to be only mildly interesting -- and difficult. It's clear that Dr. Nahin is twice the mathematician that I am, and this doesn't bother me. I do feel a little chagrined at having some difficulty following the math, when he asserts in the introduction that anyone with some calculus background, and certainly someone possessing an undergraduate degree in math, can follow the math.

But I was never a particularly good mathematician, so no surprises here.

I'll update this once I finish this tome, obviously.

Down with quadratics!

Or, more precisely, down with the term "quadratic." It gives me a headache trying to explain to students why a quadratic equation is polynomial of degree two, while quartic and quintic equations are polynomials of degree 4 and 5 respectively. While we're at it, let's also get rid of "linear" and "cubic" for the same reason. These terms, of course, represent the legacy of the geometry-crazy Greeks. A quadratic is a polynomial of degree 2 because "x squared" represents a square with four sides. Whatever. It's confusing. It's inconsistent. It's time it faded into history.

Let's call them "diptic" and "triptic" equations instead. Then everything fits neatly:

constant: polynomial of degree 0
monic: polynomial of degree 1
diptic: polnymoial of degree 2
triptic: polynomial of degree 3
quartic: polynomial of degree 4
quintic: polynomial of degree 5

Is it going to happen? No, these terms are too entrenched. And I can't get quite as upset about the term "linear" (a monic), since the locus of a monic truly is a line (whereas the locus of a quadratic is not a square).

Any thoughts? Has anyone else experienced student confusion over these inconsistent terms?

Factoring Sucks

Well, to be specific, factoring trinomial expressions with a 2nd-degree coefficient that's not one sucks. Woe be to the algebra teacher who assigns this problem:

Factor: 6x² + 16x + 10

Not only are there two distinct ways to factor this, it's a lot of work. But let's take a step back for a minute, and look at easy factoring:

Factor: x² + 8x + 12

This is easy because we can just think about the factors of 12 (1*12, 2*6, 3*4) and immediately see which two factors can be combined to make 8 (in this case, 2 and 6). So there aren't many choices (even if the constant term has a lot of factors, like 12 or 60), and it's easy to reach the solution in our heads:

x² + 8x + 12 = (x + 2)(x + 6)

Looking back at our original problem, we see that we first have to figure out how we're going to get our 2nd-degree coefficient. There are only two ways: 1*6 and 2*3, so that doesn't seem so bad. We know, then, that our solution will either look like:

(x + ...)(6x + ...)

or

(2x + ...)(3x + ...)

We also have a pretty easy time determining how to get the constant term: it's just the factors of 10 (1*10, 2*5). So we know the other numbers in the parenthesis will either be 1 and 10 or 2 and 5. So far so good.... But let's consider the possible combinations:

(x + 1)(6x + 10)
(x + 2)(6x + 5)
(2x + 1)(3x + 10)
(2x + 2)(3x + 5)

That doesn't seem too bad...but wait! We're not done yet. Unlike our simple factoring above, the ordering is important, so we also have the following possibilities:

(6x + 1)(x + 10)
(6x + 2)(x + 5)
(3x + 1)(2x + 1)
(3x + 2)(2x + 5)

So we've now got a total of 8 possible combinations. The worst part? We have to FOIL them all to figure out if we get that magical 16x term. Well, there is one shortcut we can take: we don't have to do the F part (because we know that we'll always get 6x² in every case), nor do we have to do the L part (because we know that we'll get 10 in every case). So all we have to do is OI (OI!). So we get:

(x + 1)(6x + 10) O: 10x I: 6x
(x + 2)(6x + 5) O: 5x I: 12x
(2x + 1)(3x + 10) O: 20x I: 3x
(2x + 2)(3x + 5) O: 10x I: 6x
(6x + 1)(x + 10) O: 60x I: x
(6x + 2)(x + 5) O: 30x I: 2x
(3x + 1)(2x + 1) O: 3x I: 2x
(3x + 2)(2x + 5) O: 15x I: 4x

(Forgive the ugly font; it was the easiest way to get things to line up.) Well! Finally we can glance over the OI products and figure out which of them will give us our 16x, namely 10x and 6x. So we finally have two possible factorizations:

(x + 1)(6x + 10)
(2x + 2)(3x + 5)

It's worth noting here that if we set both of these expressions to zero, we'll end up with the same solutions. So, end the end, it doesn't matter which factorization we use.

I just spent an hour trying to figure out a way to make this nonsense easier and came up with...nothing. It's just an ugly, hard slog. So I challenge you, my dear readers, to come up with a method of solving these problems that isn't so painful. You've got mnemonics? Bring 'em on. A fancy table? I'd love to see it. Something ultra-clever? Maybe it's out there....

In the meantime, should you not give your students these problems? Hell no! Of course you should assign them these problems. Maybe not all the time, but don't get the little buggers get away with easy factorizations. Sometimes math is easy and elegant, and sometimes it's a hard dig in a cold ditch, and the sooner students realize that the better. Woe be to the student who makes it to differential equations thinking that there's an easy solution to every problem.

Last week, a friend of mine told me that a mutual friend is moving off to the East Coast to teach math & was wondering if I could recommend any math text. Well, of course I jumped on this exciting opportunity, and it also got me inspired to dust of my blog and start posting again on a regular basis. So thank you, Leon, for the inspiration!

Recently, a student of mine, while learning exponentiation, asked me "but what does 'exponent' mean?" Like all questions I am unable to answer, I immediately became obsessed with answering it.

The process of exponentiation, of course, hasn't been around forever. The ancient Greeks started it all with their obsession with geometry, in which squaring and cubing numbers figured prominantly. The area of a square, for example, is x squared, and the area of a cube is x cubed (where x is the length of a side.) "x squared" was taken to mean x·x and "x cubed" was taken to mean x·x·x.

It wasn't until Diophantus of Alexandria (250 A.D.) that we saw the use of symbols to represent exponentiation, and he only had symbols for x^-6 to x⁶, inclusive. Although this was a far cry from the usefullness of exponents of arbitrary size, it certainly improved the typography of algebra. It is worth noting that Diopantus' notation employed the use of the superscript, as we do today.

No further improvements were made in algebraic notation until Nicolas Oresme in the 14^th century. This is hardly surprising considering the Middle Ages stretched between Diophantus and Oresme, casting mathematics into pitch blackness. While Oresme's notation may appear daft to us today, it was remarkable in that it allowed for arbitrary exponents (written as arabic numerals.)

The notation we use today, xⁿ, was introduced by Réne Descartes in the 17^th century, along with such important ideas as the Cartesian plane (Descartes' Latin name was Cartesius.) At first I suspected that Descartes was the first to use the term exponent, but it appears that one of the Bernoullis (Johann, Jakob, Daniel or Nicolaus II) may have used it first. At any rate, that dates the use of the term to the 17^th century.

The term itself comes from the Latin expone, "to put forth." That is, the exponent tells you how many times to "put forth" the base.