Wednesday, February 1, 2012

Unduly Noted

Tommy Ratliff—Wheaton College
When I opened the MathFest program in Lexington last summer, I took one look at the first page and nearly yelled out loud “NO! NO! NO!” The inside cover to the program contained an advertisement for an online homework system with the following example:

Find the derivative of y = 2 cos(3x − π) with respect to x.

The assertion is that a competing system marked this answer as wrong, but the advertised system identified the expression as correct, demonstrating its superiority. I assume that the intent is to show that the system can recognize equivalent, but not identical, algebraic expressions. What caused me to react so strongly, however, is that I would have also marked the given answer as wrong. The answer should have been

The parentheses matter! The expression sin(x) represents a function!

I should be clear: My irritation is not directed at this particular homework system as much as at the entire mathematics community for the sloppiness in notation that we tolerate, and even encourage, when dealing with trigonometric functions. You can pick up almost any calculus text, peek into almost any math classroom, or attend any number of talks at various MAA events to find a plethora of examples of trig functions lacking their parentheses.

Why do I think the parentheses matter so much? This is not just a pedantic preference on my part. The lack of parentheses represents an irregularity in notation that obscures the meaning of the mathematics. We often use a space to indicate multiplication, as in or 3 sin(x), so leaving off the parentheses hides the fact that we are using a trigonometric function. The confusion is compounded when we say that the derivative of “sine” is “cosine.” If we were to be consistent, this would lead to applying a distorted product rule to get

I have seen students struggle with this, even when they understand the intent of the original notation. They correctly apply the chain rule only to confuse the order of operations at the end because they did not put the constant multiple of 3 at the beginning of the expression:

After all, why should you apply the cosine function to the first 3 in the 3x but not to the trailing 3? If we always used parentheses to enclose a function’s argument, then there would be no confusion.

An even worse abuse of notation occurs in the location of the exponent when a trig function is raised to a power. I will never write sin2(x) for sin(x)2 because the first notation leads to ambiguity when discussing the inverse trig functions. Since f− 1 (x) is the standard, consistent notation for the inverse function of f(x) , we also use sin− 1(x) for arcsin(x). If we were consistent with notation, a perfectly reasonable calculation would be

Notice that I had to make a choice about the meaning of sin−2 (x) in simplifying the expression—an impossible task! Should it be sin2(x)-1 = 1/sin2(x) or sin -1(x)2 = arcsin (x)2? The point is that we should never have to make this choice! We should be taking the derivative of the inverse sine function. This is horrific. The bad notation allows at least three different interpretations of the expression

We in the mathematics community pride ourselves on the consistency and deterministic nature of our discipline. I think we do ourselves a genuine disservice when we use sloppy notation that requires another layer of interpretation to understand the intended meaning. The purpose of mathematical notation is to provide clarity and, ideally, to provide insight into the mathematics being notated.

Therefore, I implore you: The next time you use a trig function, please remember the parentheses, put the exponent on the outside, and never, ever write anything like sin3x2 cos-25x.

Tommy Ratliff is a professor of mathematics at Wheaton College in Massachusetts where he enjoys thinking about voting theory, building new science centers, and being precise in his notation.