Pure Math Or Not

[Wannabeagenius]

Hi All,

I'm currently teaching Intermediate Algebra and have taught the class how to solve algebraic problems where a denominator such as (x-2) exists. In other words, the term is not defined at x=2.

Now say that the alleged solution to the problem is x=2. In that case, the correct answer to the problem is that there is no solution.

I'm wondering whether or not a problem such as this can occur in the real scientific world. I'm tempted to tell the class that this is a purely mathematical problem with no real world significance but I want to know for sure before making such a definitive statement.

Thank you,
Bob Guercio

 

[Studiot]

Don't electrical engineers meet this situation frequently with Laplace Transforms?

(Forgive the pun)

 

[Sportin' Life]

Absolutely happens all the time. For example if anything can be described by a log function then it makes sense to look at the area under its curve, even as it approaches zero.

What you are describing is a singularity and physics (and indeed all of science) is rife with 'em.

 

[CompuChip]

Only the question is always: is it a "mathematical" (removable) singularity, like the one in f(x) = (x² - 4) / (x - 2)? Because then we can define the function at x = 2 with a value that makes sense (for example, f(x) = 4). Or is it a physical singularity, like the r = 0 in the force law F = k / r²? That is a more serious issue, as it usually indicates a boundary of the validity of the theory (for this example, you cannot put two point particles at the same point, i.e. zero separation).

Also, if it is a removable singularity, why is it there? Is it just an artefact of the way we set up our calculation, i.e. if we had done the calculation, would we have arrived at f(x) = x + 2, valid for all x, instead of f(x) = (x^{2 - 4) / (x - 2)? Or does something special indeed happen at x = 2, which your mathematics happens to "know" about.}

 

[Tac-Tics]

In that case, the correct answer to the problem is that there is no solution.

Division by zero and other oddities are really hard to explain to high school students (and more generally, to most people).

The problem is you can't just divide any two numbers. The denominator has to be zero.

But students will ask "well, what DOES happens when it's zero?" This is an unaskable question. It's like asking what's your dog's name to someone who has never owned a dog.

Now, let's ask a silly, almost stupid question to everyone in the room. The question is this:

"Does your dog's name begin with a letter of the alphabet?"

The answer is clearly yes, isn't it? All names begin with a letter of the alphabet. But we forgot one small detail: not everyone has a dog. Their dog's name is in a sense "undefined". So, asking anyone who owns a dog, the answer is "yes". For people who don't own a dog, the question doesn't make sense.

Let's look at this slightly more mathematically. We define the function

d(p) = the name of person p's dog.

It's a totally legitimate function. You give it a person p who owns a dog and it evaluates to their dog's name.

But the problem is it's domain. It's domain, as we just said is the set of people who own dogs. But in our problem above, we are sometimes providing people who don't own dogs. We are passing a parameter that doesn't belong to the domain! This isn't allowed!

There are ways to make this mathematically rigorous instead of using the vague term "undefined", but this is rarely done in algebra and is likely to just confuse the hell out of your students.

 

[CRGreathouse]

http://en.wikipedia.org/wiki/Prandtl–Glauert_singularity

 

[Mensanator]

When you hold a micro-phone next to the speaker, it tries to become infinitely loud. Of course, it can't, so it oscillates (that's that high pitched feedback squeal you hear).