Monday, February 11, 2008

0.999...=bad mathematical education

0.999... = 1

Right or wrong? An amazing number of people refuse to accept this equality. It just looks wrong, and despite assurances from the experts and proofs, some people just can't accept it.

This is such a classic example of why mathematics education is lacking, but critical, especially in the early grades.

It isn't that the experts are wrong (which really doesn't happen all that often) in making the equation but that you were TAUGHT wrong. Just like I was. The subject of mathematics is the most poorly taught and poorly understood by its instructors that you find. I have often felt horribly shortchanged by my own poor mathematical education. Not so much the naive misunderstandings of the early teachers, but the later ones who not only failed to understand, and failed to teach, but refused to brook any discontent with questions as to the underlying assumptions of what was being taught. Anyway.

The simplest proof of the equality is:

3*1/3=1 = 3*0.333...=0.999...

Simple enough, right? But you still don't like it, right? Smells funny? There's a "trick" in there somewhere, right?

No, there isn't. The "trick" is in your mind.

Children intuitively understand numbers as fluid concepts, but of course have limited vocabulary and limited range of comprehension. So we teach them to isolate and categorize and make 1=1.

The problem, you see, isn't that the equality is an approximation (one of the common counter-arguments), it's that the entire system is an approximation. That doesn't mean it's wrong or even imprecise.

It just means 0.999...=1

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