CAS: 0/0 is undefined, however 1/0 is infinite?

[Nigel (UK)]

Disclaimer: I am not a mathematician (I teach Physics to students between the ages of 13 and 18). I do own a Prime, and I am happy with its approach to 1/0, 0^0, etc.

The other day I was teaching the interesting topic of car stopping distances to students aged 14-15. A student asked why brakes cannot stop a car as soon as they are applied. To answer this, we worked out the acceleration for a series of shorter and shorter stopping times and concluded that stopping in zero time would require an infinite force, because the force was increasing without limit as the time approached zero.

I believe that the Casio calculators that my students use give "Math Error" or some such message when dividing by zero (I don't have one in front of me to check this). From my perspective, "1/0=infinity" is actually useful. It is a statement about a limit:

If \(\lim_{x\to0}f(x)=0\), then \(\lim_{x\to0}\bigl({1/f(x)}\bigr)=\infty\) (where \(\infty\) means complex infinity, i.e., its argument is unknown). So saying that \(1/0=\infty\) is reasonable, as any limiting process that leads to \(1/0\) will have \(\infty\) as its limit.

However, \(0^0\) is different. Knowing that \(\lim_{x\to0}f(x)=0\) and that \(\lim_{x\to0}g(x)=0\) is not enough to determine \(\lim_{x\to0}\bigl(f(x)^{g(x)}\bigr)\). So the calculator needs to say that \(0^0\) is undefined.

Having said this, the option to throw an error for \(1/0\) is useful. If a student finds themselves dividing by zero in an exam they have made a mistake, and their Casio calculators - or the HP Prime in Home mode - will tell them so. I think the Prime offers the best of both worlds.

To sum up: treat all of these odd expressions as statements about limits and everything works correctly. Even in Physics, knowing that a limit is infinite can be useful.

Nigel (UK)