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)

[parisse]

Very well explained!

[sasa]

Who ever worked with calculus, know that limits are the easiest part to understand, as they are quite intuitive. Where they are needed use it explicitly. However, generalizing that approach on a planes with strict definitions, simply fundamentally break everything and relativizes results, which are not desirable there, make it not quite understandable for a student in specific learning stage, knowing nothing about...

And what to do with them? That 13yo student ask math teacher to explain him higher math material in 5 minute which will fully understand? I do not think so...

This particular thread subject is just fundamental, revealing quite unpleasant CAS behaviours, I have already mentioned (limited settings and quite unexpected results). There is nothing to say more, that I did not already wrote in my first two posts.

Conclusion: if low grade student want to use modern CAS for symbolic evaluation only, should learn complete higher math in advance in order to avoid WTH questions. Or simply to avoid to use expensive CAS capable calculators and work on speed and precision to manually perform calculation in order to get valid and expected results.

[toml_12953]

(10-21-2018 09:35 AM)sasa Wrote:  Since we do not work here with limits but simply dividing, everybody know that dividing by zero is undefined - or not? CAS in Prime and Wolfram Apha claims that 1/0 is infinite...

Mathematics is exact science and clearly define that dividing by zero is undefined, thus this is clearly fundamental bug.

Dr. Parisse?

No, some branches of mathematics do define dividing by zero as infinity so it's not a bug. It's just using a different branch of mathematics than you're used to seeing.

[parisse]

to sasa: a CAS is a computer algebra system, not a math teacher. Before high school, you don't really need a CAS, except for tasks like checking expand/factor. And it's the job of the math teacher to explain that you must sometimes wait a couple of years before you can understand everything, and in the meantime there is a difference between what you learn and what a professional CAS does (same for 1/0 or sqrt(-1)). The same applies for all sciences.