Divide by Zero: How to Explain the Phenomenon

[erok81]

What is the proper term to use when you end up with a divide by zero answer? I never what to write out when I run into this situation.

I've heard; singularity, undefined, infinity, and of course divide by zero.

What are you guys using when you have to explain this phenomenon? Or does it just depend on the situation?

I am a physics major if that matters. I know you mathematicians use some wacky terms. ;)

 

[Antiphon]

It's a Norris action. Only Chuck Norris can successfully divide by zero.

 

[gb7nash]

NaN? At least, that's what the computer would say. Otherwise, I would just call it DNE (does not exist) or undefined.

Just out of curiousity, where are you encountering a division by zero?

 

[jhae2.718]

If MATLAB is anything to go by, dividing by zero gives Inf. and indeterminate forms evaluate to NaN.

 

[Hurkyl]

What is the proper term to use when you end up with a divide by zero answer? I never what to write out when I run into this situation.

I've heard; singularity, undefined, infinity, and of course divide by zero.

What are you guys using when you have to explain this phenomenon? Or does it just depend on the situation?

I am a physics major if that matters. I know you mathematicians use some wacky terms. ;)

Undefined is the term for an expression involving real number division by zero. It's pretty much analogous to an English phrase like "deputy violin daylight" -- those words simply don't go together in a meaningful way.

Some arithmetic systems do allow some division by zero. For example, in the projective numbers, $1/0=\infty$. You're probably not doing arithmetic with those, though.

One of the many uses of the word "singularity" is to refer to a point where partial function is undefined. (There may be restrictions on the kind of partial function) For example, 0 is a singularity of the partial function f defined by f(x)=1/x.



Typically, if division by zero appears in your work, you made a (possibly calculated) mistake, such as overlooked a hypothesis needed for some theorem or equation.

 

[erok81]

It's a Norris action. Only Chuck Norris can successfully divide by zero.

:rofl: see this part of my post...

I am a physics major if that matters. I know you mathematicians use some wacky terms. ;)

:tongue:

Typically, if division by zero appears in your work, you made a (possibly calculated) mistake, such as overlooked a hypothesis needed for some theorem or equation.

Just out of curiousity, where are you encountering a division by zero?

My ODE professor used the term singularity a few times when was explaining why certain things wouldn't work in calculations when divide by zero would show up.

In my quantum mechanics class it came up in a few problems where we had to show why some calculations wouldn't work - one would end up with this divide by zero and that was the answer then of course an explanation why that happened. I can't remember the exact problems though.

Undefined is the term for an expression involving real number division by zero. It's pretty much analogous to an English phrase like "deputy violin daylight" -- those words simply don't go together in a meaningful way.

Some arithmetic systems do allow some division by zero. For example, in the projective numbers, $1/0=\infty$. You're probably not doing arithmetic with those, though.

One of the many uses of the word "singularity" is to refer to a point where partial function is undefined. (There may be restrictions on the kind of partial function) For example, 0 is a singularity of the partial function f defined by f(x)=1/x.

So it seems like the term might be situation specific then. Maybe undefined could be catch all statement since that seems to be more general.

 

[Studiot]

Consider the following integral, which involves division by zero.

$$\int\limits_0^a {\frac{1}{{{r^2}}}} dr$$

This is known as an improper integral because it becomes infinite (a la Hurkyl) for the lower limit.

This integral appears in physics and has been discussed in recent threads about potential.

It is not possible to evaluate this particular improper integral, however sometimes the difficulty can be overcome and the integral can be persuaded to converge to a finite value.

 

[Pengwuino]

In my experience, whenever you get something like an $$\infty$$, you should have remembered that you are actually taking some sort of limit, be it through a derivative or through an integration. For example, let's say you're looking at the coordinate time of a particle at a Schwarzschild black hole. I can't say "what is the coordinate time of a particle AT the event horizon" because I can't just grab a particle and put it right there. What I would want to remember is that I'm actually looking at differentials, which is a process which involves limits. So what I really have is as the particle tends to the Schwarzschild radius, the coordinate-time tends to infinity, all of which has precise mathematical meaning.

 

[erok81]

Thanks for the posts everyone. It is sounding more and more like a situation specific answer. I understand a lot of the methods for encountering divisions by zero, but never what my exact answer should say. But it seems it doesn't really matter.

Let's say you have a problem where you have to prove why x doesn't work. Let x be some random situation where the end mathematical result is a division by zero. Which is what you are trying to prove.

You end your problem with the function/equation/whatever and state x isn't possible because __________. Where the blank would be something like a singularity, undefined, infinity, etc.

If I am understanding the posts correctly, any of the above choices work, it just depends on the physical situation at hand?

 

[bcrowell]

You end your problem with the function/equation/whatever and state x isn't possible because __________. Where the blank would be something like a singularity, undefined, infinity, etc.

If I am understanding the posts correctly, any of the above choices work, it just depends on the physical situation at hand?

I run into this a lot when I teach physics. What I tell my students is that saying it's undefined is lame, because it fails to give insight into why it's undefined. Saying it's infinite is preferable, because it gives insight.

For example, say you want to stretch a rope in a "vee" shape, and you want it to sag by a height of no more than h. You will need a certain minimum tension T. The h=0 case gives division by zero when you plug into the equation for T. Saying T is undefined tells us you can't get h=0, but fails to provide insight into why you can't. Saying T is infinite for h=0 tells us more: the reason you can't get h=0 is because the rope would break. You do have to realize when interpreting the statement that "T is infinite" that this doesn't mean infinity is a real number, etc.

 

[Hurkyl]

I run into this a lot when I teach physics. What I tell my students is that saying it's undefined is lame, because it fails to give insight into why it's undefined. Saying it's infinite is preferable, because it gives insight.

Possibly -- but at what cost? If you stuff students heads full of half-truths, they eventually combine to create falsehoods.

Surely it's better to make a true statement, like

the needed tension diverges to +infinity as h goes to 0​

rather than

Saying T is infinite for h=0

I'm not sure it's relevant, but one doesn't even have to deal with a limiting argument to see why you can't get a straight rope -- you simply draw the free-body diagram and see that all of the other forces are horizontal, so they cannot counteract gravity. (edit: I note this may presume too much precision from the idealization)



P.S. upon reflection, I realize that even noting that tension diverges to +infinity only implies impossibility on the presumption that the analysis would be continuous in h.

 

[bcrowell]

Possibly -- but at what cost? If you stuff students heads full of half-truths, they eventually combine to create falsehoods.

Often my students are able to combine full truths to create falsehoods :-)

But more seriously, I think the fundamental issue here is probably the extent to which the real-number system is the right model for physical reality. Historically, the reals are only one of many different ways that people have found of getting at the concept of number. There are some ways in which the reals are an extremely bad model of physical reality. For example, the distinction between rationals and irrationals can't be related to any physical observable. There isn't really any number system that is satisfactory in every way as a model of physical reality -- not the rationals, not the reals, not the hyperreals, ...

I think we should also be careful not to assume that when a student says, e.g., "3/0 is undefined," the students is being sophisticated about what that means. I would say that there are at least three levels of sophistication: (1) The student says "T is undefined for h=0," but doesn't understand what that means either physically or mathematically. The students is just repeating a memorized fact that division by zero is "undefined." (2) The student understands that the rope would break before you got to h=0. (3) The student can state all the axioms of the real-number system, including completeness, and therefore understands how to do rigorous proofs involving the reals.

Freshman physics students are essentially 100% at level 1. I have never met a freshman physics student who was at level 3. The completeness axiom isn't stated in any math textbook in the lower division, i.e., any math textbook that will ever be used by anyone who isn't a math major. Biology and engineering majors are therefore not using any formalized concept of number. They are using an informal one.