# Mathematics Weblog

## I really despair

Saturday 4 December 2004 at 8:50 pm | In Articles | 4 CommentsI came across this answer, written over a year ago, to a question about division by zero. The answer was written by a Physics instructor who seems to be rather confused.

*Division by zero is often defined as infinity. Infinity divided by zero is infinity-squared. For signs to be defined correctly, you must have a +0 and a -0. This is often accomplished through the theory of limits. The limit of x as x approaches zero from the negative side is -0. The limit of x as x approached zero from the positive side is +0. Higer (sic) level mathematics uses the concept of infinity quite often.*

Let’s gloss over the first two appalling sentences. In the following sentences I suppose he is trying to refer to the fact that one can approach 0 (or any number) from different directions. If x approaches 0 from above, that is x remains positive, then you can write, for example, . Note that the limit is written as 0 not +0 which is of course equal to 0 so there’s no point writing it! This notation allows one to show that different things can happen if x approaches 0 from above or from below and to say that if and only if (where f is a real-valued function defined on a neighbourhood of 0).

Personally, I like using and rather than and respectively, because they illustrate the approach from above or below.

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Well if you despair with the physics prof then continue to despair because my math prof said he ain’t got no problem with saying 1/0 = infinity. He’s said it multiple times probably because he’s heard some people say they don’t believe that or whatever.

So can you give a good explanation of why you think its bogus instead of just complaining?

Heck my calculator even says 1/0 = infinity.

Comment by Red Rhubarb — Friday 10 December 2004 10:08 am #

You don’t have to believe me, try John Conway, a famous methematician at http://www.mathpath.org/concepts/division.by.zero.htm. As he says:

1/0 is the number c such that 0×c = 1. But there is no such number. So 1/0 has no meaningThere are 2 problems with saying :

1. Division by zero is not allowed. It isn’t possible to define it without causing contradictions. For example, . So if it would follow that 1=2 and similarly that all real numbers are equal! Division by zero is the hidden flaw in many paradoxes; see for example Classic Fallacies

2. Since doesn’t exist and is an idea not a number, what on earth does ‘=’ mean in ? Equals is an important symbol and should only be used with objects that are of the same type and have the same value, so has no meaning here.

I have never come across a calculator that gives anything other than error when you divide 1 by 0. I asked my students today to check this and their calculators all give

erroras they should do.Comment by Steve — Friday 10 December 2004 3:22 pm #

Intuitively, I’m with you on 1/0 != infinity. It has never made sense to me bacause I cannot find any x to satisfy x*0 = 1. My prof though did have an argument for why 1/0 = infinity is valid for him. He was using infinity as not a number but an idea that you could use any numbers a and b to make an algebraic relation related to 1/0 true, therefore, 1/0 is in a sense infinity. Sorry but I don’t remember the details of his argument.

As for what does “=” mean? Mathematicians and scientists are always pulling this kind of trickery. They define some doohickey not because it makes sense but because of convention (some guy a long time ago used it in the wrong way). How many times I’ve read someone complain that the notation is ambiguous or wrong!

As for the calculator thing, I don’t know what kind your students use but mine is a Hewlett Packard calculator.

I’m beginning to suspect that this division by zero thing is more of how you define your mathematics.

Comment by Red Rhubarb — Saturday 11 December 2004 6:35 am #

Re: your last comment…

THANK YOU!

I can never remember which way it means when someone writes

x -> 0+ or x -> 0-

Comment by Ronald — Wednesday 22 December 2004 7:43 pm #