I really despair

Saturday 4 December 2004 at 8:50 pm | In Articles | 4 Comments

I 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, \displaystyle \lim_{x \to 0^+}x=0. 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 \displaystyle \lim_{x \to 0}f(x)=l if and only if \displaystyle \lim_{x \to 0^+}f(x)=\lim_{x \to 0^-}{f(x)=l (where f is a real-valued function defined on a neighbourhood of 0).

Personally, I like using x \searrow 0 and x \nearrow 0 rather than x \to 0^+ and x \to 0^- respectively, because they illustrate the approach from above or below.


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  1. 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 #

  2. 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 meaning

    There are 2 problems with saying \frac{1}{0}=\infty:
    1. Division by zero is not allowed. It isn’t possible to define it without causing contradictions. For example, \frac{a}{c}=\frac{b}{c} \implies a=b. So if \frac{1}{0}=\infty=\frac{2}{0} 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 \frac{1}{0} doesn’t exist and \infty is an idea not a number, what on earth does ‘=’ mean in \frac{1}{0}=\infty? 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 error as they should do.

    Comment by Steve — Friday 10 December 2004 3:22 pm #

  3. 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 #

  4. Re: your last comment…

    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 #

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