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Is it wrong to tell children that 1/0 = NaN is incorrect, and should be ∞?

I remember that dividing by zero is frowned upon, because it is said that there is no real answer. With the concept of limits, going from the negative direction to zero would give $-\infty$, and going towards zero from the positive direction would give $+\infty$. This is partially the reason that $\frac x0 = $ undefined, even with using limits.

But could $\frac x0$ be equal to $\pm\infty$? I suspect this is not the case, so please explain why this is incorrect.

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How does one make a frown sign? –  André Nicolas Nov 14 '12 at 6:33
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I'm sure this is a duplicate, but I'm not finding it. –  Cameron Buie Nov 14 '12 at 6:34
    
ಠ_ಠ –  The Chaz 2.0 Nov 14 '12 at 6:35
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What could $x/0=\pm\infty$ possibly mean? –  Gerry Myerson Nov 14 '12 at 6:36
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@AndréNicolas: Like so, perhaps? ☹ (U+2639 WHITE FROWNING FACE). –  Harald Hanche-Olsen Nov 14 '12 at 8:46
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marked as duplicate by The Chaz 2.0, TMM, Noah Snyder, Douglas S. Stones, Norbert Nov 14 '12 at 12:02

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3 Answers

I do not entirely agree with the answers posted so far.

First, a comment on something in the question: One should not write $\dfrac x0 =\text{undefined}$. Rather, one should say that the value of the expression $\dfrac x0$ is undefined. This is not that "is" of equality; this is the "is" of predication.

In some contexts, it makes sense to put a single $\infty$ at both ends of the real line $\mathbb R$, so that $\mathbb R \cup \{\infty\}$ is topologically a circle. That makes sense when dealing with either rational functions or trigonometric functions. It makes rational functions defined and continuous everywhere on $\mathbb R\cup\{\infty\}$ and it makes trigonometric functions defined and continuous everywhere on $\mathbb R$.

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Defining dividing by zero is much more trouble than it's worth. For example, we would expect that $\frac{x}{0}\cdot 0=x$, the same way as $\frac{x}{y}\cdot y=x$ for all other $y$, but upon dividing by zero we forget all about the $x$, thus $\pm\infty=\pm\infty\Rightarrow\frac{1}{0}=\frac{2}{0}\Rightarrow 1=2$, no good.

Sometimes, however, one might work with $\mathbb{C}\cup\{\infty\}$ (see http://en.wikipedia.org/wiki/Riemann_sphere), but then you give up some very useful niceties of working with a field.

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How does $\pm \infty = \pm \infty$? –  Thomas Nov 14 '12 at 9:12
    
Sure, you could remove reflexivity of the equality sign, but then it's not even an equivalence relation. Like I said, if you just remove some properties you probably really want, you can define division by zero. –  Max Morin Nov 14 '12 at 9:16
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Nothing can equal infinity since its not a real number.

You should read this http://answers.yahoo.com/question/index?qid=20090114213310AAsN4jB

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A link to answers.yahoo?? It just keeps getting better... –  The Chaz 2.0 Nov 14 '12 at 6:41
    
The guy basically gave a lecture on this, I just had to... :P –  Dreamer78692 Nov 14 '12 at 6:43
    
Kinda ironic saying nothing can equal to infinity when сhееsеr1 said that we know that already. Nice answer though. –  user824294 Nov 14 '12 at 6:48
    
\infty is a notation and should be understood in the context it is used. –  Gautam Shenoy Nov 14 '12 at 7:57
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