# Infinite or unknown?

If you have $0$ clients on Monday, and $5$ clients on Tuesday, how many times have the number of clients you had grown from Monday to Tuesday?

$A$ - Infinite times

$B$ - Unknown

$C$ - Undefined

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I'd go with "c - undefined". –  mrf May 3 '12 at 21:14

To ask "how many times have the number of clients you had grown from Monday to Tuesday" is the same to ask what is the value of: $$\dfrac{\mathrm{Clients}(\mathrm{Tuesday})}{\mathrm{Clients}(\mathrm{Monday})}=\frac50$$

In the context of the real numbers, or natural numbers if you prefer, this is undefined. We cannot divide by the actual number zero.

Note that it is common to say "infinity" because $\lim\limits_{n\to0}\frac5n=\infty$. However this simply tells us that the ratio is larger than any other number, it is not an actual number or ratio per se.

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@Marvis: Of course! :-) –  Asaf Karagila May 3 '12 at 21:23
This would evaluate to "infinity" for any nonzero number of clients on tuesday. Surely, a rise to 5 clients should deserve to be denoted by a larger increase than, say, 2 clients. Since I don't see a way to express this, maybe "b - unknown" is the best answer here? –  doppelfish May 3 '12 at 21:23
@doppelfish: No, this is undefined because it is not a defined operation. Defined operations are exactly operations which are not dependent of the choice of representatives ($\frac12n=\frac24n$, for example). Evaluating a division by zero is simply undefined in this context. In a broader context it might be possible and maybe even reasonable to define something like that, however this is not the sort of context that one would ask about here. Usually when you reach that point, you can do these things yourself just fine. –  Asaf Karagila May 3 '12 at 21:26
Let $x = 0$ and $y=5$. If $y=kx$, then what is $k$?
Solving this equation would involve division by zero, so the answer is undefined, i.e. there is no such $k$.