Does division by zero imply a different type of division in this situation

Firstly, I would like to apologize if this is somehow addressed in one of the many many explanations about why division by 0 is impossible that appear on this site. I have not yet found one that explains this type of situation, at least explicitly.

Let's say that you're trying to ride a roller coaster as many times as you can. You have X dollars, and it costs Y dollars for each ride. In general, you can find the number of times you can ride by X/Y. However, if the cost of the roller coaster is free, Y=0, and you would create a divide by zero situation if you attempted to apply the same formula.

Clearly X/0 is undefined, but for f(X,Y), f(X,0) will give infinity, instead of the undefined X/0. It is intuitive that when Y is 0, not only approaching 0, you can go on an unbounded number of rides, but how would this be mathematically shown if you can't use division to show it and a limit doesn't show that f(X,0) is not undefined?

• infinity is not defined as a real number Apr 3 '19 at 19:13
• @J.W.Tanner Though, it is defined as something right? Wouldn't the phrase "undefined" imply that something has no definition Apr 3 '19 at 19:20
• What does the word "kaujdulk" mean? I mean, it's a WORD, because it's made of up letters. ... Well...if "sequence of letters" is your definition of "word", then so be it. But what's the definition of that word? What does it mean? Well...nothing, because it's undefined. The same goes for $X/0$ -- it's a sequence of mathematical characters, but it's not defined to have any value. You, of course, can define it to mean something, but when you do, (1) lots of other things you think are true will start to fail, and (2) no one else is likely to adopt your definition. So it's not a wise choice. Apr 3 '19 at 20:00

If we "define" $$\frac{1}{0}=\infty$$ , we also have to "define "$$\frac{2}{0}=\infty$$ With the usual properties in the real numbers we would get $$1=0\cdot \infty=2$$ which is clearly a contradiction. To avoid this, the only possibility is to forbid division by $$0$$.

• Appreciate the response, but my question uses the undefined nature of dividing by 0 as a premise, I'm trying to figure out how you would mathematically prove that when y is 0, it does not produce X/0, which is undefined, as you have also stated, instead it produces an unbounded or infinite result. Intuition explains this, but I'm just having trouble trying to see how to mathematically justify it. Apr 3 '19 at 23:34
• @Phonzi Do you mean $$\lim_{y\rightarrow 0} \frac{x}{y}=\infty$$ when $x> 0$ ? This can be shown by the usual $\epsilon-\delta$ calcuation. That the limit exist does not mean that the expression is meaningful at the point which we approach. And to be rigirous , "$=\infty$" is a sloppy formulation meaning that we can get arbitary large values by approaching to $0$ Apr 4 '19 at 7:05

In general, you can find the number of times you can ride by $$X/Y$$.

Right there is your mistake; that sentence should read "When $$Y$$ is nonzero, you can find the number of times you can ride by X/Y; when $$Y$$ is zero, the number of times you can ride is infinite." (or better, "is unbounded.")

• So, to be clear, are you saying that the actual premise of the situation would define the value when Y=0? Apr 3 '19 at 19:16
• No. I'm saying that your logic takes the form "in general <blah blah>, therefore <blah blah>" and that the premise in that logic is false: the formula you claim is general is not in fact general, as it does not work in the case $Y = 0$. The expression $X/Y$ (which is shorthand for $XY^{-1}$) is undefined when $Y = 0$, because $0$ has no multiplicative inverse. Apr 3 '19 at 19:57
• This is simply a miscommunication then, I mean "In general" to refer to the cases other than the one I am referring to in the question. I can edit the question to make this more clear, but the gist of what I'm trying to ask is why for f(X,Y), f(X,0) will give infinity, instead of the undefined X/0. It is intuitive that when Y is 0, not just approaching 0, you can go on an unbounded number of rides, but how would this be mathematically shown if you can't use division to show it and a limit doesn't show that f(X,0) is not undefined? Apr 3 '19 at 23:17
• I don't think I can make sense of your question, so I'm going to abandon my attempt to answer. Apr 4 '19 at 3:23