x/0
 

Someone brought this up on another forum I go to.

The limit of 1/x as x approaches 0 equals infinity. But 1/0 is undefined, right? Can 1/0 be considered infinity?

What about the fact that mathematic operations should be reversible? If I have 1/2 and multiply it by two, I now have 1. If I have 1/0 and multiply it by 0... what do I have? And what did I just multiply by zero?

Do you usually learn more about this kind of thing in upper-level math classes? (I'm still trying to figure out what they fit into 4 levels of calculus)

1/0 is undefined. It is undefined because no one has yet thought up an agreeable algebraic way to describe what happens when you divide by zero in any kind of meaningful way. It just doesn't work. Even if you multiply zero by infinity you'll never get a product other than zero. If someone figures out a useful way to define it it will be thusly so. Until then it's just a big no-no.

Limits are useful because they describe what happens near a certain point - what actually happens at that point isn't very important as far as the limit is concerned. The limit of 1/x as x approaches 0 is infinity. 1/0 is undefined. This is a crucial distinction to make. What's kind of more interesting is that you can take the limit of, say x*e^(-x) as x approaches infinity. Evaluating it as you would a continuous function gives you infinity/infinity. How you explicitly evaluate a limit like this is a calc 1 kind of thing. In this case since the e^(-x) term increases in magnitude much faster than the x term the actual limit is zero.

Mathematical operations are reversible. I think what that really means is that if you can get to a from b you should be able to get back to a from b. However, dividing by zero gives you a number that is undefined, which is basically another way of saying "We don't know what the fuck to do with that so let's make an exception out of it". Algebra is good for a great many things, division by zero is not one of them. Calculus, in my experience does little to shed light on division by zero, but they give you other things to think about.

I may be wrong though, i'm no math teacher.

Careful when you talk about mathematical operations, because they don't need to always be reversible, or "invertible". But in this case you got it right, under the <em>field</em> of real numbers, multiplication is invertible <b>if</b> you exclude 0, for the reasons you already stated.

Note however that sometimes mathematicians will say things like 1/0 = infinity, but this is really just shorthand for the limit. I think a more interesting question is the value of 0^0. Calculus treats this as undefined. Not everyone agrees though. Some discrete mathematicians and logicians consider 0^0 = 1, and not just formally to make sums look pretty, but meaningfully. Fun topic.

Really? I always assumed that 0^0 was definitely 1. My calculator gives me a domain error, which rules out division by zero and infinity... hmm... The log₁₀(0) gives the same error. I guess "domain error" really means "undefined", since 10 raised to what equals zero?

Zero seems to cause a lot of problems. Who was it that invented it? The Egyptians?

Zero is old, seemingly <a href="http://en.wikipedia.org/wiki/0_%28number%29">invented</a> independently in both the Old and New Worlds, and was brought to the West through translations by the Islamic world.

Ah yes, indeterminate forms. I don't know how far you are in math, but eventually you'll get to method to deal with 0/0 and 0^0 and all sorts of other fun things such as that. But only for functions using limits. (It's called L'Hôpital's Rule). In short, math goes crazy when you try to actually divide real numbers by zero. As has already been mentioned, it's an exception. As for zero causing problems... Yes it does. But what where would we be without it? It's one of the most important concepts in pre-alegibraic math.

What exactly is the rule part of L'Hôpital's Rule? All I've ever done is factor polynomials so that I'm not dividing by zero...

"When determining the limit of a quotient f(x)/g(x) when both the numerator and denominator approach 0 or the denominator approaches infinity, <a href="http://en.wikipedia.org/wiki/L%27Hopital%27s_rule"> l'Hôpital's rule</a> states that differentiation of both the numerator and denominator does not change the limit. This differentiation, however, often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be determined more easily."

-From Wikipedia.

One of the things you misunderstand is that division is not really an operation.

When we say "t = x/y", what we're really saying is "t is equal to x multiplied by the multiplicative inverse of y, if such a thing exists." Sometimes, such a thing doesn't exist, in which case, this equation doesn't apply.

Finding a multiplicative inverse is not an "operation," in the sense that you're talking about...

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Another thing you misunderstand is that "infinity" is not a number. It's no more a number than "really big, " or "somewhere in between" are numbers. It's a property, or a description. The notion of it being a number doesn't really make any sense. For instance, if you were to define infinity as "a number greater than all other numbers," (a definition I think most lay-people can agree upon) then what do you make of the statement?
I think even lay people can agree that for all real numbers (where real can be taken in both the literary and mathemtatical sense) x < x + 1. But then, infinity isn't the largest number, is it? Inifinity plus one is even larger and that's a contradiction, isn't it? If nothing else, the truth (math) is consistent. Even if you wanted to give infinity special properties like:
Then what do you make of this?
Another clear contradiction, this time not even including infinity! No matter how much you try to hack up such a "number system," it becomes quickly apparent that infinity makes no sense as a number. You simply accept that there are an infinite number of numbers, they are well ordered, and they are unbounded.

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In answer to your last question, yes, you do learn this stuff in "upper-level math classes." However, they're really not that upper level. Unless you're in some "applied" university course, like engineering or physics, they won't bother teaching you this stuff 'cause you're only using math as a tool for making sense of the world. However, if you're studying mathematics in university, they will quickly teach you the fundamentals of logic and reasoning, because that's what mathematics is really about...

Slavakion, to give you an example of when L'Hôpital's rule is applied, take ^{(2^x - 1)}/_x when x is zero. That is a zero over zero case, but they are not necessarily the "same" zero, as it was explained to me in geometry. Graph that and look what happens when x is near zero, but evaluate it at zero.

Another example of when dealing with zeros, have ^{(2^x - 8)}/_{(x - 3)}. Evaluate it at 3, no answer, but it looks like there should be an answer as you get closer to it. You'll find that it should be approaching 8*Ln(2).

[And yes, I was that kid you threw pencils at when my hand went up with a question everyone knew had nothing to do with geometry, and more of that math the teacher called "calculus."]

I know when and how to use the rule, I just didn't know what the rule actually was. Although, the last time I used the rule was way back when a derivative was a limit of (x+h)-x/h or whatever. The definition of the derivative.

And hey, I got excited whenever calc came up in my Chemistry class. :)

I am not that well verse in math, but I always assumed that 0 was a concept much like infinity.

infinity - infinity = infinity in my mind.