Infinity may not be a natural, real, or complex number, but there are transfinite numbers, and when we're talking about "infinity" in the above discussion we refer to the smallest of these transfinite numbers, so yes, infinity is a number.

"Would it be correct to assume that 0/0 does not necessarily equal the limit as x approaches 0 of x/x? "

"Yes."

Then we have a minor problem, but it should be Undefined. Any time you divide by 0 without limits that I can think of, the answer is undefined.

Hold up, TheClark was right actually, that would be indeterminate... I'm fairly sure.

I haven't done mid-high-level math in ages though, so don't give my opinion too much weight.

I am not a mathematician, but I have read that the answer to this question is undefined/indeterminate.
As a layman my intuition would suggest that if a number is divided by 0 then it is not divided (by definition). Thus n/0 = n. I am also think that I am probably wrong. Maybe someone who knows more than me can explain the error in my logic?

No such thing as undefined. It's indeterminate. And division by zero is always infinity unless it is 0/0. When I get home, I'll pull out my applied mathematics reference and my handbook of chemistry and physics to provide proof if you like.

YOU CAN NOT DIVIDE BY 0!

It's undefined.

If, however, you were to divide by a function that aproaches 0 at some point, then you may calculate the limit of that expression aproaching that point.

That's why the limit of x/x when x -> 0 is 1. For all intents and porpouses, you can redifine that function (x/x) as simply 1.

Now, sen(x)/x can not be redefined that way, but still the limit of sen(x)/x as x->0 is 1. So usually you would redefine sen(x)/x as the function whose value is sen(x)/x for any x != 0, and 1 for x = 0.

aoe3rules, 0/0 and infinity/infinity are indeterminate. I have a degree in applied mathematics, but this was taught in 9th grade algebra and reinforced throughout Alg 2 and Calculus in high school. The reason is that 1*0=0 and 2*0=0 and so one, therefore 0/0 could be any possible number negative or positive. Infinity works the same way. These are the two absolutes in mathematics and therefore their division is indeterminate.

Spyman, n/0 is never n. It is infinity (except in this special ciircumstance). n/1 is always n, but not 0. Otherwise, n*0 would also be n and we no that isn't true.

Anytime you are dealing with the absolutes, just switch the equation around. 0/0=y switches to y*0=0. you don't have enough information to determine y, so it is indeterminate.

Draugnar, funny what you just said about n*0 <> n. That thought occurred to me about a minute after I posted that last message.
But this leads me to another thought (totally ignorant I am sure) but if the answer is indeterminate (which I thought meant the same thing as undefined. Then if n/0 = x (where x is indeterminate) therefore x * 0 = n. After all if 12/3 = 4 therefore 4 * 3 = 12. (please excuse my ignorance).

n/0 is not indeterminate, only 0/0. 1/0 is infinity. Therefore 0*infinity COULD BE 1. It also could be 2 or 3 or 4 or... x*0=0 always, which is why 0/0 is indeterminate because any value picked for the result can be translated into a valid equation.

Indeterminate (or indeterminable) is the same as undefinde nowadays, but I was taught that undefined just meant it hadn't been fuy defined to determine the value (i.e. a key component was missing to resolve the equation) where as indeterminate could NOT be clearly determined. Nowadays they seem to get lumped together.

Draugnar, you examples are great. Indeterminate is not the same as undefined. You examples really show this.

Sorry. 0 / 0 is not the same as

lim x / x
x->0

as you are only finding the result when a number very very close to zero divides itself. Since it's not zero there is no doubt the answer is 1. This logic is absurd and only shows a lack of real understanding of the concept of limits.

∞/∞ is not defined. It's because ∞ itself is undefined, unlike 0 which is as well defined as any other finite number.
Let us have a very very big number which we don't know. Let us have another very very big number which we don't know. Since we don't know whether these two very big numbers are even same we can't make sure when the divide you get one. if you put

lim x/x
x->∞
then again you are just letting a very large number divide itself. NOT ∞/∞.

This shows that your understanding of limits at infinity and the meaning of infinity with respect to limits is very limited indeed.

Wow, and I thought I was a geek.

Hmm, OK Draugnar, I'll accept your definition of the difference between undefined and indeterminate. I was taught both ways, as if they were equivalent. x/0 is indeterminate, infinity is undefined.

The important thing is that in NO circumstances can you divide by 0. When you apply a possible division by 0 to an equation, impossible things start to happen - 1=2, that kind of stuff. I hear "x/0 = infinity" a lot and it drives me crazy.

As for transinfinite numbers - Read Asimov on Numbers. Fantastic treatment on what transinfinite numbers are all about.

And I'm proud of my geek creds. :-)

The x/0=∞ is strictly from an applied mathematics point of view and you are correct, jbalcorn in that regards. Infinity is more an irrational number than an undefined number, much along the lines of SQRT(-1). But I can handle calling it undefined as well. Semantics only matter to pedants in language. We both understand the concept and that is what is important.

I'll have to read that one sometime. I like Asimov's SciFi, but have never read any of his non-fiction work.

And I have a t-shirt I bought at PDC in '05 that has "GEEK." on the front, and a lengthy definition of what a geek is on the back. I wear it proudly whenever a t-shirt is appropriate.

Varieties of the Infinite
Subject: transfinite numbers
First Published In: Sep-59, The Magazine of Fantasy and Science Fiction
Collection(s):

* 1964 Adding a Dimension
* 1977 Asimov on Numbers

Ginsburg- When was the last time you kissed a girl?

@Draugner, @Jbalcorn - Hope you've both seen xkcd.com.

I love xkcd. It's right up there with Dilbert!

more importantly... why are you trying to divide zero by zero... looks/dating opportunities? or is it the other way around...

If you are trying to figure out how much power is being used you divide the energy by time. The 0/0 could happen if there was 0 energy and 0 time.

xkcd rocks.

0/0 is undefined. Infinity/Infinity is also undefined. Technically, x/0 is also undefined though with limits you can ascertain whether it's positive or negative infinity.

"0/0 is undefined, but the limit of the function y = x/x at x = 0 is 1. Now chew on that. "

Ok, I'll chew on that. 2 x 0 = 0, therefore, the limit for function y = 2x/x at x= 0 is equal to the limit for function y = x/x at x = 0. But this is not the case, the limit is 2. You can try this for anything. 0/0 must be undefined, and the limit is pretty inconsequential.

"YOU CAN NOT DIVIDE BY 0!"

On the contrary, when dealing with L'Hopital's rule, you try your best to get the expression of the form 0/0, so you can apply the rule. Given, it's nonsensical in everyday mathematics, but limit-wise it is extremely important. Same goes for ∞/∞. Given, there's no way to tell what the limit will be without an equation (as you need an equation to talk about limits at all), but with one it's very important.

As to the people saying that the lim a->0 x/a = ∞, I present the counterexample of 1/x. Simply examine the graph as it approaches 0, and you will find that it goes to ∞, but you should also note that it goes to -∞ as well. Clearly, no such limit exists. This is due to the fact that for a limit to exist, the limit approaching from the right must match the limit approaching from the left, which it does not in this case. Now, if we're talking about a function who's domain is the positive numbers, such as in a physics experiment with time, then you're correct, but don't forget to mention your domain.

So, dividing by zero may be undefined. However, there are ways to get around it when dealing with an equation by using limits. Sometimes they work, and sometimes they don't.

In my personal opinion, within the next hundred years or so, we'll see a bright mathematician who will be able to divide by zero, though not using number systems we know about today. After all, when they couldn't solve 4-5, they invented negative numbers. When they couldn't solve 4/5, they invented rational numbers. When they couldn't solve sqrt(2), they invented irrational numbers. And when they couldn't solve sqrt(-2), they invented imaginary numbers. Who knows what they'll come up with next?

Imaginay numbers are the shit...
the next thing is mythological numbers ;]

Imaginary*

even irrational numbers don't exist in our universe. we need to stop inventing these abstractions

Mmmmmm. Interesting. I would have to say undefined.

(scratches imaginary goatee thoughtfully)

@Frelock - When the graph of some function 'func' switches from +∞ to -∞ it's called a discontinuity. If instead you plot the graph of sqrt (sqr (func)) you lose the discontinuity.

This is an abstraction that doesn't help the argume... Oh... Hang on a sec... We have Negative numbers, Rational numbers, Irrational numbers and Imaginary numbers. What we need are Abstract numbers. There. Sorted. The rest I leave as an exercise for the student.