@PUBLIQclopAccountant  

 
If you extend the reals by adding a single point at infinity it doesn’t make a difference.

Is x approaching 0 from the left or the right?

I like this comment section. It has taught me things.

@Aponty
 
Oh, well, I enjoy it. And you know, I kind of feel like with a username like this I ought to stand up when it gets mentioned. :^)
 
Man, I remember this one thread on /azu/ about 0.99999….

@mathprofbrony
 
Yes, I know! I’ve already taken Real Analysis! Recently!
 
In my proof that you shot down I originally had a variable representing a real number but copy-pasted the infinity symbol from the post I was replying to into that spot as a last-minute derp that I didn’t put any thought into.
 
There’s not much in this thread that’s worth putting thought into. We’re replying to an image containing incorrect math alongside the elation of a high school calculus student who has read too many jokes about dividing by zero and has just discovered that something “similar” (for generous application of the word “similar”) can be done.

@Aponty
 
There is something called the “extended real number system,” where in addition to all the usual real numbers we include +infinity and -infinity. This is the system we think of when we make the usual kind of statement about a value being infinite. We have to create some special rules for those symbols, though, and it does not automatically have all the properties of real numbers that we might like.
 
There is a special kind of curve called an elliptic curve. Points on elliptic curves can be added and subtracted like numbers. The point at infinity can be included in an elliptic curve and can be added and subtracted like the rest. However, I’m not familiar with any structure on the curve that gives it a multiplicative inverse.
 
Another way to think about adding infinity to the real line is to embed the reals into a circle via stereographic projection, and make both + and - infinity the same point. However, this again has a group structure (addition and subtraction), not a field structure (division).
 
For everyone: if you’re interested in questions like these, and learning all the precise definitions so that you’re able to say exactly what something like division by zero means, a college course called “real analysis” will explain these things. You should take it after the usual calculus courses.
 
Back when calculus was being invented by Newton and Leibniz, people were making statements like these without rigor – how is an infinite set of infinitesimals supposed to add up to a line, anyway? There was quite a bit of fuss over some of the claims – and sometimes, people would say something that ended up wrong. It wasn’t until some proper limit formulations that people were able to be confident in the statements they were making.
 
And then there is the whole idea of the rigorous infinite, which we didn’t really start getting a handle on until the 20th century!

@Ventig  
Aw crap, my thing is broken now.  
Have a link then  
http://www.wolframalpha.com/input/?i=lim+1%2F|x|+as+x-%3E0

@Background Pony #99F6
 
First off, I apologize if you weren’t trolling.
 
Think of it like this. Saying “0/0” has as much meaning as saying “the square root of Joe Biden.” The operation “square root” isn’t defined on Joe Biden. Likewise, division is not defined when the denominator is 0.
 
After all, being vague for a moment and going with the “repeated subtraction” definition, zero divided by anything ought to be 0, since there’s nothing to subtract; but anything divided by 0 ought to be infinity, since 0 can be subtracted infinitely.
 
So the two loose definitions conflict. When that happens in mathematics, we carefully work out how to prevent contradictions from arising. In the case of division, we decided that division is only defined for things that are (1) numbers, and (2) nonzero.
 
(I will be technical for a moment for the curious, but this paragraph doesn’t matter to the rest. Imagine we don’t know about any of the mathematical operations yet. We define a “group” to be a set of symbols with addition and subtraction rules, and a zero element which adds and subtracts nothing. We define a “ring” as a group in which you can also multiply, and there is a 1 element which doesn’t change anything when you multiply. And a “field” is a ring where all the elements except the zero element have a multiplicative inverse, i.e. for any x there is a y such that x*y = 1. For the field of real numbers, y is 1/x. But we just force in that exception for the zero element in the definition, because too many things break if you try to give zero a multiplicative inverse.)
 
If two things are getting smaller and both going toward zero, their ratio may be approaching a specific number. That’s fine. That’s a limit being used to assign a value to a ratio approaching 0/0. But you can literally get any number out of this, depending on what your starting expressions are.

@mathprofbrony  
May I ask why I was incorrect?  
It’s just that I was lead to believe that if you have nothing and divided it by nothing away, you would remain with nothing.

@mathprofbrony  
suppose we invent a new number, fnord, defined such that it is the multiplicative inverse of 1/0, much as we did with negative square roots
 
whoops, that doesn’t work, fnord doesn’t give rise to any meaningful structures
 
oh noes
 
but yes, you’re absolutely right, we can’t treat infinity as though it is a natural number

@Background Pony #99F6
 
What is this, trolling? No.
 
@Ventig
 
Best correction of Derpy’s equation.
 
@BrownieComicWriter
 
The “can be subtracted this many times” definition of division is not the greatest but let’s go with it. x/0 has no meaning unless you know what x is, and even then, as a limit, it will still depend on the direction from which you approach 0 in the denominator. 0/0 is undefined without such a context.
 
@Aponty
 
Good try, though ∞ * 0 is neither 1 nor 0. Like 0/0, it is an indeterminate form, as is ∞ - ∞, ∞/∞, 0^0, ∞^0, and many related constructions.
 
I will certainly agree with you that the inherent absurdities in attempting such operations are a good reason to deal with them only under rigorous rules.

I just like to say fnord.
 
Fnord, fnord, fnord, fnord, fnord!!

HELLO EVERYNYAN HOW ARE YOU DOING FINE SANKYOU
 
Before I continue, I point everyone to my username. Clear? Good.
 
Division by zero is not defined. A ratio of zeros is not zero, nor is it infinity. It is meaningless except as a limit that exists in a context which can cause the limit to be absolutely anything – although within such a context, the limit has a precise value, if any.
 
If I write down the mathematical notation
 
fnord(x)
 
then this has no meaning unless I define the function fnord(). If I do, then fnord(x) only has meaning if x is an object on which fnord() can act. If fnord(x) is “the number of prime factors in the integer x”, then
 
fnord(0.4)
 
is meaningless.
 
Likewise, if I write down
 
a/b
 
then this expression only has meaning if b is an invertible element, something that can be put in a denominator. Zero is not such an element.
 
Let’s now take the recent comments one by one.

@BrownieComicWriter  
for a/b = c to mean anything, it has to be true that a = b * c
 
if 1/0 = ∞, then ∞ * 0 = 1  
but ∞ * 0 = 0  
this is a contradiction
 
therefore 1/0 is not equal to ∞, QED by the simplest proof in existence, mo’fo

@BrownieComicWriter  
You are wrong on all accounts.

@BrownieComicWriter  
That’s not true because it goes towards both ∞ and -∞.

@Background Pony #311D  
If you divide three apples by one, then you get three apples.

See, that’s what I always thought.
 
If 20 / 4 = 5, that’s because 4 can be subtracted from 20 exactly 5 times before it reaches 0.
 
So x / 0 = ∞, because 0 can be subracted from x ∞ times, where x ≠ 0: i.e. no matter how many times you subract 0 from x, x will always remain..
 
The question we should be asking is, what happens when you divide zero by zero?

 
There. Fixed.

@Background Pony #99F6  
nope
 
wrong
 
0/0 doesn’t exist, but as a limit it can equal any number depending on what those zeroes are
 
also, Derpy: infinity isn’t a number.

0/0=0

Wrong, Derpy. The limit as x approaches 0 does not exist, because the limit as x approaches 0 from the right is positive infinity but the limit as x approaches 0 from the left is negative infinity.

Someone didn’t pay enough attention during their classes

@CogWeaver  
Teaching astrophysics to 5-year olds.

All’o’ yous’s gettin’s hung up o’er fancy mathematics, all while I’M o’er ‘ere wonderin’ -
 
…Why on earth is Derpy teaching limits in an elementary school? Where’s Cheerilee?