[Rant] Circular, contradicting and fallacious logic

[robly18]

This is a bit of a rant. Here's the thing. The other day, I was in a minecraft server I know. Basically it's a redstone computers server. Anyway, some stuff happened that made me slightly angry.
So I get in the server, do my stuff, and for some reason I remember to mention "0/0 is any number you want it to be", to which I was responded "no, it's undefined".
Then followed a long, 2 and a half hour argument about this. And I don't know, maybe my arguments were fallacious, but it went a bit like this:

Me: If x = a/b then bx = a. Replace b and a by zero, and you get x = 0/0 is equivalent to 0x = 0, which holds true for any number
Person: Yes, but that if b is 0 then bx is 0x, which is null and undefined
Me: I can prove that wrong.
Person: Shut up, 0/0 is undefined
Me: Is x -x undefined?
Person: No, x - x is zero.
Me: ax - x = (a-1)x right?
Person: Yes, now shut up.
Me: Then if a is 1, 1x - x = (1-1)x.Therefore, x - x = 0x. So 0x isn't undefined.
Person: Shut up, that hasn't got to do with anything.
Me: Do you admit that 0x isn't undefined though?
Person: Yes...
Me: Then ax = b holds true even if a is 0.
Person: No, because you can't divide by zero.
Me: Says who?
Person: Says everybody
Me: Source?
Person: Stephen Hawking says 0/0 is undefined, and he's considered the smartest person in the world.
Me: Source?
Person: Shut up.
Me: Well?
Person: Axioms of algebra say you can't divide by zero.
Me: I googled it, found no such thing. Found a bunch of websites with the axioms of algebra, none of them saying anything about division by zero.
Person: They're fake and incomplete.
Me: Can you lead me somewhere with the correct axioms?
Person: Shut up.
Me: How about this: an axiom is something that can't be proven wrong, correct?
Person: Yes...
Me: Therefore, there can be no proof against it
Person: Correct.
Me: Then that proof that 0/0 = x is equivalent to 0x = 0 proves it wrong.
Person: No, because the axiom says you can't do that.
Me: So let me get this straight: the axiom holds true because... It holds true?
Person: Yes, now shut up.

Note: Person stands for a bunch of other people, some more accepting than other, one of them even accepting my conclusion, while some other simply used circular logic, lacked proper arguments, and called me a troll. They also claimed I was wrong because I don't have a higher education than them. In fact, I asked them if not having taken calculus made my logic any less valid. To which they responded yes.

It went on for a while. Eventually an admin joined. And, well, it was weird. First he told me 0/0 was indeterminate, which is correct, because indeterminate means any number will do, while undefined means no number fits the bill. Then I said, yes, my point was proven.
Then, out of the blue, he told me my arguments were contradicting, while I made none while he was online and he himself had proved my conclusion.


I don't know, maybe I am just dumb to the point of missing my own arguments. But it seems like I made a pretty solid case. What do you think? Do you see any flaws in my logic?

[19683]

You are absolutely, completely, 100% CORRECT!
Every statement you used was:
1. Consistent with every other statement you used
2. Could be derived from basic axioms of algebra
3. 100% consistent with calculus!

A slightly more advanced argument would be:
x/x=1
limit as x tends to zero of x/x is 1

but limit as y tends to zero of x/(x+y) is 1
and limit as x tends to infinity of x/(x+y) is 0
so 0/0=any number between 0 and 1

Furthermore, a similar argument can be made for (x+y)/x, x/(x-y), and (x-y)/x to extend the solution to ALL numbers.

On a final note, I reserve the right to be wrong. Zeros, limits, and infinities are dangerous subjects.

I remember a time when my friends and I were presented with a complex algebraic proof that 0.9 repeating does not equal 1. All of my friends knew it was wrong, but couldn't disprove it using algebra. All it took me was a simple proof using calculus to prove that 0.9 repeating did indeed equal 1.

[A Random Player]

so 0/0=any number between 0 and 1

Limit as x->0 (3x)/x = 3.

[19683]

so 0/0=any number between 0 and 1

Limit as x->0 (3x)/x = 3.

Limit as x->0 (ax)/x = a
So 0/0=any number

Same conclusion, different method!

[testtubegames]

Whoa whoa whoa, 19683, hold the train! You are disagreeing about math with a Minecraft server admin. They channel the gods of math directly.

I mean, err, they don't. It seems there was some confusion between undefined and indeterminate. Which is understandable, since that whole terrain is a little unusual. But you are indeed correct about 0/0 being indeterminate.

As an added note, math is funky with infinities and such (of course). So that it really depends *where* you got that expression from. Maybe 0/0 came from 5x/x... or maybe it came from sin(x)/x^2, or whatever. In any case, 0/0 itself is still 'indeterminate', but depending on the case, we may be able to use limits to help us discuss the answer. That won't mean that 0/0 = 5 -- it is still indeterminate -- but it will mean that the limit of the equation 5x/x as x--->0 is 5. Slightly different questions, slightly different answers. If that makes sense.

By the way, if you get into this argument again, this site gives a very simple, clear, and complete description of what is going on. Not that the people on the minecraft servers were necessarily interested in figuring it all out...

[robly18]

Hooray, I'm not stupid.

I think I know the problem though.
Like I said, that was a redstone computers server. They were used to computers, and that was probably where they derived their mathemathics.
In computers, division by zero gives you Not a Number. Which is probably why they thought it was impossible. It's probably also why they thought you can't multiply by zero, because, and I quote:
"When you multiply a variable by zero, you are just dropping the variable, so it becomes not a number"
I don't know what sense this makes, but whatever.
Anyway, thank you for your support. This way if I ever get in such argument with someone I have a source to show them my point if they refuse to even read what I'm typing!

[exfret]

If I were you, I wouldn't be arguing in a minecraft server with people who are constantly telling you to shut up. I think the best thing to do would either be to live with the fact that someone is wrong (you or them, because you can't always be right), or to leave the server. I wouldn't have carried it on so long, because it would have probably annoyed them tremendously. Also, I would avoid people that impolite. Another thing, I like to think of the fact that 0/0 can equal any number as an OR situation, not an AND situation. Basically, when you get more than one value for an equation, like in the equation x^2-x=0, x is only one of those values, it's not all of them at once, you just don't know which one. For example, if you have the equation x=0, and you multiply it by (x-1), then the extraneous solution of 1 pops out because x may equal 1, but it turns out not to. Similarly, in x=0/0, x can still equal a specific number, it just has to be a complex number, which does you no good knowing that at all, so we just call it indeterminate. Just like Andy said, there's ways to narrow down the value of x so you know what it is.

[19683]

http://www.youtube.com/watch?v=BRRolKTlF6Q

[exfret]

http://www.youtube.com/watch?v=BRRolKTlF6Q

By the way, I bookmarked that "Problems with numberphile when they encounter 0." They didn't even explain their wrong arguments with right evidence used wrongly. They just used wrong evidence outright. I like numberphile in that they know lots of good facts and can explain them better than many, but they're not geniuses when it comes to mathematics.

[robly18]

I agree with exfret. Their arguments were not the most failproof. For instance, the 1/x argument falls short when you change it into 1/x^2 because then it always goes up to positives. They could have mentioned something about complex numbers coming in from the negatives instead.