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Hazarth22242y0 is literally not mathematically considered an integer. That's why positives integers start at 1 and negative at 1. Zero is a nonsense value we use to describe "nothingness" but it doesn't follow several other rules that integers and real numbers do. For example any positive number + any other positive number must give a result thats bigger than both of its parts. But already 0+0 fails to show this property. Its also why you are not allowed to divide by zero and why division by zero is considered undefined, its different from a division by an integer because its not one.

eeee33172y@M1sf3t there is one loop hole in that teacher's excellent explanation: 1÷0 tends to ∞, but he states that 2÷0 also tends to ∞, but he fails to mention that it does twice as fast, so we could write that as 2×∞. Not that that makes sense, but his series of fractions, 2÷n always is twice as big as 1÷n. With that reasoning, 2÷0 also should have twice the value of 1÷0, so in his reasoning: 2÷0 = 2×∞. That leads to the conclusion that 2×1 = 2, which is correct. His proof of 1÷0 ≠ ∞ is therefore flawed.
However, one could argue that 2×∞ cannot be defined, because ∞ already is the largest value, so large it cannot be measured or counted, but my mathematical knowledge is too little to truly know if mathematicians can or cannot define multiples of infinity.
His next proof is better, though, where he shows the fraction series do not converge from both directions on the xaxis. It too, however, fails to prove that the actual value is nowhere on the yaxis. 
eeee33172yThe teacher's first proof is the most intuitive, I think, and therefore I think it's the best:
If division is repeated subtraction until 0 is reached, then 1–0–0–0–0–0–... (repeated infinitely), but after an infinite number of subtractions, the value is still 1, so ∞ is not the answer to that question.
Now, to the ranter's question: 0/0=1? That cannot be defined. You could subtract 0 from 0 one time, zero times, ∞ times, –∞ times or any number in between. Therefore, there is no clear answer: any number succeeds to answer the equation correctly, but by definition of an equation that cannot be the answer. It is therefore undefinable and therefore mathematicians in general say it's undefined. 
Hazarth22242y@FrodoSwaggins is that what its called? Yeah, you're right. I don't know these things on english so thanks for pointing it out!
Point being 0 has its own rules because its specifically defined as being different from normal numbers and thats why 0/0 isn't 1 nor 0 but its undefined. 
@FrodoSwaggins Zero is definitely an integer but nobody can agree whether it's part of the natural number set. That's why there are ℕ, ℕ⁰, ℕ⁺ and probably some other notations

@FrodoSwaggins ℕ⁰ can also be ℕ ∪ {0}. That's how all the schools I went to interpreted it

Hazarth22242y@FrodoSwaggins my point is that any number x / y = z has to satisfy z * y = x.
10 / 5 = 2 and 2 * 5 = 10 for example
But 10 / 0 = z... What z * 0 = 10?.. no such z exists...
I don't know what this group of numbers are called. But I remember reading on this and the historically 0 used to not exist. Its not even in the roman numerals for example. Having a value of "not having a value" wasn't considered necessary for a long time and 0 is comparatively a new concept and quite arbitrarily defined with It's own rules and proofs. Expecting 0 to behave like a known number is always a mistake, because it might, or it might not. Thus saying "every number divided by itself is 1 thus 0 / 0 is 1" is simply nonsense 
@Hazarth you're maybe even close: in a field the zero has to be excluded from the multiplicative group (otherwise the group axioms can only be fulfilled for the trivial field with one element or 1 = 0). That's why for mathematicians all these discussions about any division with zero are just pointless, because the operation is simply not permitted.

@FrodoSwaggins you mean the integer numbers? Because whithout the negative numbers you have no inverse for addition.

@FrodoSwaggins Beware of noncommutative rings. Not sure you can show a * 0 = 0 for all a there, because you have to consider possible left an right neutral elements.

Hazarth22242y@FrodoSwaggins for all the rage and shaking you say this causes you, I think you still agree with my main point that dividing by zero is simply not reasonable by definition. And yes I didn't study maths extensively, Im trying to put together thoughts about something I read in a book once because I found it interesting and fun. And essentially the only thing I can remember from it is the conclusion that 0 is simply just a symbol and shouldn't be considered as a normal number. Dunno how you got "0 = 10" out of "z*0 = 10" that I wrote. Perhaps it was a typo on your side. The point was there is no such z that you could multiply by 0 that would lead to 10. Because no matter how many times you add 0 to 0, its never 10, but also any other number other than 0. Meaning anything multiplied by 0 is 0, which is true (tho I agree that my proof isn't true proof in maths, Its just intuition)

It's maths: you can make 0/0 whatever you want, provided it's consistent and you define your terms.
Seriously, it's just axioms. 
@Hazarth I read your argument similarly: with the test trying to show that division by zero does not give you a number. However FrodoSwaggins is right that in the crucial step you multiplied both sides with zero, which is not a valid logically equivalent transformation because from any even false assumption like 2 = 3 you will always produce a true statement 0 = 0.
(And you reduced a fraction z * 0 / 0 = z, which assumes the stupidity this whole thread started with... If you argue with the tests then clearly 0 / 0 can be anything, even infinity!) 
Hazarth22242y@FrodoSwaggins
1) I already agreed that 0 is an integer in a response to someone else. Its a language thing, I don't know these words in correct context and I admitted to it
2) take a chill pill 
Hazarth22242y@FrodoSwaggins oh wew, you're right, it wasn't someone else I responded to, it was *you* ooh ok. So I guess you just want to argue to feel better about yourself. Glad to make your day!

Hazarth22242y@FrodoSwaggins ye, exactly. That's why I responded nicely until now. Now you're getting mad and its frankly getting annoying. Have a good day tho

@FrodoSwaggins From maths, man! Yeah, in any normal system of numbers, this is undefined, and with good reason.
It's perfectly well defined in e.g. IEEE floating point semantics. 0/0 is NaN. Of course IEEE floating point arithmetic is not regular arithmetic: addition isn't even associative.
My point is, the rules depend on the often implicit context of the expression, and in many contexts, these rules aren't the same as those of ordinary arithmetic. 
@FrodoSwaggins Oh, it's certainly useful, even if that utility is to mark, literally, that this is not a number. So it's a handy invalidvalue designator, and allows the propagation of errors without traps.
From an algebraic point of view, it has consistent semantics (with the debatable exception of NaN == NaN being false). NaN op Anything is NaN, for any arithmetic operation. 
@FrodoSwaggins With integers, it's usually pretty clear: if you have a zero denominator, it's for a logical reason. For floating point though, it might just be underflow.
NaN is useful. But I have to reiterate: the rightness or wrongness in giving 0/0 a value comes solely from utility. For IEEE floating point, it's really useful, in that it allows exceptionfree computation and error propagation. For integer arithmetic, it generally isn't. 
theabbie425089dYes, 0/0 = 1, but 0/0 also equals to 2, 3, 4, .... and literally every number in the number system. 1 is one of the answers.
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