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I always felt like anything divided by zero was infinite. Like how many zeros can you fit in a number? I am also fascinated that this is a whole dimension unto itself where it never stops. Even 0/0 seems to go on forever as it never fills zero. Or is it 1 because they are equivalent? You could add infinite zeros to still have 0.

iiii94043yBecause zero breaks inversion in division.
If 6/0=0 then 0*0=6, which is false. So either multiplication by zero or division by zero would be undefined. And while multiplication by zero makes sense in common arithmetic, division does not.
You can, of course, use limit to determine the answer but you'll get infinity. 
0/a = 0
a/a = 1
a/0 = infinity
Now set a = 0 and we have 0 = 1 = infinity
That's just stupid, so /0 is "indeterminate" which is code for arithmetic has a bug so don't do this.
The real question is how you made it out of high school without learning this. 
We take 1/0 as infinity as that is what anyone would expect 1/x to be as x almost becomes 0, though it would also become infinity is we approach from negative side, It is undefined at precisely 0, but since it approaches infinity, we say it's the highest number you can possibly imagine.
There is no reason for 1/0 to be 0, it does not approach 0 at all. 
Grumm16033yDivision by 0 is undefined in normal math.
If you want to go and prove your point, just use the zero ring (https://en.wikipedia.org/wiki/...)
(or ring theory)
But then you are talking about an other level of math that is not very useful in programming. 
Hazarth90763yAs others said It's undefined. That is because 0 isn't really a normal number. I read a book on this a long time ago where it was explained, that 0 wasn't defined at all when math just started out. You could own 1 chicken, 5 chicken, but if you couldn't own none chicken. It's unintuitive to say "Hey, I have 0 chickens, wanna trade?" So initially the concept of an empty quantity wasn't even defined. Then once we figured out that It's useful to have a name for an empty quantity we started putting it into math and it turned out it doesn't work, which is why we had to define it and Its special rules and behaviours rather then just add it as another number. So now you just have to accept that that is how it is, because it just doesn't work like a traditional number.
Cont'd:... 
Hazarth90763yCont'd:
For example for Multiplication, we have the identity property that states that 1 * a = a; where a can be any real number
Indeed if you have 1 times 6 chicken, you have 6 chicken and the same works for 0... Ok cool... But this rule also states, that the inverse must exist
(1/a)*a = 1
Well ok... This works for 6... It you have a one sixth of a chicken six times, you have 1 whole Chicken... Kinda weird to think about, but its intuitively true!
Lets try that for 0 now!
If you have 1 nothingth or a chicken no times you have 1 whole Chick... Wait a second... Where did this chicken come from...
Even if you think about it in terms of a split it breaks the rules... If you split a chicken 0 times (1/a).. and then you have this unsplit Chicken 0 times (*0) you should still have an unsplit chicken... But you just said you have it 0 times... So... wait What?!
Cont'd: 
Hazarth90763yCont'd:
Ok ok.. so different approach! Lets say instead that I split this chicken into 0 sized chunks! 1/0 == Infinity right?! Well ok, you have an Infinite amount of nonchicken chunks that put together a one whole chicken somehow, but that doesn't solve the problem of having it 0 times... But you should still have one chicken if you follow the axioms of Multiplication.. (this shows you why divison by 0 is undefined. It may or may not be intuitive based on your situation entirely)
So what gives? Well simply put, 0 is a nonsense quantity, and you have to work around it yourself, because It's useful to have, but at the same time it doesn't work at all... So what do you do as a mathenatician? You definy it... You say "this will be a number, It's even an integer, but here is a set of rules of how to work with it that actually reflect the real world" 
Avyy7553yContext: The identity element(e) is an element such that for any element in the domain set, `a*e = e*a = a`
The inverse element(i) of an element is defined such that `a*i = i*a = e`
Each nonzero element has its own unique inverse(in the set R)
Not going into more theory, just convincing you by contradiction here:
Let `0/0 = k`. Then `0 = 0*k`.
Now what do you think should be the value of k here? Zero? Infinity? Something inbetween?
It can be anything, all numbers on the number line satisfy this. We say, this value is "indeterminate", it just can not be defined.
Bonus: Contrary to popular belief, 0*inf is also undefined. It is not zero, not one, not infinity; it is nothing. 
@iiii if 0*0 was 6,
That would be true for any number then 0*0 would be anything
This would basically break everything in math, right?
This is so fascinating. I think the original question in the post is very interesting. Because instead of taking the math rule as granted, it pushes logic to its limits. 
How many times does nothing go into something? An infinite number of times. Geez, this is calc 1 shit you should have learned in college. If you look at the fn 1/x, you'll see that there's a vert asymptote at x=0....

Oh shit, I figured out what 0/0 is! It is a cat! But I cannot tell if it is alive or dead...

Jedidja10203y@natesymer well kinda but an infinte number of times or a negative infinitive number of times?
the 1/x approches both depending of if you come from the negative or the positive side 
iiii94043y@rov3rand0m that's exactly why division by zero is undefined. It leads to a contradiction like that

iiii94043y@Demolishun no, 1/0 is not 0*infinity. It's 1*infinity.
If you'd define zero as limit(1/n) where n goes to infinitely large number,then 1/0 is limit(n), which equals infinity 
iiii94043yAnd, btw, limits can show why 0*infinity is undefined.
0 is lim(1/n)
Infinity is lim(m)
Then 0*infinity is lim(m/n) and here's the catch:
 if n rises faster than m, then the result is 0
 if n rises slower than m, then the answer is infinity
 if they rise equally fast then the answer is 1 
if you start doing this shit you break math and nothing makes sense anymore.
you can approach the division by zero with limit (tends towards infinity), but the actual value does not exist.
the problem you proposed is a little basic, a more interesting approach would be 0/0 = 1 
Well apparently the question is retarded.
At least a billion google searchers will now find their way to devrant.
Unfortunately they too, like myself, will be retarded.
Great answers though everyone.
I got an A on the assignment.
(Im kidding, im kidding! dont throw anything at me!) 
"(1/a)*a == 1
Well ok... This works for 6... It you have a one sixth of a chicken six times, you have 1 whole Chicken... Kinda weird to think about, but its intuitively true!"
Treat it as a mathematical object for a second.
And forgive my ignorance for moment, what I mean is this.
If (1/a)*a == 1
suppose this object carries around its identity.
Think about it
(1/a)*a doesnt really equal 1 does it?
We're actually assuming a term here. We're not showing it...
(1/a)*a == 1*1
That second term is what we're leaving off there. Its the multiplicative identity yeah?
Well in the case of a=0, its not
(1/a)*a == NaN
It would be a special case:
(1/a)*a == 1*0
@WizardOfAus
"a/0 == infinity"
No, I'm explicitly arguing that just as 0/n == 0, n/0 == 0 remainder n. 
@natesymer
"How many times does nothing go into something? An infinite number of times. "
Whats makes more sense?
That *nothing* can be divided into something infinite times, or that nothing divides exactly *zero* times into *anything*?
Last I checked zero is still zero, no matter how many times you divide it.
I actually emailed a bunch of people, and got a response from an undergrad at michigan u. he wrote out a sketch of a proof of this and accompanying code even. Idk if you want to check his work its here:
https://pastebin.com/iLB0Cz2v 
iiii94043y@Wisecrack you can define anything using any bizarre logic, but it will make no sense for algebra 🤷♂️
Related Rants
Question  is this meaningful or is this retarded?
if
2*3 = 6
2*2 = 4
2*1 = 2
2*0 = 0
2*1 = 2
then why doesnt this work?
6/3 = 2
6/2 = 3
6/1 = 6
6/0 = 0
6/1 = 6
if n/0 is forbidden and 1/n returns the inverse of n, why shouldn't zero be its own inverse?
If we're talking "0" as in an infinitely precise definition of zero, then 1/n (where n is arbitrarily close to 0), then the result is an arbitrarily large answer, close to infinite, because any floating point number beneath zero (like an infinitely precise approximation of zero) when inverted, produces a number equal to or greater than 1.
If the multiplicative identity, 1, covers the entire set of integers, then why shouldn't division by zero be the inverse of the multiplicative identity, excluding the entire set? It ONLY returns 0, while anything n*1 ONLY returns n.
This puts even the multiplicative identity in the set covered by its inverse.
Ergo, division by zero produces either 0 or infinity. When theres an infinity in an formula, it sometimes indicates theres been
some misunderstanding or the system isn't fully understood. The simpler approach here would be to say therefore the answer is
not infinity, but zero. Now 'simpler' doesn't always mean "correct", only more elegant.
But if we represent the result of a division as BOTH an integer and mantissa
component, e.x
1.234567 or 0.1234567,
i.e. a float, we can say the integer component is the quotient, and the mantissa
is the remainder.
Logically it makes sense then that division by zero is equivalent to taking the numerator, and leaving it "undistributed".
I.e. shunting it to the remainder, and leaving the quotient as zero.
If we treat this as equivalent of an inversion, we can effectively represent the quotient from denominators of n/0 as 1/n
Meaning even 1/0 has a representation, it just happens to be 0.000...
Therefore
(n * (n/0)) = 1
the multiplicative identity
because
(n* (n/0)) == (n * ( 1/n ))
People who math. Is this a yea or nay in your book?
question
division
zero
math