comment: >
  
  Zero is a placeholder and doesn’t really exist as a real number.
context: If 0/0=1, then we have:
  2*0 = 0
  (2*0)/0 = 0/0
  2*(0/0) = 0/0
  2*1=1
  2=1
  Letting 0/0 = 1 would result in a lot of contradictions with the rest of mathematics. >
  
  So why is it fine to multiply a number by 0?
  Like, it seems like the major issue is allowing the use of 0 either way for a multiplication or division. I never got the whole n*0=0 thing because it’s the only subset of multiplications that isn’t reversible, unlike a*b=c and a=c/b or b=c/a. Like 0 fundamentally beaks the symmetry between multiplication and division, but we just accept it’s okay for multiplication for reasons. >
  
  Because multiply by zero DOES give you zero. You can have 2 groups of nothing. You can’t divide something by nothing.
  I’m not going to find the actual calculus behind it, but think of numbers very close to zero in both cases.
  2*.000001=a number very close to zero.
  2/.0000001=a big number, and the more decimals you add the closer to infinity you get. It would never converge on an actual answer.
  Edit: the way I think of it is dividing by zero is kind of like multiplying by infinity. Might not be the most accurate but it’s how I’ve always thought of it. >
  
  Like how do you physically group nothing, and now how do you “have” two groups of nothing, now add those groups of nothing together, and now split that big pile back into two groups and count the things you have. Congrats: that’s multiplication AND division with 0. 
  0xGroups=0, 0/0=Groups. Groups could be any number. It breaks down with both multiplication and division. 
  It works one way but not the other because we say it does, but that’s not a why.
  2/0 makes a bit more sense, but still seem arbitrary to say it’s fine for infinitly small numbers but not for infinitely big numbers. >
  
  The reason is because defining it causes contradictions and is nonsensical. Suppose you defined a number G and said it was a number (for all intents and purposes its infinity), this division cannot be reversed via multiplication without causing contradictions, as shown multiple times in this thread. If you want to say this is a number, then many general statements about numbers can no longer be made. For example if a > b, then a + x > b + x, this is no longer true if x is G.
  Forcing x/0 to be defined as a number essentially prevents you from doing most of common algebra. So it makes way more sense to just leave x/0 undefined and let the rest of the numbers operate as per usual. >
  
  Right but a big part of why n/0 causes contradictions is because n*0 is defined. Having a multiplication that you can't reverse with division is already a contradiction. Already it means that a*b=c but c/b!=a when a or b are 0, breaking equivalence. >
  
  
  Having a multiplication that you can't reverse with division is already a contradiction.
  
  That's simply not true. A contradiction occurs when two propositions that cannot simultaneously be true are found to be true. There is no contradiction in n*0 = 0 being defined.
  
  Already it means that a*b=c but c/b!=a when a or b are 0, breaking equivalence.
  
  If you establish than n/0 is undefined, then in this instance you have one exception. If you also want to asset that n0 is undefined, you don't add anything and just create a second exception. Logically you would choose the more powerful system with n0 defined and n/0 undefined rather than the weaker one with both undefined. >
  
  You can put an infinite number of 0s into 0. 0+0+0+0+0+...=0, how can there be only one 0 in 0 when I can fit any number of 0s into 0?
  Dividing by 0 doesn't work under the standard axioms. Having 0 in the numerator doesn't change that >
  
  !delta  that… that makes sense. Thank you! >
  
  Why didn’t you just Google this instead of arguing with regards for 5 years? >
  
  Reddit would be pretty barren if we all googled things like this instead of seeking discussion here. >
  
  I disagree, Reddit has many non Google questions and even more non questions entirely. Also who cares if Reddit is barren lol >
  
  I mean, I care because I enjoy using it. I enjoy reading people's questions and the replies. Some of the questions are basic or easily searchable but not something I would have thought to ask. I don't disagree about the content of other posts and didn't think I was alluding to that. >
  
  Hey it’s cool bro I don’t truly care if you ask basic questions here or anywhere else squirt em out all over the place. I think it’s weird but that’s a personal problem >
  
  Let's do a proof by contradiction.
  We start by assuming 0/0 = 1
  Then that means 0/0 + 0/0 = 1 + 1 = 2
  But we also know x/z + y/z = (x+y)/z
  So 0/0 + 0/0 = (0+0)/0 = 0/0
  But if 0/0 = 1; and 0/0 + 0/0 = 2; and 0/0 + 0/0 = 0/0
  Then that means 2 = 1 unless our assumption that 0/0 = 1 is false. >
  
  This guy !deltas >
  
  This delta has been rejected. The length of your comment suggests that you haven't properly explained how /u/CrackaBox changed your view (comment rule 4).
  DeltaBot is able to rescan edited comments. Please edit your comment with the required explanation.
  ^Delta System Explained ^| ^Deltaboards >
  
  !delta  This is a solid mathematical proof to show that if we assume 0/0=1 that an impossible outcome occurs. Therefore the assumption of 0/0=1 is false. >
  
  Sheeiit look what you get for trying to be a nice guy >
  
  Mathematician here.
  I think that the issue at core here is understanding the division sign. Division is the opposite operation of multiplication, same as subtraction is the opposite of addition. In other words, when I write a-b, I essentially mean: “it’s a number x such that b+x=a”. So, when you write 0/0, it should be a number x such that 0*x=0. Any real number x would work here. (Remark that if there is no 0 in the denominator, the answer for x is always unique.)
  So, in principle, you could have declared 0/0 to be any number, the definition of the division operation would hold completely; but this would break the nice properties of the previous operations. For starters
  1/0 = (1+0)/0 = 1/0 + 0/0 = 1/0 + 1
  In other words, 1/0 cannot be defined as a number. We used the distributive property; we could assume that it does not work specifically for 0/0, but what’s the purpose of declaring an operation result something and then doing it an exception of all rules? You do not create anything new arithmetically speaking.
  Worse than that, as other commentators mentioned,
  1 = 0/0 = (0+0)/0 = 0/0 + 0/0 = 2, and that is a much bigger problem, as it means that you can’t define 0/0 and keep numbers staying distinct at the same time.
  Weirdly enough, this can be generalized: whatever structure you have with “nice” addition and multiplication operations, the neutral addition element (0) can never be invertible with respect to multiplication.
  Edit: trickier exercise: why would we not set 0/0=0? >
  
  You explained the group of invertible elements in a field in passing. Great answer. >
  
  the invertible elements of a field do not form a ring >
  
  Apologies. You are correct. >
  
  they do make a multiplicative set >
  
  Sure. But not a ring though. Sum of invertibles need not be invertible. They indeed form a group and I have amended my comment to say the same. >
  
  actually i guess the units of the trivial ring do form a ring >
  
  What can be divided can also be multiplied.
  1x1=1
  1x2=2
  1x0=0
  0x0=...1?
  We have somehow created something from nothing.  We shouldn't be able to do that.
  Zero is an expression of nothingness.  There is no defined quantity of nothing within nothingness.  Nothingness can't be positively expressed because it is absence. It is defined by the inability to be positively expressed.
  Put differently: how many zeros fit in zero?  The answer is all of them. >
  
  
  Implying what can be multiplied can be divided 
  
  1x1=1    
  1/1=1 
  2x2=4    
  4/2=2
  XxY=Z
  Z/Y=X
  9x0=0
  0/0=9
  7x0=0
  7/0=0
  7=9
  We say you can multiply by nothingness but can’t divide by nothingness… but why? Fundamentally 0 breaks both. >
  
  That didn't break multiplication at all >
  
  Shouldn't be able to reverse a multiplication in general?
  A*B=C -> C/A = B and C/B = A 
  0 Seems like the only case where a multiplication is irreversible. >
  
  Something being an exception doesn't mean that thing breaks something. The rules are consistent and exceptions are part of the rules. >
  
  Exceptions are inconsistencies within rules. Like, by definition you have these rules, and then exceptions that are separate and aren't consistent with those rules. >
  
  The exceptions aren't separate. They are part of the rule in the first place >
  
  If they were part of the rules then they wouldn't be exceptions to the rules. >
  
  They really aren't exceptions to the rule. They are exceptions to an equation in the rule. Without the exception you don't have a rule, just a wrong equation. >
  
  The equation is the rule though. Wrapping the equation + the exception in toilet paper and calling it a new rule is just obfuscating the fact that it's the exception to the rule. You could introduce any exception at that point and just declare that all the rules are consistent because any inconsistency is written into a new, bigger, rule. >
  
  You have zero apples and distribute them evenly to your zero friends how many apples does each friend get? >
  
  All zero of my friends get 1000 apples. >
  
  Johnny Appleseed over here. >
  
  From a physicist's perspective, the question I would ask is "how did you wind up with 0/0?"
  
  Was the numerator 0, and the denominator kept getting closer to zero?
  
  0/5 = 0
  0/4 = 0
  0/3 = 0
  0/2 = 0
  0/1 = 0
  0/(thing that's really close to 0) = 0
  
  Was the denominator 0, and the numerator kept getting closer to zero from above?
  
  5/0 = ∞
  4/0 = ∞
  3/0 = ∞
  2/0 = ∞
  1/0 = ∞
  (positive thing that's really close to 0)/0 = ∞
  
  Was the denominator 0, and the numerator kept getting closer to zero from below?
  
  -5/0 = -∞
  -4/0 = -∞
  -3/0 = -∞
  -2/0 = -∞
  -1/0 = -∞
  (negative thing that's really close to 0)/0 = -∞
  
  Were the numerator and denominator the same and getting closer to zero at the same rate?
  
  5/5 = 1
  4/4 = 1
  3/3 = 1
  2/2 = 1
  1/1 = 1
  (thing that's really close to 0)/(thing that's really close to 0) = 1
  So, 0/0 is undefined because there's a lot of different things it could be. Maybe it's 0 because the numerator is really strongly 0 while the denominator is squishy, maybe it's +∞ or -∞ because the numerator is squishy while the denominator is really strongly 0, maybe it's 1 because the numerator and denominator are the same thing and just happen to be almost 0, or maybe it's something else. There's a lot of different things it could be based on how you wound up with 0/0, so by itself, 0/0 is undefined. It represents too many possibilities to define it as any one value. >
  
  Glad you tried to bring in the real world. 
  Another physicist's perspective:
  Say you weigh two grains of rice on a bathroom scale. They are both too small to read anything except zero. 
  Now, when you calculate the ratio of their masses, you'll get 0/0. The true answer definitely isn't 0, we all know it's closer to 1. In this experiment, 0 doesn't mean "nothing", it means "very small compared to my measurement scale". 
  This happens all the time! 0/0 is undefined in the real world. >
  
  There are plenty of clear examples given already that disprove your thesis. I just want to point out that you are starting from an incorrect assertion that you have stated as a trivially true, when it is in fact abjectly false. Saying, "we all know that XYZ," doesn't make XYZ true.  
  
  A building block of basic algebra is that x/x = 1. [...] We all accept this to be the norm, anything divided by that same anything is 1.  
  
  This is not true.  
  
  But if you set x = 0, suddenly the rules don’t apply.  
  
  This is an incorrect conclusion made by trying to logically follow from a false starting premise. >
  
  a falsis principiis proficisci >
  
  
  So that's a really long winded way of saying that, even though 0/0 isn't 1, your gut is leading you in the right direction when it tells you that 0/0 feels an awful lot like 1.
  
  NONONONO. 
  What you wrote means nothing of the sort.
  Don't give a long-winded explanation of L'Hôpital's Rule and then use that to say that the answer is probably 1 because it isn't -  L'Hôpital's Rule shows in fact that the actual limit of f(x) / g(x) as x->a if f(a) = g(a) = 0 can be anything at all - positive, negative, zero. >
  
  But what I actually said, if you’ll slow down and take a breath, is that quite often in math and physics we consider 0/0 to be equivalent to 1 for the sake of understanding systems. I didn’t say anything at all like, “the answer is probably 1.” I was quiet clear that the answer is undefined, full stop. 
  This use of L'Hôpital's rule is literally a textbook example of how it’s used to make sense of indeterminate functions. It’s no coincidence that I used the natural logarithm and the linear function, this is the example presented in nearly ever analysis text I’ve ever used. >
  
  
  A building block of basic algebra is that x/x = 1. It’s the basic way that we eliminate variables in any given equation. We all accept this to be the norm, anything divided by that same anything is 1. It’s simple division. How many parts of ‘x’ are in ‘x’. If those x things are the same, the answer is one.
  But if you set x = 0, suddenly the rules don’t apply. And they should. There is one zero in zero. I understand that logically it’s abstract. How do you divide nothing by nothing? To which I say, there are countless other abstract concepts in mathematics we all accept with no question.
  
  Actually, when I still studied mathematics we were always told in such cases to add “(provided x != 0)” and for good reason. It lead to absurdity if we allowed for x to be 0.
  A simple example is proving that under Newtonian mechanics, every object in a vacuum falls with the same acceleration to another massive object such as Earth. At one point in the proof x/x does occur, where it's the mass of the body, but if we allow for the mass to be zero, we could prove that this applies even for massless objects, which is clearly false as massless objects are not attracted by gravity and don't accelerate to earth at all. But even the slightest amount of nonzero mass will cause the acceleration to be exactly the same as even the most massive object.
  Simply put, the rule that x/x=0 applies to every number but 0 for x. There are many, many rules that apply for every number but 0; 0 is in fact one of the most unique numbers that exist and that violates many laws that are universal for every other number.
  
  But 0 is somehow more abstract or perverse than the other abstract divisions above, and 0/0 = undefined. Why?
  
  Because there is no single solution in x to the æquation x*0 = 0; it's that simple. That's how division is defined. x/y is defined as the single solution to the æquation z*y = x in z [pronounced “zed”; part of the definition].
  As far as x*0=0 goes, every single number is the solution to that æquation, that makes zero unique. For every other number, say x*4=4, there is exactly one solution, that solution is 1; zero is the only case where there are an infinite number of solutions. That doesn't make it abstract, but unique in this case, and why 0/0 is not defined.
  Perhaps a more compelling reason would simply be that if we were allowed to say that 0/0=1  as I pointed out above, the mathematics by which physical laws are calculated that seem to work now, would no longer work, and we could prove that massless objects fall to earth under Newtonian mechanics, which they don't.
  A more compelling argument is that if we could rule that 0/0=1, we could prove 2=1:
  
  let a=b
  thus a²=b*a
  thus a²-b²=b*a-b²
  thus (a+b)(a-b) = b(a-b)
  thus  a+b = b ??
  thus  2*b = b
  thus  2=1
  
  The part with ?? is where the flaw lies. Since a=b, a-b=0, if 0/0=1 were to hold, we would be allowed to perform this operation, dividing  both sides by 0 and replacing the (a-b)=0 part with 1, but we cannot do this, and thank god, for if we could, two would be one and everything would be messed up. >
  
  I am perplexed by your spelling of "æquation" and your aside that z is pronounced "zed" as if that's critical to the argument. >
  
  I actually own a mathematics textbook that has something in the foreword that says something similar to “For the duration of this book, the symbol “z” shall be pronounced as “zed”.” >
  
  "zed" is just the British pronunciation of z (in general). It was probably a joke. >
  
  “zed” is the international pronunciation that's even used by mathematician's in the one country that pronounces it differently, just as scientists in that country tend to still say “aluminium” and use the metric system. It's not “British” any more than centimetres and universal healthcare are a “British” thing.
  It was obviously a joke, however, as it was in my case. >
  
  If 0/0 = 1, then it is relatively easy to use standard algebraic rules to prove that 2=1:
  1 = 0 / 0 = (0+0) / 0 = (2 * 0) / 0 = 2 * (0 / 0) = 2 * 1 = 2
  
  In the first equality, I used your assumption
  In the second equality, I used the fact that 0+0=0
  In the third equality, I used the fact that a+a=2*a for all values of a.
  In the fourth equality, I used the fact thact multiplication and division are commutative
  In the fifth equality, I again used your assumption
  
  Therefore, if you really want to claim that 0/0=1, then you have two choices:
  ​
  
  Either you also accept all the other rules of algebra, in which case you must also accept that 1=2
  Or, you must explain which of the rules of algebra (used above) you want to stop using. >
  
  Just on an intuitive level:
  5/0 = undefined
  4/0 = undefined
  3/0 = undefined
  2/0 = undefined
  1/0 = undefined
  0/0 = undefined
  
  Anything divided by zero is undefined/infinite, because an infinite number of zeros can "go into" 1. 
  0/0=undefined is more consistent with that.
  This just as consistent as your observation that x/x=0, so there's nothing to prefer there... but defining 0/0 leads to all sorts of contradictions like being able to 2=1, so better to be consistent in the undefined level. >
  
  I don't think it's ever appropriate to describe x/0 as "infinite", as depending on your approach towards the y-axis on the graph it could be +infinity or -infinity. It's ambiguously either value, or both, or neither. >
  
  I wish I could give a delta. I think the real issue here in this thread is that people need to understand is that 0/0 is not infinity under any circumstance. It's still faulty thinking on the same level that 0/0 is equal to 1 >
  
  What about limit x->0 of x/x^3? The limit goes to infinity. >
  
  A limit is not a definite way of describing a value at a certain point. Approaching a value doesn't mean that it actually holds the value at that point. This should be one of the first important rules that is taught when first entering some form of calculus.
  Take x/(x\^4). It will diverge to opposite ends of infinity when you approach 0 from both sides. But it's mathematically equivalent to 0/0 alongside x/(x\^3) when you substitute x for 0.
  But then take x/x as an example too. Using your logic, it approaches 1 from both sides and thus can be "valid" under that circumstance. However, we know that 0/0 cannot possibly be equal to 1 as already defined in the thread >
  
  I’m not arguing it does. My point was that x/x^3 evaluated at x = 0 is 0/0 which is indeterminate. If you take the limit, then it equals infinity because 1/x^3 goes to infinity much faster than x goes to infinity, so the denominator wins out in that case.
  I agree with you inasmuch as we can never assign a value to 0/0…because we can make examples where the limit equals 1, 0, and infinity. Hence the name “indeterminate.” >
  
  There is nothing dividable by zero, even zero itself, and there is a very logical explaination for that :
  We tend to think (because that's what we learned in elementary school for the sake of simplicity) that addition (+), substraction (-), multiplication (x) and division (÷) are four distinct basic math operations, when, in fact, on a purely mathematical level, substractions and divisions do not exist there are only additions and multiplications.   
  Substractions are in fact additions (of the opposite) .  
  Divisions are in fact multiplications (of the multiplicative inverse) .    
  Here' s how and why (still from a purely math point of view) :   
  The following: 10 - 3 = 7 is called a substraction since we substrat the number 3 from the number 10.
  However it is also an addition of the opposite of the number 3, which is  -3, to the number 10.  
  You can write it like this : 10 + (-3) = 7.
  Hence, every substraction of a number N is in fact an addition of its opposite -N.   
  A similar concept applies also for multiplications and divisions, althought instead of being with the opposite number, it operates with its multiplicative inverse.
  The following : 10 ÷ 2 = 5 is called a division since we divide the number 10 by the number 2.   
  However it is also a multiplication of the number 10 by  the inverse of the number 2, which is a half (1/2 or 0.5). 
  You can write it like this : 10 x 0.5 = 5. 
  Now about zero, per the rule of modern maths, the opposite of zero is zero, so you could transform a substraction using 0 into an addition using the opposite of 0, that is to say 0.   
  10 - 0 = 10 + (0) = 10    
  However, the multiplicative inverse of 0 does not exist. That's how it is, it simply does not exist, not because no one thought about inventing one but because it cannot exist mathematically speaking. It is not 0, because 0 exists. It is not infinity either (it's another topic). It just doesn't and cannot exist.     
  Since it does not exist you cannot multiply it. Which means : 10 x (1/0) is a formula that cannot exist since 1/0 does not exist.    
  Therefore, since you can't multiply by the inverse of 0, you can't divide by 0. Never. >
  
  
  A building block of basic algebra is that x/x = 1. It’s the basic way that we eliminate variables in any given equation. We all accept this to be the norm, anything divided by that same anything is 1. It’s simple division. How many parts of ‘x’ are in ‘x’. If those x things are the same, the answer is one.
  
  Not really. In fact, in division rings, a building block of an algebra of real numbers does not allow division by zero. In fact, you don't need that to eliminate a variable. If you have something like:
  x(2x+5) = 2x(3x+7)
  You can simply split the cases. You handle the case when x = 0, and another case when x != 0.
  In this case, if x=0, the equality trivially holds. So, you can just handle the case when it doesn't hold, which means x /= 0 and you can now divide by x. 
  
  But if you set x = 0, suddenly the rules don’t apply. And they should. There is one zero in zero. I understand that logically it’s abstract. How do you divide nothing by nothing? To which I say, there are countless other abstract concepts in mathematics we all accept with no question.
  
  I agree that "how do you divide nothing with nothing" is not a good counterargument. The real counterargument is that you can't have a multiplicative inverse of zero. If the multiplicative inverse of zero is w, then: 1=0w= (1-1)w = w - w = 0, showing that you're working on a trivial ring. There is another formulation of reciprocal that can work fine with 0, like in a wheel. But, there, reciprocation is no longer a multiplicative inverse. And your method of eliminating a variable no longer works.
  Losing such an algebraic structure might be acceptable if you find useful use cases for it. Unfortunately, your proposed use case, eliminating variables, not only not really works, but can be solved pretty elegantly with a powerful technique called splitting cases. >
  
  Dividing nothing doesn’t magically give you something. >
  
  Also by dividing by nothing basically means not dividing at all lol >
  
  lets look at this from a practical real worlds standpoint, because ultimately the point of math is to allow us to understand the physical world.
  10/2=5 is essentially saying you have 10 apples being split into 2 baskets.  How many apples end up in each basket?  The answer to this is simple.  There are 5 apples in each basket.  Great.  10/2=5 makes sense.
  now Imagine you have 0 apples and 0 baskets.  How many apples are in each of those baskets.  There aren't any baskets.  You don't have an empty basket.  You don't even have a basket, so saying there are zero apples in each of the zero baskets doesn't make any sense.  If you are going to somehow pretend like you can have some number of apples in a non-existent basket, you might as well say the answer is 5000 apples per non-existent basket, and that would hold true as well because if you have 0 apples divided into zero baskets, then saying there are 5000 apples in each non-existent basket would give you a total of 0 apples.
  0/1=1.  that makes sense.  0 apples in 1 basket.  So why would 0 apples in 0 baskets have the same number of apples per basket?  remember, there is no basket.  It is a nonsensical real world question, therefore it only makes sense that there is not a clear mathematical answer or else math would not reflect reality and we would not be able to trust using math to solve real world problems. >
  
  But look at the examples he offers that also make little sense when thought of in this way.
  
  Negative numbers (you can show me three apples. You can’t show me -3 apples. It’s purely representative). Yet, -3 divided by -3 is positive 1. Because there is exactly one part -3 in -3.
  “i” (the square root of negative one). A purely conceptual integer that was created and used to make mathematical equations work. Yet i/i = 1.
  0.00000283727 / 0.00000283727 = 1.
  (3x - 17 (z9-6.4y) / (3x - 17 (z9-6.4y) = 1.
  But 0 is somehow more abstract or perverse than the other abstract divisions above, and 0/0 = undefined. Why? >
  
  Multiplication and division are distributive, so (a * b) * c = a * (b * c). That has to be true or basically all of math collapses. 
  So what about 2 * 0 / 0 ? 
  If we write it as (2 * 0) / 0, that's 0 / 0, which you want to be 1, so by your logic, 2 * 0 / 0  = 1
  But if we write it as 2 * (0 / 0), then it's 2 * 1 = 2. The exact same equation has a different value depending on how it's distributed, which breaks all of multiplication. >
  
  Actually 0⁰ is undefined. And raising to the power of zero is not abstract. Raising to zero is the same as raising the to power of "1-1". And for example 2^(1-1) is the same as 2^(1)*2^(-1) which is 2/2 that is 1. So for zero it would he 0/0 which is undefined . Making 0/0 = 1 is very contradictory and othere have pointed out. But i would like to point it out graphically that on one side of the graph(positive) it goes to VERY small negative values as x approaches 0 while on the positive side it goes to very BIG positive values. So defining it as 1 would probably contradict one side >
  
  Actually 0^0 is indeterminate. As is 0/0. >
  
  Ah right. my mistake >
  
  
  So we all have agreed that if you take nothing, then raise it to the power of nothing, that equals 1 (0^0 = 1)
  
  No. 0^0 = 1 is not a universally accepted definition.
  0^x = 0 for any x, because you just multiply zeros together, so the result is 0.
  But y^0 = 1 for any y, because that's how raising something to the power of 0 is defined.
  Therefore 0^0 is considered ~~undefined~~ indeterminate, same as 0/0, ∞/∞, 0 * ∞, ∞ - ∞, 1^∞, (-1)^∞, ∞^0, 0^i and the zeroth root of x. >
  
  What you’ve described are indeterminate forms, not undefined forms. >
  
  Thanks for the clarification, I wasn't aware of this distinction. >
  
  No problem! It’s a subtle difference, but makes a lot of sense when you are aware of it. Also I had a typo on my comment, should have been “indeterminate” not “indefinite”, fixed now. >
  
  Other people haven't mentioned this, but all of math is human created. The "laws" of math are just the rules that we tend to agree on when we teach and learn math. Created by humans for humans.
  So if you want to have your own math where 0/0=1? That's fine. It isn't any less valid than the common version of math where you can't divide by 0. 
  So perhaps you mean to say that in common math, we should all agree that 0/0=1? 
  Well we use math for many purposes, and we like to have a system where we don't reach contradictions. Allowing you to divide by 0 lets you to prove things that aren't true. Let me show you:
  0 * 3 = 0 * 7 
  Now divide both sides by 0
  1 * 3 = 1 * 7
  3 = 7
  Dividing by 0 causes these problems, so we decided that we aren't allowed to divide by 0. It doesn't seem to be useful for us to be able to prove things that aren't true, so I don't see any reason for common math to allow dividing by 0. Hope that helps explain things better. >
  
  
  The "laws" of math are just the rules that we tend to agree on when we teach and learn math.
  
  Disagreed. Mathematics is a device we use to describe a certain aspect of our reality. In our reality, 0/0=Ø. You can't make up your own set of rules, as the field would then cease to describe our universe. It's like saying that one can make their own version of physics in which E=M•C^(2)•Q, where Q is the weight of an average chihuahua. Sure, you could do that, but then it would cease to be true. >
  
  
  Mathematics is a device we use to describe a certain aspect of our reality
  
  That's not true at all. What you're describing is called "science". Mathematics is by definition independent of reality. For example, we can talk about geometry in R^(2), R^(3), R^(4), R^(5), hyperbolic space, spherical space, etc, without knowing which geometry our universe actually is in. (Yes, it's still an open problem if we really lived in R^(3))
  Also, if we changed the axioms, we could actually have 0/0. It's just no longer a real number. The axiom that cannot support 0/0 is:
  
  Ring axioms
  Multiplicative inverse axiom
  0 /= 1
  
  In a trivial ring, the latter does not hold, so we can have 0/0=0=1. In a wheel theory, the first two axiom does not hold, so 0/0 is also defined, just not 1.
  
  one can make their own version of physics in which E=M•C^(2)•Q
  
  This is completely different. The difference between this and 0/0=1 is that the former is a testable hypothesis and the latter is not testable but simply contradicts field axioms.
  In fact, the whole point about math is that you can make up your set of rules, as long as the conclusion follows the axioms. Whether it is useful, though, is a different problem
  EDIT: It's science, not physics. >
  
  
  That's not true at all. What you're describing is called "physics". Mathematics is by definition independent of reality.
  
  That's not even close to correct. Where are you getting these idiosyncratic usages? >
  
  From Wikipedia:
  
  Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists of a succession of applications of deductive rules to already established results.
  
  Notice that it involves fixing an axiom. The axiom is held to be true. The axiom could be arbitrary (looks at set theorist's various set axioms)
  Meanwhile:
  
  Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe
  
  It actually describes reality by means of actual testable theories.
  (On hindsight, it's not physics. It's actually science)
  Note that while science uses math to describe reality, math itself does not care. It's up to science and science method to test whether a mathematical model conforms to reality, not math. >
  
  i/i = -1/-1 = 1. That makes perfect logical sense.  I don't see how you are using that as an argument for your theory..? >
  
  They think zero is a very basic number with very basic properties, and if something what they think is less basic (negative or imaginary numbers) can do the x/x=1 thing, zero should be able to too. >
  
  x/× only cancels to 1 in the context of an equation to remove variables and simplify where possible.  If we assign x a value of zero, that changes things. Have you factored binary code into your thought process? In its most basic form 1 = light, 0 = absence of light. Just a thought. >
  
  That's why 0/0 is not equal to 1. 0 is 0 and 1 is 1. Absence of light can never =light. And that is why setting x to 0 changes the whole meaning of an equation.  It's not abstract at all. It's very very simple. >
  
  OP should read this, but they probably know the answer by now. >
  
  I don't think OP wants to read it. I don't think they want their view changed. >
  
  Its by definition.  There are mathematicians that argue the rules should be otherwise but the bottom line is that anything divided by zero is undefined by definition. >
  
  Actually it’s indeterminate by definition. Also what mathematicians (I assume you’re referring to contemporary ones) are at odds with the definition of 0/0 being indeterminate? >
  
  A number of years back I had a similar journey as the OP and did some research.  I found that there were some that argued the for the exception for zero in x/x should be removed, which necessitates other convention changes to accommodate that.  They basically argued the rules around 0 and inf could be done differently and still yield a practical mathematics.
  It was years ago so I don't recall where I found it. >
  
  It's because your first example you show x/x = 1, but if x=0 it makes no sense, because in x/x if x=0 then x also can equal 7 and still be 0.
  it's undefined because x cannot equal 0 and 7 at the same time.  Thus it's undefined because the equation itself makes no sense.
  for instance if x=1 
  you get 1/1 = 1
  if x=4 
  you get 4/4=1
  if you do x=0
  you get sorta 0/0=undefined, because nothing divided by no people is nothing, but you cannot also define the first X as 0 because the equation is exactly the same no matter what that first X actually is and you can't define the first X as 8934, and the second X as 0 or there's no sense. >
  
  You can't divide by zero. >
  
  Essentially this is a calculus problem of limits which even going through multiple calc classes still confuses me. You've already awarded a delta to the top comment which explained it very well. I would just add that you can think about dividing by zero as dividing by smaller and smaller numbers until you are infinitely close to zero but not yet at zero 
  It doesn't matter what your numerator is, if you are dividing by 0.000...001 you are essentially going to end up with infinity. However, you are correct that x/x should equal 1 meaning that it should be 1. This means that there are two possible solutions for the same exact input which is why dividing by zero is considered undefined. >
  
  If you have 1 apple for 2 people, everyone gets 1/2 of an apple.
  If you have 0 apples for 2 people, everyone gets 0/2 of an apple (i.e. no apples)
  If you have 0 apples for 0 people, every "one* gets 1 apple??? Where did the one apple come from if there was none to begin with? >
  
  You are in credit of 3 apples, so you have -3 apples. 
  You are in credit to 3 people, so you have -3 people. 
  How many apples should each person give you ? One
  -3/-3=1 >
  
  What you have to understand is that math is something humans made up. It's not a science,  derived from the natural world. It is completely a human abstraction, and like humans it's not perfect so we have to have rules (which in math we call postulate) like 0/0 is undefined, because without those rules (postulates) math just breaks down and stops working. >
  
  one more argument I havent seen is to look at the graph of y=1/x. as x approaches 0 from the right, y goes to infinity. as x approaches 0 from the left, it goes to negative infinity. the graph would make no srnse with an aditional point at (0, 1). >
  
  in turns of conventional mathematics, it just is. Math is our ability to understand patterns and this numbering system has rules that so long as it matches said patterns will help us evolve our problem-solving skills. It's just a rule for the system. >
  
  x / y = z means that x = y * z
  For example, 6 / 3 = 2 means that 6 = 3 * 2.
  Well, if you want 0 / 0 to be 1, then 0 = 0 * 1. That checks out.
  But let's assume 0 / 0 = 2. That would mean 0 = 0 * 2. That checks out too!
  But let's assume 0 / 0 = 3. That would mean 0 = 0 * 3. That checks out too!
  But...
  So which one is it? 1? 2? 3? 41068? >
  
  The clear difference is that 0 has no value like every other number. It is an absence of value; so if you think about it, it’s not just an exception, rather, you would definitely expect it to behave differently than all other numbers. The x/x “rule” is not an inherent absolute because it is a consistent pattern for values, not the absence of a value. >
  
  if you have 30 apples and split them into groups of 3 then you have 10 groups. 30/3=20
  if you have 30 apples abd you split them into a group of 30 then you have 1 group. 30/30=1
  If you have no apples and you split them into groups of zero apples. Then you have 1 group of what?  zero apples? But you didnt split them  into groups of anything.  So 0/0=undefined >
  
  
  A building block of basic algebra is that x/x = 1
  
  The integers under division do not form a group, so what you've said isn't always true. >
  
  Let's assume 0÷0 DOES = 1
  And let's set up an arbitrary X×Y=Z
  If we set Y=0, that forces Z=0 as anything ×0=0. X can be ANY number you can imagine, it doesn't matter. We can denote this as X=C (X being our variable, C being all complex numbers, which also includes all real numbers)
  Using algebra, we can divide both sides by Y
  X×(Y÷Y)=Z÷Y
  Y÷Y=0÷0, which at the beginning I said to assume equals 1. The same for Z÷Y
  So we have X=1. However, we already found X to be ANY number. With the assumption that 0÷0=1, I just proved that ALL numbers = 1, which is obviously false. Therefore, 0÷0 CANNOT = 1 >
  
  If x/y = x * (1/y), then 0/0 = 0 *(1/0). But anything divided by zero is undefined. >
  
  0/0 is Indeterminate, not undefined. >
  
  Good catch, didn't realize there was a difference. >
  
  Let's say you are a maths teacher. There is a bunch of kids in your class. 
  One day, you notice you have zero oranges. 
  Now you decide to give those zero oranges to zero of the kids in your class.
  In what universe would the outcome of this be that suddenly one orange materializes? >
  
  I’ve got one for ya, I don’t view 0 as just “nothing” but as everything, to me 0 is the sum of every real number, negative and positive, since numbers go on forever into infinity, 0 = ♾️ 
  ♾️/♾️ is not 1 it’s ♾️. >
  
  Give zero pizza slices to zero people. How many slices per person is that? One? >
  
  Ok let’s go super simple. If I give you 1 apple and ask you to divide it into 3 pieces, you can. If I now give you an empty plate and ask that you divide the apple into 3 pieces, can you? >
  
  0/0 being 1 breaks algebra. Take something like
  y = 5x + 3
  Multiply both sides by 1
  1(y) = 1(5x) + 1(3)
  Multiply out the ones except for the one on 5x
  y = 1(5x) + 3
  Sub out that 1 for 0/0
  y = (5x) 0/0 + 3 = [0(5x)]/0 + 3 = 0/0 + 3
  we're back to 0/0 after multiplying the numerators. Sub that 1 back in for 0/0
  y = 1 + 3 = 4
  You can't just sub in 0/0 for one. it lets you break anything. >
  
  Ignore the 0 on the top of the equation, because it is irrelevant. It applies for any number including 0, that division by 0 is undefined. So change the equation to c/0, where c stands for constant, which can be any number positive and negative, including 0.
  c/0 = undefined doesn't tell us much. But we can look at the behavior of c/x as x gets closer to 0. c/1 = c; c/0.1 = 10 * c; c/0.01 = 100 * c; c/0.001 = 1000 * c; c/0.000000001 = 1000000000 * c. As x gets closer to 0, c/x gets larger. Look at the behavior as x gets closer to 0 from -1. c/-1 = -c; c/-0.001 = -1000 * c; c/-0.000000001 = -1000000000 * c. As x gets closer to 0, c/x gets larger in the negative direction.
  We can summarize the behavior as the following. As +x approaches 0, c/x approaches infinity. As -x approaches 0, c/x approaches -infinity. For c/x to equal infinity, x must equal 0, which means c/x must also equal -infinity. c/0 is undefined, because there is no number that is defined to equal both positive and negative infinity. 
  Anything divided by 0 is undefined, it doesn't make any difference for 0/0 or 1/0 or infinity/0. >
  
  
  Yet we can’t simply consistently hold the logic that anything divided by it’s exact self is one, because it’s one part itself, when it comes to zero.
  
  But then we would be breaking the consistent logic that anything divided by 0 is undefined and that 0 divided by anything other than 0 is 0.
  Think about a / b for smaller and smaller as and bs, like ½, .1/.2, .01/.02, .001/.002. Basically a limit in calculus. a and b can get arbitrarily close to 0, yet the quotient will always be 0.5 (assuming they decrease at the same rate, but not that they have to). Depending on what a and b are, the quotient will converge on wildly different values despite them all getting closer and closer to 0/0. Since no matter how close we get to 0/0 we always fail to converge on a consistent value, so this tells us that 0/0 is undefined. >
  
  a/b is the number that if multiply it by b, I get a :
  a/b x b = a
  So what is 0/0 ? A number that multiplied by 0 gives 0. Which is ...any number. Do you start to see the problem ?
  You decided to take a rule and enforce it on 0/0. But you could have chosen another rule like 0 divided by anything is 0 and so 0/0 should be 0 >
  
  In how many ways can you give no apples to no body? >
  
  It's like the dead center of a spinning object, it is turning neither right, nor left, but .000000001" to either side of center and the object is moving in opposite directions.
  That's 0 >
  
  "How many zeroes are there in zero?"
  "Exactly one".
  ... "How did you count it?" >
  
  Nothing dividing by nothing is nothing. 0 / 0 = 0
  We can eliminate the division sign by multiplying a zero on each side of the equal sign. Then we get the similar statement.
  Nothing times nothing is nothing. 0 * 0 = 0
  Remember that division and multiplication are really the same thing. Since 0*0=0, is true, then it’s similar statement 0/0=0 must be true as well. QED
  0.000000023 / 0.000000023 = 1 because there is something in the numerator and something in the denominator and they happen to be the same number so they cancel out and are exact 1 copies of each other. But 0 means absolutely nothing. There is a vertical asymptote at 0. 0 is a very powerful number. 0=nothing. >
  
  A building block of basic algebra is that 7x/x = 7. But if you set x = 0, suddenly the rules don’t apply. And they should. There are 7 zeros in zero. Indeed,
  0 + 0 + 0 + 0 + 0 + 0 + 0 = 0.
  Why isn’t 0/0 = 7? >
  
  For example, 10/2=5 therefore 52=10
  So if we assume 0/0=x it means 0x=0
  Because whatever number you multiply 0 with will be 0,
  x can be any number.
  This is weird since any whole number(like 1,2,-10,0) divided by any whole number should only have one solution. >
  
  It’s important to understanding the expression, that the one exists prior to the calculation. 
  The value derived from dividing is what you have. That is all the work it’s doing.
  So 0/0 indeed equals 1. You do have 1 zero. Of course zero has no value so you  really have nothing.
  Simply expressed you’re saying 0 = 1 or 1 = 0 or nothing I have or I have nothing. 
  All other expressions of it say 1 = 1 which of course is I have something. 
  Just update it to x/x = 1(x/x) 
  Using x\x = 1(x\x) 
  If x = 5 then 1 = 11
  If x = 0 then 0  = 10 
  That said, mathematicians say division by 0 isn’t allowed which is a pretty convenient way to avoid addressing something. 
  But, If I have 5 apples and divide them into no groups, I have 5 apples. And if I multiply them no times, I have 5 apples. 
  5/0=5 
  5*0=5
  The reason zero breaks brains is that it’s the binary expression of 1 it’s nothing to the something when multiplying and dividing. 
  I know I know who am I to say 0 times any number is that number? 
  But food for thought, it breaks nothing and fixes everything. >
  
  The number 0 doesn't even exist its just a place holder for nothing, so nothing goes into nothing never because it never existed. >
  
  I used to think this, because a/a = 1 and limit x->0 x/x = 1. If you set 0/0 = 1, you would encounter a lot of contradictions. 0 = 0 + 0 because it's the additive Identity, and we know that (a+b)/a = a/a + b/a. Let a,b = 0, then you're basically saying that 0/0 = 0/0 + 0/0, which would mean 1=2. That would completely break math and does not make any sense. You could even change the above step to set 1=3, 1=4, which obviously isn't true.
  Another example, if 0*9 = 0 this would mean that 9 = 0/0, again, a contradiction with what we just got. That is a simple explanation of why 0/0 is undefined. >
  
  0 is a representation of nothing. Much like infinity means unending or incalculable. 1/5 means 1 object (example) is being divided amongst 5 people (example). So 0/0 nothing divided amongst nobody cannot give you something, right? >
  
  Zero is less than one. If you put zero into zero groups you’re not altering the value of zero, so it remains less than one. >
  
  Zero has no value. Therefore, 0/0 can not equal 1. You have 1 group of nothing, and you divide that with nobody. You still have nothing. >
  
  Think about it as a word problem not an algebra problem. You have 0 apples and you have to divide them amongst 0 people. The answer isn’t 1 apple per person. >
  
  If you set 0/0 =1 you are free to prove any statement no matter how absurd. Want to prove that 2=4? It now does.   
  Want 4 > 8? It now is!
  There's virtually nothing you can't "prove" true if 0/0 = 1 >
  
  The limit of x/x when x approaches 0, both from the left and the right, tends to 1. That is as far as you can go. >
  
  How much nothing is in nothing? 1? 5? 61843619? 
  Or look at it like this: When you try to divide anything by 0, for example a/0, try to find a. What can you multiply by 0 to get a? Nothing
  Or, think about it like this: 0/0, now try to find out what you can multiply by 0 to get 0? Anything.
  That's why dividing anything by 0 doesn't work, you just can't give a definitive answer. >
  
  If you have 6 cookies, and 3 friends, you can give 2 cookies to each friend.
  Now, you have no cookies and no friends. You can’t really say you have 1 cookie for each of the non-existing friends, at least it’s not more correct than any other answer. >
  
  Simplified, 0 has no value.  x/x=1 implies that x is a value.  It can be positive or negative, but it has a value.  0 is neither positive nor negative and has no value. >
  
  Let's say you have two real valued continuous functions f(x) and g(x). If f(x)->a, and g(x)->b=/=0 as x->0, then the limit of x->0 f(x)/g(x)=a/b is well defined. However, ifboth f(x) and g(x) approach 0 as x->0 then we can't find the limit without additional knowledge of f(x) and g(x). For instance if f(x)=sin(x) and g(x)=x then you can show that the limit as x->0 of sin(x)/x is 1. However, if I take g(x)=2x then I will find limit x->0 f(x)/g(x)=1/2. >
  
  It must be:
  Nothing/nothing = nothing >
  
  You say there is exactly 1 zero but in 0. But there could also equally be 1000 zeros in zero. >
  
  don't take me seriously. I'm no mathematician. 
  but for me zero is not nothingness. Zero is state of no state. Zero is thing before an observation is made. Zero isn't at- empty but Just- after nothing and right- before something . 
  Imagin a pebble removed from sand. If you observe the process of removal of pebble it's subtraction if didn't see its 1 in it self from the beginning but the imprint that the pebble left must be zero. >
  
  Well what can I say. Math is inconsistent. The rules are made up and the results don't make sense. >
  
  0^0 is not 1. It is undefined. Otherwise you have the following: 0^0 = 0^(1-1) = 0^1 * 0^(-1) = 0^1 / 0^1 = 0/0  and you are back where you started your post. >
  
  Ok so there are 5 main operations we learn in math right: Addition, multiplication, subtraction, division, exponents.
  Addition is you taking numbers and combining them if I have 9 apples and you have two apples, we have 11 apples.
  Subtraction is similar to addition too. If I have 9 apples and I give you 2 apples, I have 7 apples.
  If every day I collect 9 apples and it’s been 2 days, I have 18 apples.
  Division is I have 9 apples and I’m dividing it into 2 groups. Each group gets 4 1/2 apples.
  An exponent just makes multiplication easier 9^2 is me saying I collected 9 apples for 9 days, so I have 81 apples. (9^2)^2 is me saying I collected 9^2 apples for 9^2 days, which is 81 apples for 81 days so I have 6561 apples.
  Now replace all of that with 0.
  Adding and subtracting 0. Keeps the same number
  If I collect 9 apples every day, but I don’t go collect apples today how many new apples do I have? 0 new apples.
  If I have 9 apples and 0 groups to divide them in, each group gets nothing. Therefore it’s undefined.
  9^0 is 1 because the answer has to be 9 * a multiple of 9. This also works with fractions for example 9^(3/2) is the square root of 9^3 which is 27. 9 * 1 = 9 that’s the same as me saying 9* (9/9) = 9. Therefore 9^0 must equal 1.
  Now 0 is different because every number could be a multiple of 0. 0 * 0 = 0. 0^1 = 0 because it’s the same as me saying 0* (0/0) = 0. If I have 0 apples in 0 groups, how many apples would be in each group? Nothing, it’s not quantifiable. It’s indeterminate.
  Another way to think of it is an undefined slope on graphs are lines like x= 4. Normally a line on a graph is written in the form of y = mx+b. Y, here is the answer you get based on how you manipulate x.
  Y = 2x + 4
  X = 9 y = 22
  X = 13 y = 30
  Slope or “m” is calculated by taking the difference of two y points and dividing it by the difference of two x points (30-22)/(13-9) = 8/4 = 2
  What happens to y when x = 4? It could be any number. Therefore the slope is undefined.
  Y = 4, on the other hand means that some operation is done to always make the answer 4, but the only way that’s possible is y = 0x + 4. Therefore, the slope will be 0.
  undefined vs indeterminate. >
  
  I see a lot of awesome proofs here, but from a perspective outside of mathematics, ‘0’ is a symbol to represent nothing. 
  You are correct to say that 0/0=1 in the sense that nothing is an outcome/possibility. I feel you are using ‘1’ to represent nothing, which is incorrect since 0 already represents nothing. 
  1 represents a quantity, but you’re using it to represent an outcome. All equations have an outcome, but it doesn’t mean those outcomes are realistic, tangible quantities.
  So 0/0 = 1 outcome, but 0/0 ≠ 1 quantity.
  I hope that makes some sense from a weird perspective. >
  
  You say anything dividend by the same anything is 1. 
  In the same therms we also say 0 divided by anything is 0.
  Choosing 0 as anything breaks one.
  Also maybe I’m wrong but I don’t believe 0^0=1 is a generally accepted definition. But lim x->0: x^x=1 but those are not the same statements. >
  
  This was an interesting thread, thanks >
  
  0/0 is undefined because 0x=0y regardless of x and y, if it is equal to 
  1, it means that x=y regardless of x and y, this we have 1=9 and so on. >
  
  I’m the furthest thing from a mathematician but if you accept that:
  0 = nothing
  Then you should be able to accept that:
  Nothing / nothing = nothing
  Not:
  Nothing / nothing = 1 >
  
  I’m pretty sure it’s because there is an infinite amount of nothing in any given nothing. 
  So like, one 0 can have an infinite amount of 0s in that one. Think like, it could be one 0, or 0x where x = infinity, or any other number between 1 and infinity. 
  And any of that could be represented by 0x where x = whatever, or it could just be 0, because 0x is always equal to 0. 
  So the reason 0/0 isn’t 1 is because 0 is always nothing, and logically, you can’t divide by nothing. If you divide by nothing then you’re not dividing at all. And when you’re dividing nothing by nothing, welp, that is undefined. >
  
  0/0 is infinity.  
  0/0 = 0/(0+0) = 0/(0+0+0) and so on.  This doesn't work with any non zero number.  You can divide an infinite number of zeros into zero. >
  
  Ive seen a lot of good answers, another thing to consider that I didnt see at quick glance, 0 divided by any number is 0, so that causes another issue to deal with >
  
  You're using the logic that if x/x = 1, then 0/0 = 1. Decent logic.
  By similar logic 0/x = 0, therefore 0/0 = 0. Hmm.
  And by similar logic x/0 = infinity, therefore 0/0 = infinity.
  So, we have three different logical statements that show that 0/0 is equal to 0, 1, and infinity simultaneously. Maybe it's just better to call it undefined and let's all go home and eat pizza. >
  
  Identity politics. Any 0 can easily identify as 1 (and vice versa). >
  
  If I remember correctly. 0 in the denominator effectively translates to "(   ) divided by (    )."   but that is as far as the axiom can  get because zero "runs away" from the "divide by" operator. so the operator "divide by" is never able to execute it's command, which is why dividing by zero is undefined rather than any other value. The term "undefined" essentially translates to "the instruction cannot execute" or "there is no output" if there is no output, then it cannot be equal to anything. Hence "Undefined" >
  
  You're dividing nothing by nothing and saying that equals to something, which is false. >
  
  Another rule we use in algebra would be 0/x=0.
  So, turning x=0 should also mean that 0/0=1 and 0 at the same time. >
  
  HAS TO BE AMERICAN >
  
  It’s one of the indeterminate forms. Those are forms which there is no answer. Zero times infinity, infinity minus infinity, infinity, divided by Finley, one to the power of infinity, and 0÷0 >
  
  I think 0/0 is just simply 0 >
  
  The problem is 5*1 = 5 but 5*2 isn't 5 any more.
  But with 0 you can say 0*0 = 0, 0*1 = 0, 0*2 = 0 and so on. On general you could say that 0*x = 0. Thats why you aren*t allowed to divide with 0, because it doesn't matter with what you multiply it, it can never be increased. >
  
  So x/x = 1
  But also 0/x = 0 
  And because we have the general that x/0 is undefined, so 0/0 is also undefined. >
  
  Multiply both sides of the equation by "0" and see what happens. >
  
  It comes down to how you can reverse the equations to get the initial number. Math should work the same way for both directions for addition/subtraction and multiplication/division. 
  If you have 3 apples and 3 people, each person gets one apple. 
  1*3 = 3
  3/3 = 1
  If you are owed 6 apples total from 3 people, each person owes you 2 apples. 
  -2*3 = -6
  -6/3 = -2
  If you have zero apples and split them equally among 12 people, each person has 0 apples. 
  0*12 = 0
  0/12 = 0
  If you have 12 apples and give them to zero people... how many apples does each person have?
  12/0 = ???
  ??? * 0 = 12
  Is there a number that works for both of these equations? How many apples do ZERO people need to give you each so that you have 12 apples total?
  We would need a number that can be multiplied by zero to get 12, but also, we would run into the same problem for any starting number.
  There isn't a way to 'undo' dividing by zero in math, so we just list it as 'undefined'. >
  
  Why are you placing X/X = 1 as the corner stone of your argument?
  0 / x = 0 is also a universal truth in algebra. >
  
  
  There is one zero in zero
  
  This is where you mislead yourself. You're arbitrarily assigning it as a single zero. Assigning a finite value to something that isn't really finite. When really zero is the absence of value (which theoretically is infinite nothing, so infinite zero).
  I think you would have a stronger argument for 0/0 = 0 since this would be "nothing divided by nothing remains, well, nothing". There appears no starting value, and  no operative to change the value so presumably it stays 0. Essentially positing that "zero divided by anything is still zero" is overruled by "the inability to divide by zero". (However this would still be the same mistake of effectively defining zero when in reality the answer should still be undefined). >
  
  You say one building block is that x/x = 1
  Then I say another building block is that 0/x = 0.
  For 0/0 these building blocks conflict, a clear hint that 0/0 = undefined rather than either 0 or 1. >
  
  yes but also any number divided by 0 is (positive or negative) infinity and 0 divided by any number is 0. if you take calculus, you learn how to "solve" different versions of this by finding the number that is being approached. for example, to find the limit of (2x³+5x²)/(5x³+1) as x approaches infinity you can take some derivatives and find out that, if the line didn't stop existing there, the answer would be 2/5. >
  
  OP suppose you were to argue that you can't add zero to zero or multiply it. Would you still be convinced that 0/0 is not 1?
  Maybe we are supposed to divide by zero and get that it equals to 1, and we are not supposed to add or multiply it. 
  After all, it is we who made up the rules and made them fit into a model we built >
  
  0/0 = 1 <=>
  2(0/0) = 21 <=>
  (2*0)/0 = 2 <=>
  0/0 = 2
  If we assume that 0/0 = 1, then it can be shown that 0/0 = 2
  Since 0/0 = 0/0, then it must hold that 1 = 2, which is not the case, therefore our initial assumption is wrong >
  
  man i’m just tryna eat some hot cheetos and chill >
  
  0 is not a number. It's a lack of a number. >
  
  0/0 is indeterminate but not undefined for a reason.
  By the definition of division, some x is equal to 0/0 if 0x=0, this property is true for all real numbers.
  So you can set 0/0 to be equal to any number including 1, making 0/0 indeterminate, or not uniquely defined.
  For any other variations, it's not the case.
  1/2 = x implies 2x=1 which only one real number satisfy the property, so 1/2 is defined
  1/0 = x implies 0x=1 which no real number satisfy the property, so 1/0 is undefined
  It isn't illegal move for you to set 0/0=1 but it is illegal move for you to prove 1=2 or something like that using dividing by zero.
  This is because multiplying a number by 0 is an "irreversible move" and dividing by 0 does not give the information of which number you came from.
  In another word, f(x)=0x is not a one-to-one function, implying that, the inverse map g(x)=x/0 is not a function and does not have unique outputs. So you can't reason that x=y implies g(x)=g(y)
  e.g. 2*0=3*0 does not imply (2*0)/0=(3*0)/0 >
  
  
  A building block of basic algebra is that x/x = 1. It’s the basic way that we eliminate variables in any given equation. We all accept this to be the norm, anything divided by that same anything is 1. It’s simple division. How many parts of ‘x’ are in ‘x’. If those x things are the same, the answer is one.
  
  This isn't a building block of basic algebra. The building block should be: 
  
  x/x = 1 (assuming x ≠ 0)
  
  Others have shown how that leads to absurd, so I won't go into that here. I bring this up to show how your original message doesn't correspond to how we do math:
  In systems/equations with multiple solutions, we have to consider the denominator=0 possibility separately. Here's a simple example:
  
  x^2 - 2x = 0 
  x - 2 = 0 (divide by x)
  x = 2  
  
  That leaves out the solution x=0, which is also valid for the original equation. If we wanted to be rigorous, line 2 would include "assuming x≠0" and we'd check x=0 separately, since that assumption is not necessarily true. The x's that satisfy that equation are 0 and 2.
  In calculus, the derivative of a function f at a point x is defined as "the limit as h approaches 0 of (f(x+h)-f(x))/h"
  One of the reasons why we have to carefully talk about the limit and not simply the value (h=0) is because we would be dividing by 0 otherwise. But sometimes we would get 0/0 (when f(x+h)=f(x)=0). Still, that doesn't make 0/0=1 true: you can find many functions where f(a) = 0 and f'(a) ≠ 1.
  Overall, dividing by 0 always leads to absurd, even if you're dividing 0 by 0.