comment: >
  
  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
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.