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Divide by zero
Houshalter:
No, 0/0 can't be undefined. I think it could have multiple answers, but it can be zero for example. See, if you take 0 and multiply it zero times you get zero. So in reverse you get the same thing. Heres the real problem: x/0=y. If you use traditional math on it, you get x=0. But what if x isn't equal to zero? Its because 0(anything) is always equal to zero, even if its x/0. But then that means that the rule where you multiply both sides by the same thing is invalid. Now where back to where we started. Its a circular problem that depends on its self for an answer. Good luck though.
Numsgil:
--- Quote from: Houshalter ---No, 0/0 can't be undefined. I think it could have multiple answers, but it can be zero for example. See, if you take 0 and multiply it zero times you get zero. So in reverse you get the same thing.
--- End quote ---
If you do x/x, it's 1 for all real numbers except for 0. So another way to look at it is to take the limit and then 0/0 becomes 1. Point is there are lots of different rationalizations you can use to define 0/0, but they don't all agree with each other. That's what it means to be "undefined". Or rather, undefinable. There are multiple competing and contradictory definitions.
Houshalter:
Huh. You made me think .
I had an idea today while thinking about this problem. Its simple: Zero isn't a number. Its a symbol, a visual aid to represent a concept. Its not an actual number. We invented it as a useful place holder so that we didn't have to invent a new symbol for 10, 50, 100, etc, like most ancient people did. Think about it, it litteraly represents the lack of a number, not a number itself. Its own properties conflict with the properties numbers have. Take any number and add zero, it doesn't change because zero isn't a number. Its like not adding in the first place. Take an number and multiply it by zero, you get nothing because you litteraly take that number zero times, like not taking it at all. You can't have zero of something, because if you did its the same as not having it at all anyways. Its because zero is a symbol, not a number, it gets confusing when you treat it like one. Its like trying to dividing 1,000 by the at (@) symbol. It just doesn't make sense. So its no wonder what happens when you try to find out how many times nothing fits into something, even if that something is nothing in the first place. It can't be done.
bacillus:
If it helps, take the opposite as an example - when you solve a polynomial, for example x(x-2)(x+5)=0 , then it would be arithmetically correct to divide by any of the factors eg. 0 = x(x-2) could be considered equivalent and logical after dividing through by (x+5). But you'll have gotten rid of a solution, and thus altered the formula, and by carrying on further, you'll eventually end up with 1=0. The point is that wierd things happen at zero, even though they seem to make sense-you're essentially modifying the equation in a geometric sense, and if zero is the 'pivot', it's difficult to realize when the result becomes illogical.
Numsgil:
--- Quote from: Houshalter ---Huh. You made me think .
I had an idea today while thinking about this problem. Its simple: Zero isn't a number. Its a symbol, a visual aid to represent a concept. Its not an actual number. We invented it as a useful place holder so that we didn't have to invent a new symbol for 10, 50, 100, etc, like most ancient people did. Think about it, it litteraly represents the lack of a number, not a number itself. Its own properties conflict with the properties numbers have. Take any number and add zero, it doesn't change because zero isn't a number. Its like not adding in the first place. Take an number and multiply it by zero, you get nothing because you litteraly take that number zero times, like not taking it at all. You can't have zero of something, because if you did its the same as not having it at all anyways. Its because zero is a symbol, not a number, it gets confusing when you treat it like one. Its like trying to dividing 1,000 by the at (@) symbol. It just doesn't make sense. So its no wonder what happens when you try to find out how many times nothing fits into something, even if that something is nothing in the first place. It can't be done.
--- End quote ---
You can, by definition, remove 0 from the real numbers. But then you have an unfortunate problem with addition, since there wouldn't be an additive identity.
Which means real numbers under addition wouldn't be a "group", because it wouldn't have closure or an identity element. Without group status it loses a lot of the nice properties we need and expect in algebra, so it suddenly becomes much less useful.
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