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Because dividing by 0 is a logical paradox, akin to "This statement is False". Sure x=1/0 looks like a valid syntactic construction, it doesn't comply with the
by tekromancr 9y ago
Because dividing by 0 is a logical paradox, akin to "This statement is False". Sure x=1/0 looks like a valid syntactic construction, it doesn't comply with the logical system it exists within. Try looking at it using elementary algebra;
x=1/0 must be equivalent to x0=1. And since one of the axioms on which the system is built is that any value taken zero times must be zero, we know that x0=1 is a false statement for any value of x.
- soperj 9y agoNow try looking at square root of -1 using elementary algebra...
- tprice7 9y agoThe two situations are not analogous, there is a crucial difference. Like tekromancr was getting at, you get a logical contradiction if you assume the existence of 1/0, if you also assume the ring axioms. To elaborate, if 0x = 1, then 0 = 1. It is also an axiom that 0 != 1, and even if you abandon this axiom you only get one extra ring that only has one element. Boring! You also get a logical contradiction if you add a square root of -1, if you are assuming, say, the ordered ring axioms. But you can lose the order structure and then there is no more contradiction. If you want to add a new number x with 0x = 1, then you also need to propose how one might alter the notion of a ring so that you don't get a logical contradiction, and you are still left with something interesting. Just like how we altered the notion of an ordered ring, by removing the ordering, so that you don't get a contradiction when adding a square root of -1. So the crucial difference is, as far as I know, nobody has succeeded in doing this when it comes to division by zero.