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If it didn't exist then we'd invent it. Pretty much like we invented sqrt(-1)...
by d--b 9y ago
If it didn't exist then we'd invent it. Pretty much like we invented sqrt(-1)...
- westoncb 9y agoWe already invented it in that same sense—it was just a more obviously useful thing to invent ;)
- soperj 9y agowhy haven't we invented divide by 0 then?
- d--b 9y agoWe probably would have if it proved to be useful to anything.
- tim333 9y agoJavascript has it. 1/0==Infinity
- tekromancr 9y agoBecause 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.
- dragonwriter 9y agoWe did, see IEEE 754.
- opportune 9y agoThe non-handwavy answer is that mathematics is defined by a set of axioms through which all mathematics follow via logic and defining new structures / relations. If there were no 6, given the same set of axioms we could simply define something representing the same concept of 6 with a different symbol, because the concept of 6 is consistent with our axioms. Dividing by 0 is inconsistent with our axioms, so even if we define x = 1/0, we would be able to arrive at conclusions which contradict our axioms.
- cperciva 9y agoWe have... in a sense. There are number systems wherein 0 is treated rather like dx in terms of being a formal entity upon which algebraic manipulations can be performed; in such systems we can say for example that sin(0)/0 = 1 while sin(0^2)/0 = 0.
- tprice7 9y agoDo you have a reference for this? I would believe you if you said "an infinitesimally small value" rather than 0. If it's true then I'm very surprised. For one, 0 = -0, so if sin(0)/0 = 1, then -1 = -sin(0)/0 = sin(0)/-0 = sin(0)/0 = 1
- cperciva 9y agoI'm talking about various forms of infinitesimals, yes. In a sense they're zero, but in another sense they're not. For the reasons you mention, these models have 0 != -0. :-)
- tprice7 9y agoBut don't these models all have a true zero, distinct from the infinitesimals, which is its own negative?
- chongli 9y agoBecause we don't need it: we have limits! The constant i and its associated field, ℂ, is needed so we can solve the half of the quadratic equations which have no real roots. Dividing by zero has no use case (that I know of) that isn't covered by taking the limit as x->0 of 1/x.
- kobeya 9y agoSee: limits in calculus.
- edanm 9y agoWe can if we want. The problem is that there is no useful way to define division by zero, and have it work well with all the other rules of arithmetic that we want to continue working.