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 Posted: Wed Oct 11, 2006 4:58 am
Lieutenant_Charon AvenirLegacy A Lost Iguana Lieutenant_Charon So 0/0 = 0. Not "undefined" or "infinity." Unfortunately, mathematics is a system of logic and only needs to be follow from the axioms and be deductively valid. It's 1/0 that makes your head explode, yet 0/0 makes perfect mathematical and logical sense. pirate sweatdrop The formula's made my head explode, logic is something I believe I have a good grasp of. Actually, 1/0 is a bit more complicated, because it's having something, and dividing it into no parts. But that would also (technically, and I suppose logically) remove the part that contained the something, so I suppose you'd be left with 0 there as well. On the other hand, where did your something go? Same follows with any number over 0, except of course 0, since logic should be able to solve that. It is not about dividing into parts, it is about the inverse of a pre-defined mathematical relation. Real numbers are elements of a [closed?] field which have addition and multiplication defined as Associative: a (b c) = (a b) c ; a+(b+c) = (a+b)+c [for all members]Commutive: a b = b a ; a+b = b+a [for all members]Distributive: a (b + c) = a b + a c [for all members]There exists identity elements for both multiplication (1) and addition (0) that leave the number unchanged for all members. Addition: a + 0 = a; multiplication: 1a = aThe existence of an additive inverse, i.e. a + (-a) = 0 [for all members]The existence of a multiplicative inverse, i.e a (a^{-1}) = 1 [for all members except zero] If you allow division by zero to be defined then it has knock-on effects for all the other operations. There are other sets of real numbers that do define division by zero. One extension [affinely extended reals] is to define x/0=+∞ and -x/0=-∞ [where x>0]. This then means you need to come up with axioms about how the new +∞ and -∞ elements behave under addition and multiplication. Another way [projectively extended reals] is to define -∞=+∞ and that x/0=∞ for all x. This makes your number line become a circle, which I think is cute.  http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.htmlhttp://mathworld.wolfram.com/ProjectivelyExtendedRealNumbers.html[For any watching mathematician that wants to correct my simplifications — seeing as I'm only a physicist and I know that there are better ways to define the reals — then do so by all means]
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Posted: Wed Oct 11, 2006 9:52 pm
Huh, I suppose that's why I dislike math.
Logic does not apply.
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Posted: Fri Oct 13, 2006 3:51 am
Lieutenant_Charon Huh, I suppose that's why I dislike math. Logic does not apply. That's not quite fair. Maths is very logical, maybe it is not so much intuitive and easy to grasp [at least for me] but there is a lot of formal logic.
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Posted: Sat Oct 14, 2006 10:54 pm
Lieutenant_Charon Huh, I suppose that's why I dislike math. Logic does not apply. -[ domokun ]-
Logic applies. It's a different sort of logic than other sciences though. Math is more absolute than anything else until you find unknowns like infinity. Only then does math become illogical.
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Posted: Mon Oct 16, 2006 8:26 pm
Lieutenant_Charon Huh, I suppose that's why I dislike math. Logic does not apply.
Math is extremely logical. Except for the fact that I cannot comprehend the logics of math at times, each mathematical fact has a logicial and resonable explanation to back it up. Hence, those scary annoying proofs I'm doing in honors geometry. Of course, when math cannot explain something like 0 and infinity, only then does it really become illogical. But most of it is logical, when you think about it at least.
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Posted: Fri Oct 20, 2006 7:12 am
Azn Mochi Gurl Lieutenant_Charon Huh, I suppose that's why I dislike math. Logic does not apply.
Math is extremely logical. Except for the fact that I cannot comprehend the logics of math at times, each mathematical fact has a logicial and resonable explanation to back it up. Hence, those scary annoying proofs I'm doing in honors geometry. Of course, when math cannot explain something like 0 and infinity, only then does it really become illogical. But most of it is logical, when you think about it at least.
Sure you could say that. But then there is the probability that the logic in which Math sets it's foundation could be fundamentally wrong. I.E The way Math and it's logic view and describe the world about us is wrong. =O But that chances of that being true...much less proven.... >_>
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Posted: Mon Oct 23, 2006 7:00 am
AvenirLegacy Darken_mortal No, zero isn't equil to infinity. Well 0 is a numerical digit like Infinity. But thats about it. xp No way dude. Infinity is NOT a number. It is not a number. I want to say it is undefined, but I'm not sure I can get away with that. When you get infinity as an answer it means nonsense. Either that answer doesn't exist or you did something wrong. Infinity is a concept to help us understand certain phenomena, it is not a numerical value.
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Posted: Mon Oct 23, 2006 7:03 am
AvenirLegacy Lieutenant_Charon AvenirLegacy So I did some research on the number 0. Here's what I dug up. ""0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. Zero is the last digit to be incorporated in most numeral systems. In the English language, zero may also be called nil when a number, o/oh when a numeral, and nought/naught in either context."" "" * Multiplication: x · 0 = 0 · x = 0. * Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse, a consequence of the previous rule. For positive x, as y in x / y approaches zero from positive values, its quotient increases toward positive infinity, but as y approaches zero from negative values, the quotient increases toward negative infinity. The different quotients confirms that division by zero is undefined. * Exponentiation: x0 = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0x = 0. The expression "0/0" is an "indeterminate form". That does not simply mean that it is undefined; rather, it means that if f(x) and g(x) both approach 0 as x approaches some number, then f(x)/g(x) could approach any finite number or ∞ or −∞; it depends on which functions f and g are. See L'Hopital's rule."" I find this last part very interesting, as it suggests that 0/0 is indeed some sort of value. The problem is is that this value has to real definition, but it must exists as according to L'Hopital's rule [which I checked out] The derivative of such a equation must exists in calculus. ""In simple cases, L'Hôpital's rule states that for functions f(x) and g(x): if or then where the prime (') denotes the derivative. Among other requirements, for this rule to hold, the limit must exist "" Getting past the sheer meaning of the calculus, I came to understand that for the derivative of this interger to exist, there must be a value for 0/0. Since the integer can come to be any finite number [as stated in the law], or either of the two infinities. Cite/Sources The number 0.L'Hopital's rule.EDIT: To link this to the OP's post, the sources I mentioned provide no arguement as to why 0x0 would not = 0. It seems its just a mathematical law. My head espolode. All it is sayins is that since the Derivative of 0/0 exists, then 0/0 must have a value. [ L'Hopital's Law ]. Whoa. No way again. A derivative of 0/0 does not exist. When you take derivatives you are no longer looking at the value of the actual function, derivatives give you an approximation to the slope of the function at a certain point. It's been few years since I've done limits and L'Hopital's rule but being able to find a value for the limit of 0/0 does not imply that 0/0 has a value. The limit and the value are not the same. I'm not sure if this is clear, but I'll come back and repost when I get my thoughts together.
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Posted: Mon Oct 23, 2006 11:26 am
linxx85 AvenirLegacy Lieutenant_Charon AvenirLegacy So I did some research on the number 0. Here's what I dug up. ""0 (zero) is both a number — or, more precisely, a numeral representing a number — and a numerical digit. Zero is the last digit to be incorporated in most numeral systems. In the English language, zero may also be called nil when a number, o/oh when a numeral, and nought/naught in either context."" "" * Multiplication: x · 0 = 0 · x = 0. * Division: 0 / x = 0, for nonzero x. But x / 0 is undefined, because 0 has no multiplicative inverse, a consequence of the previous rule. For positive x, as y in x / y approaches zero from positive values, its quotient increases toward positive infinity, but as y approaches zero from negative values, the quotient increases toward negative infinity. The different quotients confirms that division by zero is undefined. * Exponentiation: x0 = 1, except that the case x = 0 may be left undefined in some contexts. For all positive real x, 0x = 0. The expression "0/0" is an "indeterminate form". That does not simply mean that it is undefined; rather, it means that if f(x) and g(x) both approach 0 as x approaches some number, then f(x)/g(x) could approach any finite number or ∞ or −∞; it depends on which functions f and g are. See L'Hopital's rule."" I find this last part very interesting, as it suggests that 0/0 is indeed some sort of value. The problem is is that this value has to real definition, but it must exists as according to L'Hopital's rule [which I checked out] The derivative of such a equation must exists in calculus. ""In simple cases, L'Hôpital's rule states that for functions f(x) and g(x): if or then where the prime (') denotes the derivative. Among other requirements, for this rule to hold, the limit must exist "" Getting past the sheer meaning of the calculus, I came to understand that for the derivative of this interger to exist, there must be a value for 0/0. Since the integer can come to be any finite number [as stated in the law], or either of the two infinities. Cite/Sources The number 0.L'Hopital's rule.EDIT: To link this to the OP's post, the sources I mentioned provide no arguement as to why 0x0 would not = 0. It seems its just a mathematical law. My head espolode. All it is sayins is that since the Derivative of 0/0 exists, then 0/0 must have a value. [ L'Hopital's Law ]. Whoa. No way again. A derivative of 0/0 does not exist. When you take derivatives you are no longer looking at the value of the actual function, derivatives give you an approximation to the slope of the function at a certain point. It's been few years since I've done limits and L'Hopital's rule but being able to find a value for the limit of 0/0 does not imply that 0/0 has a value. The limit and the value are not the same. I'm not sure if this is clear, but I'll come back and repost when I get my thoughts together. That is the rule and that is how it is stated. :nod:
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Posted: Mon Jul 09, 2007 10:10 am
Quote: Whoa. No way again. A derivative of 0/0 does not exist. When you take derivatives you are no longer looking at the value of the actual function, derivatives give you an approximation to the slope of the function at a certain point. It's been few years since I've done limits and L'Hopital's rule but being able to find a value for the limit of 0/0 does not imply that 0/0 has a value. The limit and the value are not the same. I'm not sure if this is clear, but I'll come back and repost when I get my thoughts together.
But the baseline is, the limit of a function is not a value of it. The simplest of examples: 1/x->0, but there is no value of x to which 1/x=0. The only functions which take the value of their limits are constant ones, I think...
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