
In mathematics, a division (mathematics) is called a division by zero if the divisor is 0 (number). Such a division can be formally expressed as , where ''a'' is the dividend. Whether or not this expression can be assigned a meaningful (well-defined) value depends upon how the expression is interpreted. ==Algebraic interpretation== It is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations. Under the standard rules for arithmetic on integers, rational numbers, real numbers and complex numbers, the value of a division by zero is undefined. The reason is that division (mathematics) is defined to be the inverse operation of multiplication. This means that the value of : is the solution ''x'' of the equation : whenever such a value exists and is unique. Otherwise the expression is undefined. For ''b'' = 0, the equation ''bx'' = ''a'' can be rewritten as 0''x'' = ''a'' or simply 0 = ''a''. Thus, in this case, the equation ''bx'' = ''a'' has ''no solution'' if ''a'' is not equal to 0, and has ''any'' ''x'' as a solution if ''a'' equals 0. In either case, is undefined. Conversely, for the number systems mentioned above, the expression is ''always'' defined if ''b'' is not equal to zero. ===Fallacies based on division by zero=== It is possible to disguise a division by zero in an algebraic argument, leading to invalid proofs that 2 = 1 such as the following: * For any real number ''x'': :: * Factoring both sides in two different ways: :: * Dividing both sides by ''x'' − ''x'': :: * Since this is valid for any value of ''x'', we can plug in ''x'' = 1. :: The fallacy is in the assumption that division by ''x'' − ''x'' = 0 is defined. In practice, division by a term in any algebraic argument will require either an explicit assumption that the term is not zero, or a separate justification showing that the term can never be zero. ===Abstract algebra=== Similar statements are true in more general algebraic structures, such as ring (mathematics)s and field (mathematics)s. In a field, every nonzero element is invertible under multiplication, so as above, division only poses problems when attempting to divide by zero. However, in other rings, division by nonzero elements may also pose problems. Consider, for example, the ring Z/6Z of integers mod 6. What meaning should we give to the expression : This should be the solution ''x'' of the equation : But this equation has two distinct solutions, ''x'' = 1 and ''x'' = 4, so the expression is undefined. The problem occurs because 2 is not invertible under multiplication. == Limits and division by zero == At first glance it seems possible to define by considering the limit (mathematics) of as ''b'' approaches 0. For any nonzero ''a'', it is known that : and : Therefore, we might consider defining as for positive ''a'', and for negative ''a''. However, this definition is not generally useful, because positive and negative infinity are not real numbers, and the equation : still has no solution for any finite ''a''. Furthermore, there is no obvious definition of that can be derived from considering the limit of a ratio. The limit : does not exist. Limits of the form : in which both ''f(x)'' and ''g(x)'' approach 0 as ''x'' approaches 0, may converge to any value or may not converge at all. See l'Hopital's rule for discussion and examples of limits of ratios. ==In mathematical analysis== In distribution theory one can extend the function : to a distribution on the whole space of real numbers (in effect by using Cauchy principal values). It does not, however, make sense to ask for a 'value' of this distribution at ; a sophisticated answer refers to the singular support of the distribution. ==Other number systems== Although division by zero is undefined with real numbers and integers, it is possible to consistently define division by zero in other mathematical structures, for instance on the Riemann sphere (see also pole (complex analysis) in complex analysis). In hyperreal numbers and surreal numbers, division by non-zero infinitesimals is possible. If a number system forms a commutative ring, as do the integers, the real numbers, and the complex numbers, for instance, it can be extended to a wheel theory in which division by zero is always possible, but division has then a slightly different meaning. == Division by zero in computer arithmetic == IEEE floating-point standard specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. In IEEE 754 arithmetic, ''a''/0 is positive infinity when ''a'' is positive, negative infinity when ''a'' is negative, and NaN ("not a number") when ''a'' = 0. These definitions are derived from the properties of limits of ratios, as discussed above. At present, IEEE 754 is the most common floating point specification, as it is implemented by Intel processors and others. Integer division by zero may be handled differently than floating point. Intel processors generate an interrupt when an attempt is made to divide an integer by zero. The usual result is aborting whatever program it occurred in. To ensure that every operation yields a finite, numerical result (floating point), and avoids an interrupt (integer), a computer program may refuse to execute a division if the divisor is zero. == Other == *Division by Zero is a company that designs computer games. [http://www.division-by-zero.de/dbz/index.asp] Elementary arithmetic Computer arithmetic Fractions
''In particular, it is incorrect to say that a ÷ 0 is infinity. The argument that any number a, divided by a very small one, becomes extremely large is unconvincing: a negative number a divided by a small positive number does not become large, and neither does a positive number a divided by a small negative number.'' Fine. But how about, a ÷ 0 is infinity if a is positive, and negative infinity if a is negative? I have no idea what 0 ÷ 0 would be, of course. User:Evercat 23:19 6 Jul 2003 (UTC) :(Disclaimer: IANAM; poorly remembered high school algebra follows!) Lemme try to show why that doesn't work (aside from the definitional problem already explained in the article)... If you plot y = a/x (where a is positive) you'll see the trouble: |* <- approaching 0 from positive x |* we get a limit of positive infinity |* | * | * | ** | ************** ------------------+------------------ ************** | ** | * | * | *| *| approaching 0 from negative x *| <- we get a limit of negative infinity :If a is negative, the curve is upside-down but has the same split. a/x approaches positive infinity when approaching x=0 from negative x, and negative infinity when approaching from positive x. :If we try to just come straight out with x=0, we can't fake it with the limit, since the curve is discontinuous. There's no reason to favor positive infinity over negative infinity, whatever the sign of a. :Now, depending on what you're doing, positive or negative infinity may be a useful way to ''treat'' a divide-by-zero case, but which is appropriate probably depends on which side of 0 you're approaching from in the denominator, as well as the sign of the numerator. :It may be useful to have a pretty graph in the article to illustrate this, as the quoted sentence tries to say it in words and I think is even more confusing than my attempt here. :) --User:Brion VIBBER 23:45 6 Jul 2003 (UTC) ::What you say may be insightful (and the italicized passage quoted at the beginning of this thread mentions this briefly.) However, I don't think a lengthier discussion than the passage above is warranted, and this why. ::The reasoning that this explanation attempts to demolish only applies to a problem which is tangential to the question "what's 1/0?" What the above does say is: "the limit of 1/x as x tends to zero does not exist, even in the extended sense when limits are allowed to be +∞ or -∞." ::The remainder of the article makes this clear. Division is the inverse of multiplication. 0*x is 0 for any x, therefore 0*x isn't invertible (because 0*x isn't injective.) This is the core of the matter. ::When you ask a politician a question they don't like ("will you provide transmogrifiers to all the citizens?") their reply is sometimes tangential, sometimes even nonsensical ("I will work so that all citizens are treated justly. My opponent doesn't support the fluxification of space, but I will put a flux on the moon if it kills me. Remember the Alamo.") You then have two choices. You can take that long statement, and make a longer statement explaining how that's nonsense and doesn't answer the question, or you can try to give a more correct answer. To the question "what is 1/0?", the answer "if x is small and positive, 1/x is large, thus 1/0 is +∞" is at best tangential, and at worst nonsensical, and I don't think it's worth a lengthy rebuke. ::Since it is in fact a widespread fallatious reasoning, it is worth mentioning as it is now, but wasting electrons on a long exposition about what is essentially bollocks (as I'm doing now) isn't ideal. ::Ok, I'll shut up now. -- User:Loisel 05:16 7 Jul 2003 (UTC) ----------------------------------------- Wow. all that over a simple idea like 1/0. Seriously, guys, I always made the assumption that x/0 was either 0 or null since if you have any quantity "x" and you place equal fractions of x in a number of containers equal to 0 then how many pieces of x are there in each containter? Zero. But, you actually don't even have any containers, so really "null". And I never believed in the concept of infinity so the other statements about 1/0 = ∞ were insane at the get-go. Regardless, the question I always wanted to know was: what's the practical application (if any) of the concept of x/0 anyway? No mathematician has ever been able to answer that, and I doubt they ever would. :Um, integration. User:Morwen 22:51, Dec 18, 2003 (UTC) ::While true, one must wonder if this is just a result of our mathematical system. If x*0 is 0 (defined), why then isn't 0*0=x ?(as it should be if you reverse the process) Yet, x/0=(undefined) because the undefined * 0 also doesn't = x. Almost a little broken at points. Maybe one day (certainly not in my lifetime) it will be fixed. :::I'm not sure if I understand what you mean; modifying x*0 = 0 would lead to 0/0 = x, which makes x undefined, which is why one can't divide by zero. Er.. I dunno. User:Evil saltine 23:01, 18 Dec 2003 (UTC) ---- I look at this ''question'' as is being similar to how for example : : yet : the bold W meaning whatever or simply "?" indicating ''unknowledge''. Using sets, however, it can be argued that : Should this analogy be applied to division by zero : such that : In other words zero divided by zero is the most ambigious ''number''; it could be any number of any type: scalar, vector, or tensor; set or matrix; real or imaginary; differential (Rn); albiet combinations thereof. Zero is an amazing number. As for the division of any nonzero number by zero, : this being due in part, if not mainly, to the fact that there is more than one ''infinity'': positive, negative, imaginary, complex; others perhaps. I recommend avoiding the usage of the words "possible" or "impossible" in formulating conjectures about division by zero. To be a step ahead of human knowledge, one should acknowledge the possiblity of that which we accept as true being false. --User:Lindberg G Williams Jr 19:30, 22 May 2004 (UTC) == 0/0=0 == Try reading http://members.lycos.co.uk/zerobyzero and see how certain logical approach can result in solution to real life problems otherwise un solvable due to your error theory. 0 / 0 = 0 Try this simple formula for average speed. Total Distance Traveled / Hours Traveled = Average Speed 10 Kilometers / 2 Hours = 5 Kilometers Per Hour What happens when you're not traveling 0 Kilometers / 0 Hours = 0 kl/hr 0/0=0 There is no other solution. We have created a world of limited mathematics, eliminated the limit of zero, and told the world it was unlimited, providing you did not divide by zero which limited it. Seems you all forget there are other applications and formulas that ABSOLUTELY NEED IT. Anyhow that is just one of thousands of them. ZBZ :Well, so much for the theory that 0/0 belongs here rather than at indeterminate form. User:Charles Matthews 18:25, 24 Sep 2004 (UTC) : world of limited mathematics? User:CSTAR 19:44, 24 Sep 2004 (UTC) This is actually a good example of why 0/0 is not defined! If you go at 0 kilometers for any amount of time (say 1 hour), the 0/1=0, as you would expect. Now imagine you are going at X km/hr, for any X. How far will you go in 0 hours? you will go 0 kilometres. Therefore 0 kilometers / 0 hours = X km/hr for any X! Think of it this way, - say you ran 20 kilometres. To define the speed, if you ran over it in 2 hours, it would be 10 km/h. If I took a "snapshot" of a zero-second timeframe (nil), and you ran 0 km during this period, it does not mean your speed is 0 km/h. Because the timeframe taken is insufficient/invalid and isn't an accurate reflection of actual speed, and therefore the value given is undefined because it could be anything. -- User:Natalinasmpf 09:41, 25 Apr 2005 (UTC) == let me end this dumb attractor == Two two branches converge at ∞. ∞ is nonpositive and nonnegative, just like 0. There is no sense in equivocating its sign with its side; there are only ∞+ and ∞-, not +∞ and -∞. So 1/0 = (1)R∞, where R is any real number. Zero distance over zero time does ''not'' produce zero speed; it produces ''any'' speed. If any mathematicians disagree with me, and they do because they don't know me, they're wrong. User:Lysdexia 19:44, 13 Oct 2004 (UTC) == my 2 cents == I look at this question as essentially one of general topology, in which case the answer depends on which topology you choose. In this case, "1/0" is just shorthand for "what is the limit of 1/x as x approaches 0?" It depends on the space and topology. If you're in the ''real numbers'' R, then 1/0 doesn't "equal" anything, since 1/x has no limit as x approaches zero. If you throw in plus and minus infinity as separate points (the so-called "extended real number system") with the obvious neighbourhoods, then it still doesn't have a limit, so 1/0 still is meaningless. If, however, you identify plus and minus infinity to a single point, then it's perfectly legitimate to say that 1/0 = infinity. The same thing goes for the case of the Riemann sphere. The limit of 1/z as z approaches infinity (z ''complex'') is infinity. Here, infinity is a well-defined "point" that exists just as surely as 0, 1, or pi. My whole point is, it doesn't really make much sense to ask, "what does 1/0 mean??". It's a meaningless question, a red herring. Stop whoever is asking it, and demand that they ask a more well-defined question which has an answer. User:Revolver 09:11, 15 Oct 2004 (UTC) : (1) I guess I see it as an algebraic problem -- that is, it has mostly to do with sets & operations. It's not necessary to bring limits into play; the motivation to do that is to attempt to guess a value for 1/0, but whether the assigned value was found by a limit or some other means is ultimately irrelevant, since what you want to investigate is whether 1/0 = x implies 1 = 0 x and that sort of algebraic thing. A beneficial side effect is that the article becomes accessible to anyone with some knowledge of algebra, if we can avoid making limits central to the discussion. (2) I'm not sure that assigning 1/0 = infinity on a Riemann sphere solves the problem entirely. Since (presumably) a/0 also = infinity for any nonzero a, one can't conclude a = 0 infinity. But perhaps there is more that can be said here. (3) I'm not convinced that 1/0 = ? is any more "meaningless" than any other question in mathematics. In any event, "stop whoever is asking it" seems a counterproductive tactic for an encyclopedia. User:Wile E. Heresiarch 14:23, 15 Oct 2004 (UTC) Looking at the current article, it would be suitable to add a short section on the algebraic geometry case - where rational functions to a complete variety extend to honest functions on a blowing-up. This effectively says the maximum possible, about the case of polynomials which vanish. It also includes the Riemann sphere = complex projective line case, without privileging it unduly. User:Charles Matthews 14:49, 15 Oct 2004 (UTC) : Sounds good to me. Have at it! User:Wile E. Heresiarch 16:17, 15 Oct 2004 (UTC) Wile, by "stop whoever is asking it", I didn't mean "silence the person", I meant "demand whoever is speaking to ask a better question". I do think that "1/0 = ?" is a meaningless question, simply because, as our responses indicate, ''it has multiple interpretations''. I interpreted it as a topological question, you as algebraic. I don't think there's anyway for either of us to "prove" the other's interpretation is "wrong". This is precisely what I mean by "1/0 = ?" not being a "well-defined" question. You say that "what you want to investigate is whether 1/0 = x implies 1 = 0x, etc." but ''that is presupposing an algebraic interpretation''. Your interpretation comes from viewing the problem algebraically. Mine comes from viewing it from a calculus point of view (i.e. removing discontinuities, etc.) Charles' interpretation is even more broad and fundamental, and further supports my contention that the question must be recast before addressing it. As for the "1 = a for any nonzero a implies contradiction" argument, again that's ''presuming an algebraic perspective''. There is no contradiction from the calculus perspective, in fact in complex analysis classes, the function 1/z is often defined from the Riemann sphere to itself, in the usual way for finite complex numbers, with the stipulation that 1/0 = infinity and 1/infinity = 0. This is not a statement about binary algebraic operations, but a statement about extending the function 1/z on C to be continuous on the Riemann sphere. User:Revolver 21:22, 15 Oct 2004 (UTC) : Well, I could quibble about various things. Instead I'll just ask how you would like to improve the article. I think that will help us stay focused. Regards, User:Wile E. Heresiarch 22:36, 16 Oct 2004 (UTC) OK, time for some mathematics. For X and Y compact spaces, and F: X → Y a function but not everywhere defined, one can take the graph G of F and its closure G* in XxY. The main idea is to project G* back onto X and say it must be a closed set. Which is true if spaces are Hausdorff. For something like 1/x this suggests Y should be some compactification. What we are looking for is that G* should actually be the graph of a function. In this purely topological case one can't really argue that compactifying the real line to the extended real line is better or worse that to a circle (i.e. we could add two points or one to compactify R). What the algebraic geometers do is a bit more definite, in that projective space acts as a compactification (non-Hausdorff, but everything is covered by the complete variety/proper morphism properties of projective space, plus the fact that the Zariski topology on XxY is not defined as the product topology). The added feature here is that there is more control of G* when it has a 'vertical component' projecting down to X. User:Charles Matthews 08:22, 16 Oct 2004 (UTC) : OK, thanks for the information. How would you like to use this to improve the article? User:Wile E. Heresiarch 22:36, 16 Oct 2004 (UTC) I would improve it by first noting that there are multiple ways of addressing the problem. Then, I would address it by interpretation, starting with the algebraic (binary operation) way. This includes much of the article (e.g. "incorrect arguments in dividing by zero" is essentially a misuse of algebra) and is easiest to understand. Then, maybe a mention of the other interpretations, but clearly set off in other sections. Again, I don't object to what's there, or object to the usual algebraic interpretation, just to its priveledged status. User:Revolver 22:50, 16 Oct 2004 (UTC) : Well, it doesn't seem like what you want is all that different from what's already in the article. Why not just go ahead and make the changes you want -- at least it will give us something different to talk about. User:Wile E. Heresiarch 16:36, 17 Oct 2004 (UTC) I worry a bit about going back and forth between "concrete" and "abstract" discussions, that it might be something that only a relatively few who read it will be interested in. (Or worse, that it might confuse... "2*4 = 2"?) The only way to do this seems to be to break up the organisation by interpretation. User:Revolver 06:12, 19 Oct 2004 (UTC) == 0/0 and differentiation == I can see a great deal of serious thought went into this article and it is extremely well and formally reasoned. However, it perpetuates the (formally accepted) statement that 0/0 is "just as" undefined as, say, 127/0. I'm quite sure this is formally correct, and I'll thank you not to spank me for saying it is not correct ''at a lower level'', somewhere down among primitive life-forms such as engineers. Here's the text: ''Limits of the form'' : ''in which both'' f(x) and g(x) ''approach 0 as'' x ''approaches 0, may converge to any value or may not converge at all.'' Okay, this is all true, but still, what you are doing, in a sneaky sort of sliding-up fashion, is 0/0. Please don't throw a blizzard of complicated math at my statement; I know what you're after. My point is that 0/0 is what differentiation ''feels'' like when I imagine the limit approaching. It is useful -- to me, if to nobody else -- to believe in 0/0 as the limit: the point at which anything becomes possible. Look at this from the amphibian point of view of an engineer. Think of the tangent to the curve as a long, massive stick; at an initial approximation, it is fixed to the curve at two points, (x,y) and (x1,y1). These points are widely spaced and the tangent does not wiggle. But, as the limit is approached, the distance between the two points decreases and the stick is liable to wiggle more and more; at the limit itself, the tangent is only attached to the curve by a single point and is free to rotate throughout a complete circle. However, we know better than to apply any torque to the tangent-stick as we approach the limit; we ''very carefully'' edge (x1,y1) up to (x,y), so when we finally reach the limit, the tangent is left pointing in the correct direction. All I'm saying here is that it is a useful tool to the student to ''suspend disbelief'' and ''imagine'' 0/0 = the tangent. It makes differentiation real. By the same token, when integration is demonstrated by area under a curve, first numerical integration approximates the area as a large number of thin slices. As the limit is approached, the student can ''think'' of an infinite number of zero-width slices, which add up to the total area. This is ∞ × 0 = the area. This may not be formally correct, but it is a good and useful way to ''think'' about it. I remember I once listened to my 8th grade math teacher lecture my class on this subject. He said it was "impossible" to divide any number by zero, since no number could be multiplied by zero to yield anything except zero. I raised my hand and said, What about 0/0? He said that was "undefined" and looked pissed off. I accepted that 0/0 was mathematical nonsense. Later, when I got into the calculus, I had a tough time of it, despite being very good at algebra, until I freed my mind and allowed myself to simply imagine the process as a clever way of defining the undefined. I would like to see if there is anyone bold enough to step away a moment from the purely theoretical and formally correct. Who will join me in wondering where these thoughts belong? I truly believe they are of value to the beginning student of the calculus, who is ill-served by adamant denial of 0/0. --User:Xiong 06:27, 2005 Mar 11 (UTC) ==Difference between 1/0 and 1/infinitesimal and x/0== Can someone explain in simple terms, if 1/0, 1/infinitesimal, or 1000000/0 or 1000000/infinitesimal or any different?
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Division_by_zero
Division_by_zero
Division by zero
In mathematics, a division (mathematics) is called a division by zero if the divisor is 0 (number). Such a division can be formally expressed as , where ''a'' is the dividend. Whether or not this expression can be assigned a meaningful (well-defined) value depends upon how the expression is interpreted. ==Algebraic interpretation== It is generally regarded among mathematicians that a natural way to interpret division by zero is to first define division in terms of other arithmetic operations. Under the standard rules for arithmetic on integers, rational numbers, real numbers and complex numbers, the value of a division by zero is undefined. The reason is that division (mathematics) is defined to be the inverse operation of multiplication. This means that the value of : is the solution ''x'' of the equation : whenever such a value exists and is unique. Otherwise the expression is undefined. For ''b'' = 0, the equation ''bx'' = ''a'' can be rewritten as 0''x'' = ''a'' or simply 0 = ''a''. Thus, in this case, the equation ''bx'' = ''a'' has ''no solution'' if ''a'' is not equal to 0, and has ''any'' ''x'' as a solution if ''a'' equals 0. In either case, is undefined. Conversely, for the number systems mentioned above, the expression is ''always'' defined if ''b'' is not equal to zero. ===Fallacies based on division by zero=== It is possible to disguise a division by zero in an algebraic argument, leading to invalid proofs that 2 = 1 such as the following: * For any real number ''x'': :: * Factoring both sides in two different ways: :: * Dividing both sides by ''x'' − ''x'': :: * Since this is valid for any value of ''x'', we can plug in ''x'' = 1. :: The fallacy is in the assumption that division by ''x'' − ''x'' = 0 is defined. In practice, division by a term in any algebraic argument will require either an explicit assumption that the term is not zero, or a separate justification showing that the term can never be zero. ===Abstract algebra=== Similar statements are true in more general algebraic structures, such as ring (mathematics)s and field (mathematics)s. In a field, every nonzero element is invertible under multiplication, so as above, division only poses problems when attempting to divide by zero. However, in other rings, division by nonzero elements may also pose problems. Consider, for example, the ring Z/6Z of integers mod 6. What meaning should we give to the expression : This should be the solution ''x'' of the equation : But this equation has two distinct solutions, ''x'' = 1 and ''x'' = 4, so the expression is undefined. The problem occurs because 2 is not invertible under multiplication. == Limits and division by zero == At first glance it seems possible to define by considering the limit (mathematics) of as ''b'' approaches 0. For any nonzero ''a'', it is known that : and : Therefore, we might consider defining as for positive ''a'', and for negative ''a''. However, this definition is not generally useful, because positive and negative infinity are not real numbers, and the equation : still has no solution for any finite ''a''. Furthermore, there is no obvious definition of that can be derived from considering the limit of a ratio. The limit : does not exist. Limits of the form : in which both ''f(x)'' and ''g(x)'' approach 0 as ''x'' approaches 0, may converge to any value or may not converge at all. See l'Hopital's rule for discussion and examples of limits of ratios. ==In mathematical analysis== In distribution theory one can extend the function : to a distribution on the whole space of real numbers (in effect by using Cauchy principal values). It does not, however, make sense to ask for a 'value' of this distribution at ; a sophisticated answer refers to the singular support of the distribution. ==Other number systems== Although division by zero is undefined with real numbers and integers, it is possible to consistently define division by zero in other mathematical structures, for instance on the Riemann sphere (see also pole (complex analysis) in complex analysis). In hyperreal numbers and surreal numbers, division by non-zero infinitesimals is possible. If a number system forms a commutative ring, as do the integers, the real numbers, and the complex numbers, for instance, it can be extended to a wheel theory in which division by zero is always possible, but division has then a slightly different meaning. == Division by zero in computer arithmetic == IEEE floating-point standard specifies that every floating point arithmetic operation, including division by zero, has a well-defined result. In IEEE 754 arithmetic, ''a''/0 is positive infinity when ''a'' is positive, negative infinity when ''a'' is negative, and NaN ("not a number") when ''a'' = 0. These definitions are derived from the properties of limits of ratios, as discussed above. At present, IEEE 754 is the most common floating point specification, as it is implemented by Intel processors and others. Integer division by zero may be handled differently than floating point. Intel processors generate an interrupt when an attempt is made to divide an integer by zero. The usual result is aborting whatever program it occurred in. To ensure that every operation yields a finite, numerical result (floating point), and avoids an interrupt (integer), a computer program may refuse to execute a division if the divisor is zero. == Other == *Division by Zero is a company that designs computer games. [http://www.division-by-zero.de/dbz/index.asp] Elementary arithmetic Computer arithmetic Fractions
Division by zero
''In particular, it is incorrect to say that a ÷ 0 is infinity. The argument that any number a, divided by a very small one, becomes extremely large is unconvincing: a negative number a divided by a small positive number does not become large, and neither does a positive number a divided by a small negative number.'' Fine. But how about, a ÷ 0 is infinity if a is positive, and negative infinity if a is negative? I have no idea what 0 ÷ 0 would be, of course. User:Evercat 23:19 6 Jul 2003 (UTC) :(Disclaimer: IANAM; poorly remembered high school algebra follows!) Lemme try to show why that doesn't work (aside from the definitional problem already explained in the article)... If you plot y = a/x (where a is positive) you'll see the trouble: |* <- approaching 0 from positive x |* we get a limit of positive infinity |* | * | * | ** | ************** ------------------+------------------ ************** | ** | * | * | *| *| approaching 0 from negative x *| <- we get a limit of negative infinity :If a is negative, the curve is upside-down but has the same split. a/x approaches positive infinity when approaching x=0 from negative x, and negative infinity when approaching from positive x. :If we try to just come straight out with x=0, we can't fake it with the limit, since the curve is discontinuous. There's no reason to favor positive infinity over negative infinity, whatever the sign of a. :Now, depending on what you're doing, positive or negative infinity may be a useful way to ''treat'' a divide-by-zero case, but which is appropriate probably depends on which side of 0 you're approaching from in the denominator, as well as the sign of the numerator. :It may be useful to have a pretty graph in the article to illustrate this, as the quoted sentence tries to say it in words and I think is even more confusing than my attempt here. :) --User:Brion VIBBER 23:45 6 Jul 2003 (UTC) ::What you say may be insightful (and the italicized passage quoted at the beginning of this thread mentions this briefly.) However, I don't think a lengthier discussion than the passage above is warranted, and this why. ::The reasoning that this explanation attempts to demolish only applies to a problem which is tangential to the question "what's 1/0?" What the above does say is: "the limit of 1/x as x tends to zero does not exist, even in the extended sense when limits are allowed to be +∞ or -∞." ::The remainder of the article makes this clear. Division is the inverse of multiplication. 0*x is 0 for any x, therefore 0*x isn't invertible (because 0*x isn't injective.) This is the core of the matter. ::When you ask a politician a question they don't like ("will you provide transmogrifiers to all the citizens?") their reply is sometimes tangential, sometimes even nonsensical ("I will work so that all citizens are treated justly. My opponent doesn't support the fluxification of space, but I will put a flux on the moon if it kills me. Remember the Alamo.") You then have two choices. You can take that long statement, and make a longer statement explaining how that's nonsense and doesn't answer the question, or you can try to give a more correct answer. To the question "what is 1/0?", the answer "if x is small and positive, 1/x is large, thus 1/0 is +∞" is at best tangential, and at worst nonsensical, and I don't think it's worth a lengthy rebuke. ::Since it is in fact a widespread fallatious reasoning, it is worth mentioning as it is now, but wasting electrons on a long exposition about what is essentially bollocks (as I'm doing now) isn't ideal. ::Ok, I'll shut up now. -- User:Loisel 05:16 7 Jul 2003 (UTC) ----------------------------------------- Wow. all that over a simple idea like 1/0. Seriously, guys, I always made the assumption that x/0 was either 0 or null since if you have any quantity "x" and you place equal fractions of x in a number of containers equal to 0 then how many pieces of x are there in each containter? Zero. But, you actually don't even have any containers, so really "null". And I never believed in the concept of infinity so the other statements about 1/0 = ∞ were insane at the get-go. Regardless, the question I always wanted to know was: what's the practical application (if any) of the concept of x/0 anyway? No mathematician has ever been able to answer that, and I doubt they ever would. :Um, integration. User:Morwen 22:51, Dec 18, 2003 (UTC) ::While true, one must wonder if this is just a result of our mathematical system. If x*0 is 0 (defined), why then isn't 0*0=x ?(as it should be if you reverse the process) Yet, x/0=(undefined) because the undefined * 0 also doesn't = x. Almost a little broken at points. Maybe one day (certainly not in my lifetime) it will be fixed. :::I'm not sure if I understand what you mean; modifying x*0 = 0 would lead to 0/0 = x, which makes x undefined, which is why one can't divide by zero. Er.. I dunno. User:Evil saltine 23:01, 18 Dec 2003 (UTC) ---- I look at this ''question'' as is being similar to how for example : : yet : the bold W meaning whatever or simply "?" indicating ''unknowledge''. Using sets, however, it can be argued that : Should this analogy be applied to division by zero : such that : In other words zero divided by zero is the most ambigious ''number''; it could be any number of any type: scalar, vector, or tensor; set or matrix; real or imaginary; differential (Rn); albiet combinations thereof. Zero is an amazing number. As for the division of any nonzero number by zero, : this being due in part, if not mainly, to the fact that there is more than one ''infinity'': positive, negative, imaginary, complex; others perhaps. I recommend avoiding the usage of the words "possible" or "impossible" in formulating conjectures about division by zero. To be a step ahead of human knowledge, one should acknowledge the possiblity of that which we accept as true being false. --User:Lindberg G Williams Jr 19:30, 22 May 2004 (UTC) == 0/0=0 == Try reading http://members.lycos.co.uk/zerobyzero and see how certain logical approach can result in solution to real life problems otherwise un solvable due to your error theory. 0 / 0 = 0 Try this simple formula for average speed. Total Distance Traveled / Hours Traveled = Average Speed 10 Kilometers / 2 Hours = 5 Kilometers Per Hour What happens when you're not traveling 0 Kilometers / 0 Hours = 0 kl/hr 0/0=0 There is no other solution. We have created a world of limited mathematics, eliminated the limit of zero, and told the world it was unlimited, providing you did not divide by zero which limited it. Seems you all forget there are other applications and formulas that ABSOLUTELY NEED IT. Anyhow that is just one of thousands of them. ZBZ :Well, so much for the theory that 0/0 belongs here rather than at indeterminate form. User:Charles Matthews 18:25, 24 Sep 2004 (UTC) : world of limited mathematics? User:CSTAR 19:44, 24 Sep 2004 (UTC) This is actually a good example of why 0/0 is not defined! If you go at 0 kilometers for any amount of time (say 1 hour), the 0/1=0, as you would expect. Now imagine you are going at X km/hr, for any X. How far will you go in 0 hours? you will go 0 kilometres. Therefore 0 kilometers / 0 hours = X km/hr for any X! Think of it this way, - say you ran 20 kilometres. To define the speed, if you ran over it in 2 hours, it would be 10 km/h. If I took a "snapshot" of a zero-second timeframe (nil), and you ran 0 km during this period, it does not mean your speed is 0 km/h. Because the timeframe taken is insufficient/invalid and isn't an accurate reflection of actual speed, and therefore the value given is undefined because it could be anything. -- User:Natalinasmpf 09:41, 25 Apr 2005 (UTC) == let me end this dumb attractor == Two two branches converge at ∞. ∞ is nonpositive and nonnegative, just like 0. There is no sense in equivocating its sign with its side; there are only ∞+ and ∞-, not +∞ and -∞. So 1/0 = (1)R∞, where R is any real number. Zero distance over zero time does ''not'' produce zero speed; it produces ''any'' speed. If any mathematicians disagree with me, and they do because they don't know me, they're wrong. User:Lysdexia 19:44, 13 Oct 2004 (UTC) == my 2 cents == I look at this question as essentially one of general topology, in which case the answer depends on which topology you choose. In this case, "1/0" is just shorthand for "what is the limit of 1/x as x approaches 0?" It depends on the space and topology. If you're in the ''real numbers'' R, then 1/0 doesn't "equal" anything, since 1/x has no limit as x approaches zero. If you throw in plus and minus infinity as separate points (the so-called "extended real number system") with the obvious neighbourhoods, then it still doesn't have a limit, so 1/0 still is meaningless. If, however, you identify plus and minus infinity to a single point, then it's perfectly legitimate to say that 1/0 = infinity. The same thing goes for the case of the Riemann sphere. The limit of 1/z as z approaches infinity (z ''complex'') is infinity. Here, infinity is a well-defined "point" that exists just as surely as 0, 1, or pi. My whole point is, it doesn't really make much sense to ask, "what does 1/0 mean??". It's a meaningless question, a red herring. Stop whoever is asking it, and demand that they ask a more well-defined question which has an answer. User:Revolver 09:11, 15 Oct 2004 (UTC) : (1) I guess I see it as an algebraic problem -- that is, it has mostly to do with sets & operations. It's not necessary to bring limits into play; the motivation to do that is to attempt to guess a value for 1/0, but whether the assigned value was found by a limit or some other means is ultimately irrelevant, since what you want to investigate is whether 1/0 = x implies 1 = 0 x and that sort of algebraic thing. A beneficial side effect is that the article becomes accessible to anyone with some knowledge of algebra, if we can avoid making limits central to the discussion. (2) I'm not sure that assigning 1/0 = infinity on a Riemann sphere solves the problem entirely. Since (presumably) a/0 also = infinity for any nonzero a, one can't conclude a = 0 infinity. But perhaps there is more that can be said here. (3) I'm not convinced that 1/0 = ? is any more "meaningless" than any other question in mathematics. In any event, "stop whoever is asking it" seems a counterproductive tactic for an encyclopedia. User:Wile E. Heresiarch 14:23, 15 Oct 2004 (UTC) Looking at the current article, it would be suitable to add a short section on the algebraic geometry case - where rational functions to a complete variety extend to honest functions on a blowing-up. This effectively says the maximum possible, about the case of polynomials which vanish. It also includes the Riemann sphere = complex projective line case, without privileging it unduly. User:Charles Matthews 14:49, 15 Oct 2004 (UTC) : Sounds good to me. Have at it! User:Wile E. Heresiarch 16:17, 15 Oct 2004 (UTC) Wile, by "stop whoever is asking it", I didn't mean "silence the person", I meant "demand whoever is speaking to ask a better question". I do think that "1/0 = ?" is a meaningless question, simply because, as our responses indicate, ''it has multiple interpretations''. I interpreted it as a topological question, you as algebraic. I don't think there's anyway for either of us to "prove" the other's interpretation is "wrong". This is precisely what I mean by "1/0 = ?" not being a "well-defined" question. You say that "what you want to investigate is whether 1/0 = x implies 1 = 0x, etc." but ''that is presupposing an algebraic interpretation''. Your interpretation comes from viewing the problem algebraically. Mine comes from viewing it from a calculus point of view (i.e. removing discontinuities, etc.) Charles' interpretation is even more broad and fundamental, and further supports my contention that the question must be recast before addressing it. As for the "1 = a for any nonzero a implies contradiction" argument, again that's ''presuming an algebraic perspective''. There is no contradiction from the calculus perspective, in fact in complex analysis classes, the function 1/z is often defined from the Riemann sphere to itself, in the usual way for finite complex numbers, with the stipulation that 1/0 = infinity and 1/infinity = 0. This is not a statement about binary algebraic operations, but a statement about extending the function 1/z on C to be continuous on the Riemann sphere. User:Revolver 21:22, 15 Oct 2004 (UTC) : Well, I could quibble about various things. Instead I'll just ask how you would like to improve the article. I think that will help us stay focused. Regards, User:Wile E. Heresiarch 22:36, 16 Oct 2004 (UTC) OK, time for some mathematics. For X and Y compact spaces, and F: X → Y a function but not everywhere defined, one can take the graph G of F and its closure G* in XxY. The main idea is to project G* back onto X and say it must be a closed set. Which is true if spaces are Hausdorff. For something like 1/x this suggests Y should be some compactification. What we are looking for is that G* should actually be the graph of a function. In this purely topological case one can't really argue that compactifying the real line to the extended real line is better or worse that to a circle (i.e. we could add two points or one to compactify R). What the algebraic geometers do is a bit more definite, in that projective space acts as a compactification (non-Hausdorff, but everything is covered by the complete variety/proper morphism properties of projective space, plus the fact that the Zariski topology on XxY is not defined as the product topology). The added feature here is that there is more control of G* when it has a 'vertical component' projecting down to X. User:Charles Matthews 08:22, 16 Oct 2004 (UTC) : OK, thanks for the information. How would you like to use this to improve the article? User:Wile E. Heresiarch 22:36, 16 Oct 2004 (UTC) I would improve it by first noting that there are multiple ways of addressing the problem. Then, I would address it by interpretation, starting with the algebraic (binary operation) way. This includes much of the article (e.g. "incorrect arguments in dividing by zero" is essentially a misuse of algebra) and is easiest to understand. Then, maybe a mention of the other interpretations, but clearly set off in other sections. Again, I don't object to what's there, or object to the usual algebraic interpretation, just to its priveledged status. User:Revolver 22:50, 16 Oct 2004 (UTC) : Well, it doesn't seem like what you want is all that different from what's already in the article. Why not just go ahead and make the changes you want -- at least it will give us something different to talk about. User:Wile E. Heresiarch 16:36, 17 Oct 2004 (UTC) I worry a bit about going back and forth between "concrete" and "abstract" discussions, that it might be something that only a relatively few who read it will be interested in. (Or worse, that it might confuse... "2*4 = 2"?) The only way to do this seems to be to break up the organisation by interpretation. User:Revolver 06:12, 19 Oct 2004 (UTC) == 0/0 and differentiation == I can see a great deal of serious thought went into this article and it is extremely well and formally reasoned. However, it perpetuates the (formally accepted) statement that 0/0 is "just as" undefined as, say, 127/0. I'm quite sure this is formally correct, and I'll thank you not to spank me for saying it is not correct ''at a lower level'', somewhere down among primitive life-forms such as engineers. Here's the text: ''Limits of the form'' : ''in which both'' f(x) and g(x) ''approach 0 as'' x ''approaches 0, may converge to any value or may not converge at all.'' Okay, this is all true, but still, what you are doing, in a sneaky sort of sliding-up fashion, is 0/0. Please don't throw a blizzard of complicated math at my statement; I know what you're after. My point is that 0/0 is what differentiation ''feels'' like when I imagine the limit approaching. It is useful -- to me, if to nobody else -- to believe in 0/0 as the limit: the point at which anything becomes possible. Look at this from the amphibian point of view of an engineer. Think of the tangent to the curve as a long, massive stick; at an initial approximation, it is fixed to the curve at two points, (x,y) and (x1,y1). These points are widely spaced and the tangent does not wiggle. But, as the limit is approached, the distance between the two points decreases and the stick is liable to wiggle more and more; at the limit itself, the tangent is only attached to the curve by a single point and is free to rotate throughout a complete circle. However, we know better than to apply any torque to the tangent-stick as we approach the limit; we ''very carefully'' edge (x1,y1) up to (x,y), so when we finally reach the limit, the tangent is left pointing in the correct direction. All I'm saying here is that it is a useful tool to the student to ''suspend disbelief'' and ''imagine'' 0/0 = the tangent. It makes differentiation real. By the same token, when integration is demonstrated by area under a curve, first numerical integration approximates the area as a large number of thin slices. As the limit is approached, the student can ''think'' of an infinite number of zero-width slices, which add up to the total area. This is ∞ × 0 = the area. This may not be formally correct, but it is a good and useful way to ''think'' about it. I remember I once listened to my 8th grade math teacher lecture my class on this subject. He said it was "impossible" to divide any number by zero, since no number could be multiplied by zero to yield anything except zero. I raised my hand and said, What about 0/0? He said that was "undefined" and looked pissed off. I accepted that 0/0 was mathematical nonsense. Later, when I got into the calculus, I had a tough time of it, despite being very good at algebra, until I freed my mind and allowed myself to simply imagine the process as a clever way of defining the undefined. I would like to see if there is anyone bold enough to step away a moment from the purely theoretical and formally correct. Who will join me in wondering where these thoughts belong? I truly believe they are of value to the beginning student of the calculus, who is ill-served by adamant denial of 0/0. --User:Xiong 06:27, 2005 Mar 11 (UTC) ==Difference between 1/0 and 1/infinitesimal and x/0== Can someone explain in simple terms, if 1/0, 1/infinitesimal, or 1000000/0 or 1000000/infinitesimal or any different?
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