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| No | 46% | 590 votes | Total: 1280 votes | |
| Yes | 54% | 690 votes |
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Add to FavoritesDivision by zero is done on the Riemann sphere in the complex plane and its result tends to infinity whereas zero divided by any other number tends to one. It is also important to note that in the real number system (and on better graphing calculators), division by zero is not impossible but remains undefined. http://en.wikipedia. org/wiki/Riemann_sph ere
Georg Friedrich Bernhard Riemann, German mathematician, student of Carl Friedrich Gauss, and mathematical genius developed numerous theories in mathematics such as Riemann partitions, The Generalized Riemann Hypothesis, Non-Euclidean Geometry, and extensive work in the realm of imaginary (complex) numbers, and was the mathematician Einstein selected for use in the General Theory of Relativity. Riemann's work in Non-Euclidean geometry just happens to fit exactly with what is actually experienced when taking measurements on the surface of the Earth.
That is, triangles laid upon the surface of the Earth add to 182 degrees instead of the Euclidean 180 degrees. This is due to the failure of the Euclidean fifth axiom, the Parallel Postulate. Its failure is immediately evident to anyone looking at a spherical model of planet Earth where longitude lines at the equator cross at the North Pole proving failure of the parallel postulate (parallel at the equator and crossing at the North Pole). Carl Friedrich Gauss suspected this as well as the existence of non-Euclidean geometry, and the number of primes less than a given magnitude yet did not disclose choosing to let his students discover and receive credit for the items. His most significantly accomplished student was Georg Friedrich Bernhard Riemann.
It is fascinating to note that real numbers generally tend to give imaginary results and imaginary numbers tend to give real results. Meaning that Euclidean results tend to give ideal answers verses real answers and non-Euclidean geometry gives answers consistent with what is really observed in our natural world. This is not to detract from Euclidean mathematics merely that a perfect triangle is merely an idealization and not typically encountered in the real world. Certainly, man can create one but trying to find one in the real world would be as challenging as trying to find a straight line. A straight line laid out on the surface of the Earth would tend to describe a geodesic. Nevertheless, this remains for another topic.
Since the consequences of Riemann's interpretation of non-Euclidean geometry tend to reflect what is actually observed here on Earth, it is tantalizingly imminent that Einstein would indeed use this mathematical model for the description of gravitation. How else could one conceivably proceed? This also provides compelling evidence of Riemann's stature as a founding father of modern physics, a circumstance that will inevitably become more prevalent upon a conclusive solution to the Riemann hypothesis.
In order to provide some degree of logical reasoning for the necessity of division by zero in some (but not all) circumstances, examine the incompleteness theorem presented by Kurt Godel.
"Godel showed that truth cannot be contained within the limits of strict logic. Only if we allow paradox can truth completely reveal itself in form. These two sides of Godel's proof represent the apophatic (via negativa) and cataphatic (via positiva) approaches to truth, respectively. In the apophatic approach, one adheres to strict logic to show that any attempt to represent or speak truth necessarily failsthe truth is beyond all rational comprehension. In the cataphatic approach, on the other hand, one embraces paradox and the coincidence of opposites to demonstrate the tangible presence of truth in all its limitless expressions. Like the ancient mathematics of Pythagoras, Godel's mathematical proof can be seen as a symbol of profound truths about the relationship between the limited and the unlimited, form and formlessness, transcendence and immanence, Godel's postmodern mathematics undermines any attempt to fixate on any totalizing axiomatic system for mathematical discourse, and reveals the essential ambiguity, openness, and emptiness of mathematical activity." Thomas J. McFarlane Spring 2000 Revised and Edited for the Web March 2004.
Inherent to Godel's proof is the conclusion that if a system is consistent, then it is incomplete. Conversely, if a system is complete, then it must be inconsistent. Considering this fact when examining such a precept as the impossibility of division by zero, we would be relegated to an inconsistent mathematics. Alternatively, entertaining the possibility of division by zero generates consistency, yet leaves us with an incomplete understanding of mathematics, which is precisely the circumstance with which we are dealing. Therefore, in some circumstances, one can in fact divide by zero.
http://en.wikip edia.org/wiki/Bernha rd_Riemannnn
http://e n.wikipedia.org/wiki /Riemann_sphere
Mathe matical Poetics of Enlightenment Thomas J. McFarlane Spring 2000, Revised and edited for the web March 2004. http://www.integrals cience.org
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Yes
Division by zero is impossible in ordinary, everyday number systems such as the rational numbers (whole number fractions), algebraic numbers (solutions to polynomials with whole number coefficients) or real numbers (numbers on the number line). If it were possible, zero would be equal to one, moreover, all numbers would be zero. This contradicts the definition of these number systems, and moreover renders them useless for computation.
It is somewhat astonishing that this question is being asked, and that several hundred voters somehow disagree with trained mathematicians and with logic itself. More astonishing still are the excuses given by well over thirty Helium.com writers for this position. The worst of them are nonsequiturs, bordering on glossolalia, about mirrors or the distribution of wealth. The best make an appeal to "infinity" which I will address at the end of this article. In between are many that indicate severe problems in the way we teach mathematics: they're touchy-feely arguments about physical objects or the words "something" and "nothing" as though division of numbers is a physical process or a matter of semantics. Although manipulation of cubes, strings, and other objects is used to teach math to schoolchildren, numbers are not physical objects or flimsy metaphors. Talk arguments about pieces of pie or pseudophilosophical reflections on the nature of nothing don't help us to understand numbers. When we must address a subtle question-and from Everyman's perspective division by zero is a subtle question even if it is trivial for the mathematically trained-we must turn to the rigorous definition of the numbers and operations in question: a set of axioms.
I prove below that the ability to divide by zero would imply 1=0 for rational, real, and algebraic numbers by proving a more general result for algebraic structures called fields. The proof is elementary in nature and should be understandable even by readers for whom this is the first exposure to formal mathematics. Technical limitations of Helium.com make notation somewhat awkward; this article will be updated if and when Helium upgrades its back-end to allow use of a better character set such as Unicode.
Rational, reals, and algebraic numbers are but familiar examples of a formal algebraic structure known as a "field", a field being a set of numbers F and two operations + and * satisfying what we usually expect from addition and multiplication, defined by the following axioms, which I am taking at the moment from Wikipedia but can be found in any good book on abstract algebra. Zero is one of the two special numbers in a field; it is the additive identity.
(1) Closure of F under + and *:
For every a and b in a field F, a+b and a*b are both in F.
(2) Associativity of both operations:
For all a, b, and c in F, a+(b+c)=(a+b)+c and a*(b*c)=(a*b)*c.
(3) Commutativity of both operations:
For all a and b in F, a+b=b+a and a*b=b*a
(4) * is distributive over +:
For all a, b, and c in F, a*(b+c)=a*b+a*c.
(5) Existence of an additive identity:
There exists an element 0 in F such that, for all a in F, a+0=a.
(6) Existence of a multiplicative identity:
There exists an element 1 in F, different from 0, such that, for all a in F, a*1=a.
(7) Existence of additive inverses:
For any element a in F there exists an element -a such that a+-a=0.
(8) Existence of multiplicative inverses:
For any element a in F, except perhaps zero, there exists an element a^-1 such that a*a^-1=1.
We can show that the existence of a multiplicative inverse for zero renders these axioms contradictory, as it results in 0 and 1 being the same number.
Note that the field axioms do not say anything about the multiplicative properties of 0. We must first prove that a*0=0 for all a in F. We begin by proving that, for any nonzero a in F, a*0=0. One proof is as follows:
Consider two nonzero numbers from F, a and b, such that a=b^-1.
We have from Axiom 5 above that b+0=b.
Thus a*(b+0)=a*b.
It follows from Axiom 4 that a*b+a*0=a*b. Now we add -(a*b), which must exist given Axiom 7, immediately giving us Lemma 1, a*0=0, for all nonzero a in F.
We can now prove Lemma 2, that 0*0 must equal 0.
Consider two nonzero numbers, a and b, from F.
a*b=a*b
(a+0)*(b+0)=a*b (Axiom 5:a+0=a and b+0=b).
a*b+0*a+0*b+0*0=a*b (Axiom 4 applied twice, with the result rearranged with the help of axiom 3: (a+0)*(b+0)=a*(b+0)+ 0*(b+0)=a*b+a*0+b*0+ 0*0)
a*b+0*0=a*b (By Lemma 1, 0*a=0*b=0. From Axiom 7, a*b+0+0+0*0=a*b+0*0)
0*0=0 QED (Applying Axiom 7, -(a*b)+a*b+0*0=-(a*b )+a*b.)
Now let us prove by contradiction that there exists no multiplicative inverse for 0. Consider its negation, that there exists a multiplicative inverse for 0, 0^-1, such that 0*(0^-1)=1.
For any number a in F, axiom 1 would have there exist some b in F such that
a*0^-1=b.
This would imply that
a*(0^-1)*0=b*0.
From our assumption,this implies
a=b*0.
Lemmas 1 and 2 above imply that for all b in F, b*0=0.
We have thus proved that
a=0 for all a in F.
Let a be 1. We have proved that 1=0.
Since, from our axioms, 1 is not 0, this is a contradiction. QED
It can be shown by further application of the field axioms and Lemma 1 that this result implies that all nonzero numbers are zero. Not only can a field where division by zero is allowed not be a field by contradiction, it can also contain at most one element! That is hardly a number system at all!
There are axiomatic systems in which division by zero is defined. Called "wheels", they redefine division as being a unary operation, resulting in subtly different behavior from what we expect from number systems. In particular, where x is an element of the wheel, 0*x=0 is not generally true.
Attempts have been made since the flourishing of mathematics in medieval India to permit division by zero by adding a number "infinity" to the set of numbers in question such that for all a in the field, a/0=infinity. The real projective line and (more familiar still) the complex sphere are the most common examples. Paradoxes arise concerning the case 0/0; 0/0 can be shown to be required to be both 1 and infinity at once and is thus undefined. A multiplicative inverse for zero cannot exist in any field at all.
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