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	Addition - ExampleProblems.com
		Addition is commutative, meaning that one can reverse the terms in a sum left-to-right, and the result will be the same.
		The fact that addition is commutative is known as the "commutative law of addition".
		However, many binary operations are not commutative, such as subtraction and division, so it is misleading to speak of an unqualified "commutative law".
	www.exampleproblems.com /wiki/index.php/Addition (2022 words)

	Could the Photon be a Quasi-boson? (Site not responding. Last check: 2007-10-22)
		Quasi-bosons are composite particles formed of fermion pairs, and they obey commutation relations similar to those of bosons with extra terms.
		The theoretical arguments (Bose commutation relations follow from canonical quantization procedures, identical particles must be symmetric or antisymmetric under interchange, and spin-statistics theorem) are shown to be non-compelling reasons.
		The commutation relations for the electromagnetic field of quasi-boson photons do not satisfy space-like commutivity, but this is true for the fields of all composite particles formed of fermions (e.g., the deuteron).
	flux.aps.org /meetings/YR99/CENT99/abs/S7900011.html (172 words)

	Day 2 - Math 290
		Thus we have at least four different descriptions of what could be considered otherwise similar groups.
		All these enjoy elements with the same orders and the group binary operation is commutative.
		The property of commutivity, which is not present in all mathematical groups, makes such groups "abelian," a term in honor of the mathematician Neils Abel of Norway who died around 1820 at the age of 21.)
	soucc.southern.cc.oh.us /home/jdavidso/Math/Math290Project/Day2.html (956 words)

	[No title]
		For the operation to be guaranteed commutative, then the chi-squares need to have X2_n = X2_1*Sum(AllWeights)/Sum(WeightsOf the nodes m>=n) This suggests doing XMATCHes in decreasing weight, but that's likely to be counter to the desire for the first queries to return the fewest results.
		However this would mean that nodes need to be careful if they think about reorganizing the query (some piece of documentation suggests that a node can re-order a query to suit itself).
		There are three things that affect the X2 value that an actual XMATCH uses: the user desired limit, the number of unit vectors that have been averaged into current vectors, and need to accommodate potentially low errors in subsequent queries to assure commutivity.
	www.us-vo.org /summer-school/2005/proceedings/nvoss2005/java/dev/fullskynode/issues (2788 words)

	Matrices - Introducing Mathematics - Page 7 (Site not responding. Last check: 2007-10-22)
		Although in general matrix multiplication does not obey the commutivity law, it does always obey the associativity law, that is, if A, B and C are matrices that can be multiplied together (i.e.
		which commutes with any other square n by n matrix.
		will commute with any 5 by 5 matrix.
	www.introducingmathematics.com /matrices/07.html (139 words)

	Ring laws (Site not responding. Last check: 2007-10-22)
		When an operation is shown as a grid where the entry i, j in the grid is the element (i op b), I can see immediately what the commutative property looks like (grid symmetric about diagonal from top left) and what a unit, a zero, or a unity looks like.
		For addition, the relation {((1, 2), 2), ((2, 2), 2), ((2, 1), 2), ((1, 1), 2)} satisfies commutivity and associativity but it lacks a zero, so a proof based only on commutivity and associativity cannot be what I want.
		I mean that the laws of commutivity and associativity of addition are blind to that addition rule; they couldn't discover it not to be valid ring addition even though it isn't.
	www.physicsforums.com /showthread.php?t=71523 (1207 words)

	Logical Identities
		This means that in certain instances, the order of the expressions is important, in contrast to the mathematical commutivity of logical-and.
		Since one true result causes the entire logical-or to evaluate true, there is no reason to continue evaluation.
		Once again, while logical-or is commutative in pure mathematics, in many programming languages, including Java, the order of the expressions is important.
	www.cs.miami.edu /~burt/learning/Csc120.011/Notes/LogicIdent.html (430 words)

	Operator Product
		Defined as the sequential application of two operations, there is no reason to expect that the operator product be commutative, that the net affect of two operations be independent of the sequence in which they are applied.
		The drive to give this basic principle of quantum physics a direct expression in our mathematical formalism is a key force in our choice of the definition of operator product in terms of sequential application.
		While true for quantum operators, it is not true for general operators and depends on a property known as linearity which we shall discuss in detail in section (4.1).
	people.ccmr.cornell.edu /~muchomas/8.04/Lecs/lec_ave_oper/node9.html (484 words)

	Searching for Patterns in Pascal's Triangle (Site not responding. Last check: 2007-10-22)
		It would also be a nice complement in an abstract algebra aourse.
		These pages could also be used in a mathematics course for prospective elementary school teachers in which the properties (associativity, commutivity, etc.) of various number systems are being covered.
		They could also be read by students in mathematics appreciation courses.
	www.joma.org /images/upload_library/4/vol3/pascalpatterns/sect2fn.htm (129 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		For example, a commutative ring, and ring with identity, an integral domain, a field.
		One way to show that two rings are not isomorphic is to compare things that get preserved by isomorphisms.
		Such things include zero-divisors, units, commutivity, etc. Chapter 4 Know the division algorithm and Euclidean algorithm for polynomial rings EMBED Equation over a field.
	math.smsu.edu /~wickham/Math532/study.doc (1095 words)

	Multiplying Binomials: Instructional Notes (Site not responding. Last check: 2007-10-22)
		Have students build the problems on the worksheet.
		Ask questions about commutivity, associativity, and the distributive property.
		Ask students how they might tackle the issue of negative numbers using this model.
	math.la.asu.edu /~carlson/algebratiles.html (254 words)

	2 x 2 Singular Matrices (Site not responding. Last check: 2007-10-22)
		All that is left is to prove the elements commute, or A^p x A^q is the same as A^q x A^p.
		A group that has this quality is said to be Abelian.
		By manipulating both expressions we see they yeild the same results:
	www.ohioriverpottery.com /sec24.html (43 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		So what does this mean for our original question?
		One the one hand, overloading an operator as a non-member function gives you automatic type conversion for all arguments, which is needed for commutivity of binary operators.
		On the other hand, overloading an operator as a member function allows us to directly access private member data without the use of accessors or mutators.
	www.cs.umbc.edu /courses/undergraduate/202/fall03/Lectures/Overloading/Commutivity.shtml (262 words)

	Aristotle and Mathematics
		For example, Euclid provides separate definitions of proportion (v def.
		Compare the rule above (alternando),which is proved at v.16, while the rule follows trivially for numbers from the commutivity of multiplication and vii.19: ad = bc ⇔ a : b = c : d.
		In Plato's Academy, some philosophers suggested that lines are composed of indivisible magnitude, whether a finite number (a line of indivisible lines) or a infinite number (a line of infinite points).
	plato.stanford.edu /entries/aristotle-mathematics (9432 words)

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