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Topic: Commutative law

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History of Singapore

	SAMPLE LESSON 4
		A social law is a rule that forbids or requires a given conduct of all members of a community.
		Laws are established by custom or through adoption by a legislature, that is, a group of people given the power to make laws.
		The laws of mathematics were developed by professional and highly expert mathematicians who staked their careers on the fact that, if followed conscientiously, their mathematical procedures would result in arithmetic and algebraic solutions that were correct in every way.
	members.tripod.com /learnmath/id19.htm (1889 words)

	Commutative operation Summary
		Every element commutes with itself and, in a group, every element commutes with the identity, with its own inverse, and with its powers.
		The subset of the domain on which an operation is commutative is sometimes called the center in algebra.
		A commutative ring is a ring whose multiplication is commutative.
	www.bookrags.com /Commutative_operation (789 words)

	Commutative ring - Wikipedia, the free encyclopedia
		In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law.
		Given a commutative ring R and an ideal I of R, the factor ring R/I is the set of cosets of I together with the operations (a+I)+(b+I)=(a+b)+I and (a+I)(b+I)=ab+I.
		The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigating the theory of its modules.
	en.wikipedia.org /wiki/Commutative_ring (949 words)

	Business Law, 8e: Chapter 1
		Commutative justice is restricted in its application to the criminal justice system, while distributive justice applies in all aspects of the legal system.
		Commutative justice places everyone on an equal footing, while distributive justice recognizes that all persons are not equal, and tries to "distribute" justice accordingly.
		A ______________ is a decision by a court that becomes the law in future cases that involve the same legal problem.
	www.swlearning.com /blaw/davidson/davidson8e/quiz/ch01/quiz.html (364 words)

	commutative law - HighBeam Encyclopedia
		More generally, in addition, for any two numbers a and b the commutative law is expressed as a + b = b + a.
		In general, any binary operation, symbolized by [symbol], joining mathematical entities A and B obeys the commutative law if A [symbol] B = B [symbol] A for all possible choices of A and B.
		Not all operations are commutative; e.g., subtraction is not since 2-5≠5-2, and division is not since 2/5 ≠ 5/2.
	www.encyclopedia.com /doc.aspx?id=1E1:commutat-lw (303 words)

	Highbeam Encyclopedia - Search Results for commutative
		commutative law in mathematics, law holding that for a given binary operation (combining two quantities) the order of the quantities is arbitrary; e.g., in addition, the numbers 2 and 5 can be combined as 2+5=7 or as 5+2=7.
		ring in mathematics, system consisting of a set R of elements and two binary operations, such that addition makes R a commutative group and multiplication is associative and distributes over addition (see commutative law ; associative law ; distributive law).
		Set theory not only is involved in many areas of mathematics but has important applications in other fields as well, e.g., computer technology and atomic and nuclear physics.
	www.encyclopedia.com /SearchResults.aspx?Q=commutative (625 words)

	[No title]
		(Commutative Law) If a, b are integers, then a + b = b + a, a • b = b • a.
		(Law of Identity Elements) For all integer a, a + 0 = 0 + a = a; a • 1 = 1 • a = a.
		(Law of Additive Inverse) For all integer a, a + (—a) = (—a) + a = 0.
	www.cs.ucf.edu /courses/cot3100.fall00/section1/chap1defs.doc (550 words)

	Matrices
		Laws 1-4 concern addition and since matrix multiplication is the same as regular addition in each position, it is easy to check that it satisfies the same rules.
		The associative law of multiplication isn't as obvious, but multiplying things out shows that law 5 is satisfied as well.
		The commutative law of multiplication, law 6, fails for matrices.
	www.math.ksu.edu /math511/notes/1004.html (1046 words)

	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)

	commutative - Search Results - MSN Encarta
		Commutative Property (mathematics), trait of mathematical operations that are independent of the order of the numbers or symbols involved.
		Commutative Law of Addition, Commutative Law of Multiplication, extension beyond real numbers
		defined, commutative property, quaternions, complex numbers that break the commutative law
	encarta.msn.com /commutative.html (125 words)

	commutative law. The Columbia Encyclopedia, Sixth Edition. 2001-05
		in mathematics, law holding that for a given binary operation (combining two quantities) the order of the quantities is arbitrary; e.g., in addition, the numbers 2 and 5 can be combined as 2+5=7 or as 5+2=7.
		In general, any binary operation, symbolized by [symbol], joining mathematical entities A and B obeys the commutative law if A[symbol]B=B[symbol]A for all possible choices of A and B.
		Not all operations are commutative; e.g., subtraction is not since 2-5[symbol]5-2, and division is not since 2/5[symbol]5/2.
	www.bartleby.com /65/co/commutat-lw.html (161 words)

	All Elementary Mathematics - Study Guide - Arithmetics - Laws of addition and multiplication...
		Commutative law of addition: m + n = n + m.
		Commutative law of multiplication: m · n = n · m.
		This law expands the rules of operations with brackets (see the previous section).
	www.bymath.com /studyguide/ari/ari4.html (146 words)

	Algebraic Laws
		The Trinary Algebraic Laws are virtually the same for other branches of mathematics.
		The commutative law says that the order in which variables are written makes no difference to the outcome.
		As stated in the previous Laws, the Magnitude operator does not follow all of the conventions of normal algebra because it is a comparison of one magnitude (A) to another (B).
	www.trinary.cc /Tutorial/Algebra/Laws.htm (212 words)

	AllRefer.com - ring, mathematical system (Mathematics) - Encyclopedia (Site not responding. Last check: 2007-10-08)
		ring, in mathematics, system consisting of a set R of elements and two binary operations, such that addition makes R a commutative group and multiplication is associative and distributes over addition (see commutative law; associative law; distributive law).
		A commutative ring is one in which the commutative law also holds for multiplication.
		Examples of commutative rings are the sets of integers (see number) and real numbers.
	reference.allrefer.com /encyclopedia/R/ring2.html (186 words)

	Boolean Functions
		The commutative law of addition for two variables is algebraically expressed as
		The commutative law of multiplication for two variables is expressed as
		A (B+C) = AB + AC This law states that ORing several variables and ANDing the result is equivalent of ANDing the single variable with each of the variables in the grouping, then ORing the result.
	scitec.uwichill.edu.bb /cmp/online/P10F/boolean.htm (268 words)

	Natural Numbers (Site not responding. Last check: 2007-10-08)
		Associative law for addition (a + b) + c = a + (b + c)
		Commutative law for addition a + b = b + a
		Distributive law a ´ (b + c) = a ´ b + a ´ c
	www.faculty.sfasu.edu /cproctor/sect32.html (250 words)

	Commutative Law of Addition on the Abax
		Now we will use the above law (the one about 5 + 3 = 8, as well as 3 + 5 = 8) to explain why it doesn't matter in which order you add the columns in addition on an abax (or on paper, or anywhere else, for that matter).
		By the way, the law which describes that it doesn't matter which order you add in is called the Commutative Law of Addition, or the Commutative Property of Addition.
		You may know that to "commute" is to go travel back and forth, like commuting from home to work.
	www.mathmojo.com /abacus/abax/abaxaddition/commutativeaddabax.html (854 words)

	PlanetMath: ring
		Cross-references: formula, distributes over, abelian group, distributive law, inverse, identity, additive, commutative, associative, binary operations
		Object id is 354, canonical name is Ring.
		Can't a ring (R,+,) be described as a commutative* group (R,+) and a * (closed) operation?
	planetmath.org /encyclopedia/Ring.html (255 words)

	A7 Key Properties - Vector and Complex Numbers:
		NonZero Product Law: If Z and W have lengths r and s both greater than 0 then their product has length rs > 0 (By the methods of decimal arithmetic with, the product of two positive numbers or length is positive.
		Because multiplication commutes (that is, AB = BA), the left and right forms of the distributive law are equivalent.
		Because they are equivalent, we the adjectives left and right may some be omitted, and we may talk about a distributive law instead of distributive laws, a harmless variation in language.
	whyslopes.com /etc/ComplexNumbers/ch06A.html (943 words)

	What is a Lie Algebra? (Site not responding. Last check: 2007-10-08)
		Another important law is the associative law: (a + b) + c = a + (b + c).
		Taking the commutative law, associative law, and two other laws, we get a particular structure we call an abelian group.
		The laws of commutativity seem somewhat intuitive, as one is used to doing addition and subtraction from grade school.
	www.math.rutgers.edu /~nacin/def1.html (422 words)

	Special cases for laws of signs (Site not responding. Last check: 2007-10-08)
		the laws of signs may be used to advantage.
		The word "commute" means to change, substitute or move from place to place.
		To verify the distributive law, we note that 2(3 + 4 + 5) is the same as 2(12) or 24.
	www.tpub.com /math1/4c.htm (985 words)

	Factoring.html
		Objective: To learn how to use the distributive law in reverse, a useful tool that will become handy when factoring is important.
		Recall 2: The process of applying the distributive law of multiplication over addition in reverse is usually referred to as factoring out the common factor or just plain factoring.
		By the commutative law for multiplication (in each term), the last expression may be written as
	www.marlboro.edu /academics/study/mathematics/courses/factoring (1016 words)

	Chapter 1 Table of Contents
		The Commutative Law: An easy way to remember the commutative law is to think of what people are doing when they travel from a subway to the city for work—they are COMMUTING or changing positions.
		Example 2. 4(5)= 5(4) Is and example of the commutative property of multiplication.
		Write a problem using the commutative property of addition using the numbers 8 and 4.
	www.mtsu.edu /~smcdanie/080_Fall_2001/Chapter_1/Section7/Section1_7.htm (436 words)

	Starting to Prove Something
		Next, use ``ruleintro'' to propose application of the commutative law to both the complete term and the subterm at the same time, then use the ``top'' command to return to the complete term, and only then issue the ``execute'' command.
		Both applications of the commutative law will be carried out at once, returning the term to its original shape.
		As the last exercise illustrates, the commutative law of addition is completely symmetrical in its effect.
	math.boisestate.edu /~holmes/mark2/node80.html (994 words)

	II Fundamentals of General Algebra
		By the law of transitivity, each element of (a) and of (c) belongs to (b), and each element of (b) belongs to (a) and to (c).
		As in every field multiplication is commutative, a ring which is a sub-ring of a field must be a commutative ring; moreover, as the product of two elements which differ from 0 differs itself from 0, the same holds in every sub-ring of a field.
		The two commutative laws and the associative law of multiplication are obvious.
	mpec.sc.mahidol.ac.th /radok/physmath/MAT5/algebra21.htm (8565 words)

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