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Topic: Multiplicative inverse

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	Multiplicative inverse - Wikipedia, the free encyclopedia
		In mathematics, the reciprocal, or multiplicative inverse, of a number x is the number which, when multiplied by x, yields 1.
		In modular arithmetic, the multiplicative inverse of x is also defined: it is the number a such that (a·x) mod n = 1.
		It is important to distinguish the reciprocal of a function f in the multiplicative sense, given by 1/f, from the reciprocal or inverse function w.r.t.
	en.wikipedia.org /wiki/Multiplicative_inverse (986 words)

	Inverse element - Wikipedia, the free encyclopedia
		In mathematics, the idea of inverse element generalises the concepts of negation, in relation to addition, and reciprocal, in relation to multiplication.
		A square matrix M with entries in a field K is invertible (in the set of all square matrices of the same size, under matrix multiplication) if and only if its determinant is different from zero.
		If the determinant of M is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one.
	en.wikipedia.org /wiki/Inverse_element (428 words)

	Multiplicative Inverse
		The second factor is the multiplicative inverse of the first factor.
		The numerator of the multiplicative inverse will be the denominator of the first factor.
		The numerator of the multiplicative inverse is 2 and the denominator is 3.
	www.visualfractions.com /Inverse.html (225 words)

	inverse - Search Results - MSN Encarta
		Inverse Function, function that exactly reverses the transformation produced by a function f.
		Given any real number a, there is a number one (1) called the multiplicative identity, such that a(1) = 1(a) = a.
		Arc Cosecant, in trigonometry, the inverse of the cosecant function.
	ca.encarta.msn.com /inverse.html (156 words)

	NUMBER THEORY AND CRYPTOGRAPHY (Site not responding. Last check: 2007-10-21)
		If b is multiplicative inverse of a then a is multiplicative inverse of b.
		Multiplicative inverse of a is denoted by a
		In all elements 1, 2, 3, 4, 5, 6 have multiplicative inverse.
	www.math.ohiou.edu /~sustain/summer00/12.html (97 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-21)
		Date: 11/07/2000 at 07:12:20 From: mmcl Subject: Multiplicative Inverse in GF(2^8) I have a 4x4 matrix of bytes: [B0 B4 B8 B12] [B1 B5 B9 B13] [B2 B6 B10 B14] [B3 B7 B11 B15] I need to get the multiplicative inverse of this matrix in GF(2^8).
		"Addition" is the usual addition of polynomials (reducing coefficients modulo 2), and "multiplication" is the usual multiplication of polynomials (reducing coefficients modulo 2), followed by a reduction modulo f(x) (and further coefficient reduction modulo 2) until the result has degree less than 8.
		Given a polynomial a(x) whose inverse you seek, perform the Extended Euclidean Algorithm on a(x) and f(x).
	mathforum.org /library/drmath/view/51675.html (673 words)

	PlanetMath: ring
		Additive inverses are unique, and one can define subtraction in any ring using the formula
		multiplicative identity, multiplicative inverse, ring with unity, unit, ring addition, ring multiplication, ring sum, ring product, unital ring, unitary ring
		additive inverse of one element times another element is the additive inverse of their product
	planetmath.org /encyclopedia/Ring.html (246 words)

	Modular division via the multiplicative inverse of the denominator
		The multiplicative inverse of a number b is the number c such that b times c is 1.
		In ordinary arithmetic, the multiplicative inverse of b is the reciprocal of b, namely 1/b.
		Looking at the table, we see that the multiplicative inverse of 1 is 1, the multiplicative inverse of 2 is 4 (and vice versa), the multiplicative inverse of 3 is 5 (and vice versa), and the multiplicative inverse of 6 is 6.
	www.cs.brown.edu /courses/cs007/modmult/node2.html (549 words)

	Inverse Activity (Site not responding. Last check: 2007-10-21)
		I expect students to know that the inverse of 2/3 is 3/2, but I want to make sure they understand that this is because when you multiply 2/3 by its inverse, 3/2, you get 1, the multiplicative identity.
		This activity is meant to remind them what an inverse is using an operation with which they are very familiar.
		See that the inverse of an unrestricted parabolic function is not a function.
	www.tcnj.edu /~golazes2/inverse_activity.htm (614 words)

	Day Three: Multiplicative Inverses (Site not responding. Last check: 2007-10-21)
		The basic way of finding multiplicative inverse is to take a fraction and reverse the numerator and denominator.
		For instance, the multiplicative inverse of 3/4 is 4/3.
		Likewise the multiplicative inverse of 1 1/3 is 3/4.
	jwilson.coe.uga.edu /emt669/student.folders/Lewis.Millard/unit/DayThree (310 words)

	Operations on Matrices (Site not responding. Last check: 2007-10-21)
		is a very special matrix which is equal to its multiplicative inverse.
		This matrix is of order greater than 2; thus, the inverse of A is obtained by using a graphing calculator.
		Since multiplication of matrices is associative, this equation can be rewritten.
	www.faculty.sfasu.edu /cproctor/sect62.html (758 words)

	Section 5.2: Multiplicative Inverses
		Finding integer solutions to linear equations is a bit more complicated because the only integers which have integer multiplicative inverses are ±1.
		In this context, the multiplicative inverse (or just "inverse" for short) of a modulo n, which is denoted by a
		In the first research question, we address the question of which integers a have a multiplicative inverse modulo n.
	www.math.mtu.edu /mathlab/COURSES/holt/dnt/lincong2.html (451 words)

	Mathwords: Multiplicative Inverse of a Number
		Multiplicative inverse means the same thing as reciprocal.
		For example, the multiplicative inverse (reciprocal) of 12 is
		Note: The product of a number and its multiplicative inverse is 1.
	www.mathwords.com /m/multiplicative_inverse_of_a_number.htm (49 words)

	Which mod m numbers have multiplicative inverses? (Site not responding. Last check: 2007-10-21)
		It is an important theorem that a number b between 0 and m-1 has a mod m multiplicative inverse if b is relatively prime to m.
		Thus if m is prime, the only number between 0 and m-1 that does not have a mod m multiplicative inverse is 0.
		We saw that the numbers 1, 2, 3, 4, 5, and 6 all have mod 7 multiplicative inverses.
	www.cs.brown.edu /courses/cs007/modmult/node4.html (195 words)

	SparkNotes: Matrices: The Inverse of a Matrix
		The multiplicative inverse of a real number is the number that yields 1 (the identity) when multiplied by the original number.
		Then, row reduce to convert the matrix to reduced row-echelon form--that is, to get an m×m identity matrix on the left.
		The new m×m matrix on the right is the multiplicative inverse of the original matrix.
	www.sparknotes.com /math/algebra2/matrices/section5.rhtml (167 words)

	tutOR: Matrix Inverse
		This centre was established to provide help to visitors who are not familiar with the ideas of the inverse of a matrix and/or with a method of finding an inverse using row operations.
		There are other simple ways of showing whether a matrix has an inverse, see for example the discussion on Determinants.
		We describe here two distinct sequences of row operations, which we will see lead us to the same inverse, as of course they should, since, as we stated earlier, the inverse is unique.
	www.tutor.ms.unimelb.edu.au /matrix/matrix_inverse.html (775 words)

	[No title]
		We thought of subtraction as "just adding the (additive) inverse;" we can think of division as "just multiplying by the (multiplicative) inverse." The multiplicative inverse of a number m is denoted EMBED Equation.DSMT4 and is the number that when multiplied by m results in 1.
		Because in cryptology, when we encipher we want an inverse process to decipher, we must be careful using multiplication modulo n for enciphering.
		For each nonzero number that does not have a multiplicative inverse modulo 8, show that it is a zero divisor.
	www.nku.edu /~christensen/fivem.doc (1749 words)

	The Multiplicative Inverse Eigenvalue Problem over an Algebraically Closed Field (ResearchIndex)
		The multiplicative inverse eigenvalue problem asks for the construction of a matrix Z 2 Z such that the product matrix MZ has characteristic polynomial p().
		In this paper we provide new necessary and sufficient conditions when Z is an affine variety over an algebraically closed field.
		Keywords: Eigenvalue completion, inverse eigenvalue problems, dominant morphism...
	citeseer.ist.psu.edu /375605.html (478 words)

	[No title]
		Using multiplication to make an encryption cipher led us to the question of whether we can find multiplicative inverses in modular arithmetic, in order to find the decryption function.
		Surprisingly, the answer to the multiplicative inverse problem goes all the way back to the ancient Greeks, and to the Euclidean algorithm for finding the greatest common divisor for a pair of numbers.
		1a) Use the two multiplication tables on the next page to construct multiplicative inverse tables, when the modulus is n=7 and when it is n=15.
	www.math.utah.edu /~korevaar/ACCESS2005/WedJune15.doc (1511 words)

	Citebase - The Multiplicative Inverse Eigenvalue Problem over an Algebraically Closed Field
		The Multiplicative Inverse Eigenvalue Problem over an Algebraically Closed Field
		Let M be a square matrix and let p(t) be a monic polynomial of degree n.
		The multiplicative inverse eigenvalue problem asks for the construction of a matrix in Z such that the product matrix MZ has characteristic polynomial p(t).
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0009163 (145 words)

	Multiplicative Inverse Notes (Site not responding. Last check: 2007-10-21)
		Home > CS178 > Homework 6 > Multiplicative Inverse
		Curby's Guide to Calculating the GCD and Multiplicative Inverse
		So the last value is our multiplicative inverse (green number)!
	www.cs.ucsb.edu /~kirbysdl/cs178/multinv.html (445 words)

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