1. Digital Logic There is a group of useful theorems of Boolean algebra which help in developing the logic for a given operation. Boolean algebra theorems. http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/diglog.html

Digital Logic For two binary variables (taking values and 1) there are 16 possible functions . The functions involve only three operations which make up Boolean algebra: AND, OR, and COMPLEMENT. They are symbolically represented as follows: These operations are like ordinary algebraic operations in that they are commutative associative , and distributive . There is a group of useful theorems of Boolean algebra which help in developing the logic for a given operation. Digital Logic Theorems Digital Logic Functions Index Electronics concepts ... Electricity and magnetism R Nave Go Back Boolean Algebra Theorems The applications of digital logic involve functions of the AND, OR, and NOT operations. These operations are subject to the following identities: These theorems can be used in the algebraic simplification of logic circuits which come from a straightforward application of a truth table DeMorgan's Theorem Basic Gates Index ... Electricity and magnetism R Nave Go Back Binary Functions of Two Variables Digital logic involves combinations of the three types of operations for two variables: AND, OR, and NOT. There are sixteen possible functions: This is an active graphic. Click on any of the functions for further details.

2. AMCA: Hu's Primal Algebra Theorem Revisited By Hans-E. Porst Conference Homepage. Hu s Primal algebra theorem Revisited presented by HansE. Porst University of Bremen, Germany Various proofs http://at.yorku.ca/c/a/e/e/79.htm

AMCA Document # caee-79 AAA60: Workshop on General Algebra (60. Arbeitstagung Allgemeine Algebra) June 22-25, 2000 Dresden University of Technology Dresden, Germany Conference Organizers View Abstracts Conference Homepage Hu's Primal Algebra Theorem Revisited presented by Hans-E. Porst University of Bremen, Germany Various proofs of Hu's Theorem characterizing the variety of Boolean algebras up to equivalence (in the categorical sense) have been obtained over the last decades. We add another one to this list which not only is of striking simplicity (it is essentially a three line proof) but at the same time classifies the varieties in question up to equivalence in the sense of Universal Algebra. The proof is based on the categorical fundamentals of Universal Algebra as provided by Lawvere and the representation theorem for finite Boolean algebras only. Date received: May 28, 2000 The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts

3. AMCA: Hu's Primal Algebra Theorem Revisited By Hans-E. Porst Hu s Primal algebra theorem Revisited by HansE. Porst University of Bremen, Germany. Various proofs of Hu s Theorem characterizing http://at.yorku.ca/cgi-bin/amca/caee-79

Atlas Mathematical Conference Abstracts Conferences Abstracts Organizers ... About AMCA AAA60: Workshop on General Algebra (60. Arbeitstagung Allgemeine Algebra) June 2225, 2000 Dresden University of Technology Dresden, Germany Organizers Reinhard Pöschel, Manfred Droste, Bernhard Ganter View Abstracts Conference Homepage Hu's Primal Algebra Theorem Revisited by Hans-E. Porst University of Bremen, Germany Various proofs of Hu's Theorem characterizing the variety of Boolean algebras up to equivalence (in the categorical sense) have been obtained over the last decades. We add another one to this list which not only is of striking simplicity (it is essentially a three line proof) but at the same time classifies the varieties in question up to equivalence in the sense of Universal Algebra. The proof is based on the categorical fundamentals of Universal Algebra as provided by Lawvere and the representation theorem for finite Boolean algebras only. Date received: May 28, 2000 Atlas Mathematical Conference Abstracts . Document # caee-79.

4. Theorem Proving And Algebra Theorem Proving and Algebra. To be published by MIT Press, someday. This draft textbook is intended to introduce general (universal http://www.cs.ucsd.edu/users/goguen/pubs/tp.html

5. The Fundamental Theorem Of Algebra The multiplicity of roots. Let's factor the polynomial . We can "pull out" a term Can we do anything else? The Fundamental theorem of algebra. The multiplicity of roots. http://www.sosmath.com/algebra/factor/fac04/fac04.html

The Fundamental Theorem of Algebra The multiplicity of roots. Let's factor the polynomial . We can "pull out" a term Can we do anything else? No, we're done, we have factored the polynomial completely; indeed we have found the four linear (=degree 1) polynomials, which make up f x It just happens that the linear factor x shows up three times. What are the roots of f x )? There are two distinct roots: x =0 and x =-1. It is convenient to say in this situation that the root x =0 has multiplicity 3 , since the term x x -0) shows up three times in the factorization of f x ). Of course, the other root x =-1 is said to have multiplicity 1. We will from now on always count roots according to their multiplicity. So we will say that the polynomial has FOUR roots. Here is another example: How many roots does the polynomial have? The root x =1 has multiplicity 2, the root has multiplicity 3, and the root x =-2 has multiplicity 4. All in all, the polynomial has 9 real roots! Irreducible quadratic polynomials. A degree 2 polynomial is called a quadratic polynomial. In factoring quadratic polynomials, we naturally encounter three different cases: Case 1A: Consider the quadratic polynomial depicted below.

6. ThinkQuest : Library : Seeing Is Believing Discrete algebra. Binomial theorem. The binomial theorem is a useful formula for determining the algebraic expression that results from raising a binomial to an integral power. http://library.thinkquest.org/10030/11binoth.htm

Index Education Seeing is Believing Need a primer on math, science, technology, education, or art, or just looking for a new Internet search engine? This catch-all site covers them all. Maybe you're doing your homework and need to quickly look up a basic term? Here you'll find a brief yet concise reference source for all these topics. And if you're still not sure what's here, use the search feature to scan the entire site for your topic. Visit Site 1997 ThinkQuest Internet Challenge Languages English Students Peter Oakhill College, Castle Hill, Sydney, Australia Suranthe H Oakhill College, Sydney, Australia Coaches Tina Oakhill College, Castle Hill, Sydney, Australia Tina Oakhill College, Castle Hill, Sydney, Australia Want to build a ThinkQuest site? The ThinkQuest site above is one of thousands of educational web sites built by students from around the world. Click here to learn how you can build a ThinkQuest site. Privacy Policy

7. KryssTal : Pascal's Triangle Pascal's Triangle and its uses in probability. The Binomial theorem used for algebraic expansions and finding roots. combinations algebra probability binomial theorem roots. Introduction http://www.krysstal.com/binomial.html

Pascal's Triangle combinations : algebra : probability : binomial theorem : roots Introduction to Pascal's Triangle Mathematics is the language of science. Something that may be difficult to picture, may be easy to understand mathematically. A mathematical equation may take up a single line whereas the same thing written in words may take up a large paragraph. In this essay I'd like to introduce some clever ways of doing algebra and of calculations using a simple calculator. By simple calculator I mean one that does only the basics (plus, minus, multiply and divide). When you have finished this essay you should be able to calculate quite difficult roots with this calculator. But before picking up a calculator, let us look at Mr Pascal. Blaise Pascal (1623 - 1662) was a French mathematician. His surname is used as the unit of pressure. One of his sayings was to note that 'had Cleopatra's nose been differently shaped, the history of the world would have been different'. He is most famous for the triangle named after him, Pascal's Triangle . It's not a geometrical triangle but a triangle of numbers. Here it is below: Study these numbers and see if you can figure what the next line should be before reading on . . .

8. Fundamental Theorem Of Algebra The applet on this page is designed for experimenting with the fundamental theorem of algebra, which state that all polynomials with complex coefficients (and hence real as a special case) have a complete set of roots in the complex plane. http://www.math.gatech.edu/~carlen/applets/archived/ClassFiles/FundThmAlg.html

The applet on this page is designed for experimenting with the fundamental theorem of algebra, which state that all polynomials with complex coefficients (and hence real as a special case) have a complete set of roots in the complex plane. The applet is designed to impart a geometric understanding of why this is true. It graphs the image in the complex plane, through the entered polynomial, of the circle of radius r. For small r, this is approximately a small circle around the constant term. For very large r, this is approximately a large circle that wraps n times around the origin, where n is the degree of the polynomial. For topological reasons, at some r value in between, the image must pass through the origin. When it does, a root is found. This applet lets you vary the radius and search out these roots. The real and imaginary parts of the polynomial must be entered separately in the function entering panels at the bottom of the applet in this version. There are instructions for how to enter other functions into these applets, but probably you should just try to enter things in and experiment always use * for multiplication, and ^ for powers, and make reasonable guesses about function names, and you may not need the instructions. Also, when you click to go to the radius entering panel, click again after you get there. For reason unbeknownst to me, the canvas on which the radius and such is reported erases itself after being drawn in. But a second click brings it back. The second click makes the exact same graphics calls, so this shouldn't happen. In any case, a second click cures it. If you know how to solve this the source is available on-line please let me know.

9. Fund Theorem Of Algebra The fundamental theorem of algebra. The Fundamental theorem of algebra (FTA) states Every polynomial equation of degree n with complex http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fund_theorem_of_algebra.html

The fundamental theorem of algebra Algebra index History Topics Index The Fundamental Theorem of Algebra (FTA) states Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. In fact there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early studies of equations by al-Khwarizmi (c 800) only allowed positive real roots and the FTA was not relevant. Cardan was the first to realise that one could work with quantities more general than the real numbers. This discovery was made in the course of studying a formula which gave the roots of a cubic equation. The formula when applied to the equation x x + 4 gave an answer involving -121 yet Cardan knew that the equation had x = 4 as a solution. He was able to manipulate with his 'complex numbers' to obtain the right answer yet he in no way understood his own mathematics. Bombelli , in his Algebra , published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'. Descartes in 1637 says that one can 'imagine' for every equation of degree n n roots but these imagined roots do not correspond to any real quantity.

10. Nightlife On The Chalkboard A basic walkthrough that covers some prealgebra/algebra concepts such as solving and graphing linear equations, Pythagorean theorem, square roots, permutations, linear combinations, and much more. http://rachel5nj.tripod.com/NOTC/main.html

var cm_role = "live" var cm_host = "tripod.lycos.com" var cm_taxid = "/memberembedded" Check out the NEW Hotbot Tell me when this page is updated Nightlife on the Chalkboard So, what is this? It's just a place with a some tutorial things, not that much, in the way of math. Since I am now in 8th grade and in Algebra I , i t will cover some pre-algebra and algebra. Enjoy, and please sign my guest book! Guestbook Credit goes to my teacher for most, if not all, of the definitions, and the "Balance" principle in linear equations. Links to My Pages: Combining Like Terms Solving Linear Equations Graphing Linear Equations Square Roots ... Factoring Polynomials Main ((subtopic below)) F actoring Q uadratic ... graph s Solving Systems of Equations using algebra (substitution and elimination) Permutations and Combinations Solving Quadratic Equations Graphing Parabolas (Quadratic Equations) Converting Bases ... Functions

11. Fundamental Theorem Of Algebra -- From MathWorld Fundamental theorem of algebra. 101103, 1996. Krantz, S. G. The Fundamental theorem of algebra. §1.1.7 and 3.1.4 in Handbook of Complex Variables. http://mathworld.wolfram.com/FundamentalTheoremofAlgebra.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Algebra Polynomials Fundamental Theorem of Algebra Every polynomial equation having complex coefficients and degree has at least one complex root . This theorem was first proven by Gauss It is equivalent to the statement that a polynomial P z ) of degree n has n values (some of them possibly degenerate) for which Such values are called polynomial roots . An example of a polynomial with a single root of multiplicity is which has z = 1 as a root of multiplicity 2. Degenerate Frivolous Theorem of Arithmetic Polynomial Polynomial Factorization ... search Courant, R. and Robbins, H. "The Fundamental Theorem of Algebra." §2.5.4 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 101-103, 1996. Krantz, S. G. "The Fundamental Theorem of Algebra." §1.1.7 and 3.1.4 in

12. Emmy Noether Modern physics is largely defined by fundamental symmetry principles and Noether's theorem. It requires little more than high school algebra to understand and manipulate these concepts. We prescribe a symmetry module to insert into the curriculum, of a few weeks length. http://www.emmynoether.com/

Emmy Noether Welcome to EmmyNoether.com Our website devoted to teaching "Symmetry" in the introductory physics curriculum is currently undergoing renovation. But, we are pleased to announce our forthcoming book, which will deal with the topics of Symmetry and Physics, and the Life of Emmy Noether (and many additional interesting subjects). If you are interested in learning more about Relativity, Superstrings, Quantum Cosmology, Quarks, and Black Holes, then Start with this Book!!! Look for it at your local bookstore in the Fall of 2004: click on book jacket for more information Symmetry and the Beautiful Universe Prometheus Books, expected release in Fall, 2004 Authors: (click on author's names for homepages.) Christopher T. Hill, Theoretical Physics Department, Fermi National Accelerator Laboratory, MS. 106, P.O. Box 500, Batavia, Illinois, 60510, USA email: hill@fnal.gov Leon M. Lederman, (Nobel Laureate in Physics ) Illinois Math and Science Academy Resident Scholar 1500 W. Sullivan Rd.

13. Lie Algebra -- From MathWorld of some where the associative algebra A may be taken to be the linear operators over a vector space V (the PoincaréBirkhoff-Witt theorem; Jacobson 1979, pp. http://mathworld.wolfram.com/LieAlgebra.html

INDEX Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics ... Alphabetical Index ABOUT THIS SITE About MathWorld About the Author DESTINATIONS What's New MathWorld Headline News Random Entry ... Live 3D Graphics CONTACT Email Comments Contribute! Sign the Guestbook MATHWORLD - IN PRINT Order book from Amazon Algebra Group Theory Lie Theory ... Lie Algebra Lie Algebra A nonassociative algebra obeyed by objects such as the Lie bracket and Poisson bracket . Elements f g , and h of a Lie algebra satisfy and (the Jacobi identity ). The relation implies For characteristic not equal to two, these two relations are equivalent. The binary operation of a Lie algebra is the bracket An associative algebra A with associative product xy can be made into a Lie algebra by the Lie product Every Lie algebra L is isomorphic to a subalgebra of some where the associative algebra A may be taken to be the linear operators over a vector space V (the ; Jacobson 1979, pp. 159-160). If L is finite dimensional, then V can be taken to be finite dimensional ( Ado's theorem for characteristic p Iwasawa's theorem for characteristic The classification of finite dimensional simple Lie algebras over an algebraically closed field of characteristic can be accomplished by (1) determining matrices called Cartan matrices corresponding to indecomposable simple systems of roots and (2) determining the simple algebras associated with these matrices (Jacobson 1979, p. 128). This is one of the major results in Lie algebra theory, and is frequently accomplished with the aid of diagrams called

14. Dave's Math Tables Features common formulas for arithmetic, algebra, geometry, calculus, and statistics. theorem, Also, has forum board to ask questions. Available in both English and Spanish. http://www.math2.org/index.xml

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15. Fundamental Theorem Of Algebra Fundamental theorem of algebra. The Fundamental theorem of algebra establishes this reason and is the topic of the discussion below. http://www.cut-the-knot.org/do_you_know/fundamental.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Fundamental Theorem of Algebra Complex numbers are in a sense perfect while there is little doubt that perfect numbers are complex. Starting from the tail, perfect numbers have been studied by the Ancients ( Elements, IX.36 ). Euler (1707-1783) established the form of even perfect numbers. [Conway and Guy, p137] say this: Are there any other perfect numbers? ... All we know about the odd ones is that they must have at least 300 decimal digits and many factors. There probably aren't any! Every one would agree it's rather a complex matter to write down a number in excess of 300 digits. Allowing for a pun, if there are odd perfect numbers they may legitimately be called complex. What about complex numbers in the customary sense? There is at least one good reason to judge them perfect. The Fundamental Theorem of Algebra establishes this reason and is the topic of the discussion below. In the beginning there was counting which gave rise to the natural numbers (or integers ): 1,2,3, and so on. In the space of a few thousand years, the number system kept getting expanded to include fractions, irrational numbers, negative numbers and zero, and eventually complex numbers. Even a cursory glance at the terminology would suggest (except for fractions) the reluctance with which the new numbers have been admitted into the family.

16. Fund Theorem Of Algebra The fundamental theorem of algebra. algebra index. History Topics Index. The Fundamental theorem of algebra (FTA) states. Every polynomial equation of degree n with complex coefficients has n roots http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Fund_theorem_of_algebra.h

The fundamental theorem of algebra Algebra index History Topics Index The Fundamental Theorem of Algebra (FTA) states Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. In fact there are many equivalent formulations: for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early studies of equations by al-Khwarizmi (c 800) only allowed positive real roots and the FTA was not relevant. Cardan was the first to realise that one could work with quantities more general than the real numbers. This discovery was made in the course of studying a formula which gave the roots of a cubic equation. The formula when applied to the equation x x + 4 gave an answer involving -121 yet Cardan knew that the equation had x = 4 as a solution. He was able to manipulate with his 'complex numbers' to obtain the right answer yet he in no way understood his own mathematics. Bombelli , in his Algebra , published in 1572, was to produce a proper set of rules for manipulating these 'complex numbers'. Descartes in 1637 says that one can 'imagine' for every equation of degree n n roots but these imagined roots do not correspond to any real quantity.

17. Fundamental Theorem Of Algebra Fundamental theorem of algebra. Statement and Significance. This is indeed so. But the Fundamental theorem of algebra states even more. http://www.cut-the-knot.org/do_you_know/fundamental2.shtml

CTK Exchange Front Page Movie shortcuts Personal info ... Recommend this site Fundamental Theorem of Algebra Statement and Significance We already discussed the history of the development of the concept of a number. Here I would like to undertake a more formal approach. Thus, in the beginning there was counting. But soon enough people got concerned with equation solving. (If I saw 13 winters and my tribe's law allows a maiden to marry after her 15th winter, how many winters should I wait before being allowed to marry the gorgeous hunter who lives on the other side of the mountain?) The Fundamental Theorem of Algebra is a theorem about equation solving. It states that every polynomial equation over the field of complex numbers of degree higher than 1 has a complex solution. Polynomial equations are in the form P(x) = a n x n + a n-1 x n-1 + ... + a x + a where a n is assumed non-zero (for why to mention it otherwise?), in which case n is called the degree of the polynomial P and of the equation above. a i 's are known coefficients while x is an unknown number. A number a is a solution to the equation P(x) = if substituting a for x renders it identity : P(a) = 0. Coefficients are assumed to belong to a specific set of numbers where we also seek a solution. The polynomial form is very general but often studying P(x) = Q(x) is more convenient.

18. Gauss One of the alltime greats, Gauss began to show his mathematical brilliance at the early age of seven. He is usually credited with the first proof of The Fundamental theorem of algebra. http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Gauss.html

Johann Carl Friedrich Gauss Born: 30 April 1777 in Brunswick, Duchy of Brunswick (now Germany) Died: Click the picture above to see thirteen larger pictures Show birthplace location Previous (Chronologically) Next Biographies Index Previous (Alphabetically) Next Main index At the age of seven, Carl Friedrich Gauss In 1788 Gauss began his education at the Gymnasium binomial theorem and the arithmetic- geometric mean, as well as the law of quadratic reciprocity and the prime number theorem. Kaestner , whom Gauss often ridiculed. His only known friend amongst the students was Farkas Bolyai . They met in 1799 and corresponded with each other for many years. ruler and compasses This was the most major advance in this field since the time of Greek mathematics and was published as Section VII of Gauss's famous work, Disquisitiones Arithmeticae Gauss returned to Brunswick where he received a degree in 1799. After the Duke of Brunswick had agreed to continue Gauss's stipend, he requested that Gauss submit a doctoral dissertation to the University of Helmstedt. He already knew Pfaff , who was chosen to be his advisor. Gauss's dissertation was a discussion of the

19. Robbins Algebras Are Boolean been solved Every Robbins algebra is Boolean. This theorem was proved automatically by EQP, a theorem proving program developed at Argonne http://www.mcs.anl.gov/~mccune/papers/robbins

Robbins Algebras Are Boolean William McCune Automated Deduction Group Mathematics and Computer Science Division Argonne National Laboratory Posted on the Web October 15, 1996. Last updated September 24, 2003. These Web pages contain some information on the solution of the Robbins problem. A paper on this topic appears in the Journal of Automated Reasoning [W. McCune, "Solution of the Robbins Problem", JAR 19(3), 263276 (1997)]. Here is a preprint . The JAR paper has simpler proofs than the ones below on this page. Here are the input files and proofs corresponding to the JAR paper A draft of a press release , intended for a wider audience, is also available. Introduction The Robbins problem-are all Robbins algebras Boolean?-has been solved: Every Robbins algebra is Boolean. This theorem was proved automatically by EQP , a theorem proving program developed at Argonne National Laboratory. Historical Background In 1933, E. V. Huntington presented [1,2] the following basis for Boolean algebra: x + y = y + x. [commutativity] (x + y) + z = x + (y + z). [associativity] n(n(x) + y) + n(n(x) + n(y)) = x. [Huntington equation] Shortly thereafter, Herbert Robbins conjectured that the Huntington equation can be replaced with a simpler one [5]:

20. Orbital Library A class library providing objectoriented representations and algorithms for logic, mathematics and artificial intelligence. It comprises theorem proving, computer algebra, search and planning, as well as learning algorithms. http://www.functologic.com/orbital/

Orbital library The Orbital library is a class library providing object-oriented representations and algorithms for logic, mathematics and artificial intelligence. It comprises theorem proving, computer algebra, search and planning, as well as machine learning algorithms. Generally speaking, the conceptual idea behind the Orbital library is to provide extensional services and components for Java, which surround the heart of many scientific applications. Hence the name Orbital library. In order to satisfy the requirements of high reusability, the design of this foundation class library favors flexibility, conceptual simplicity and generalisation. So many sophisticated problems can be solved easily with its adaptable components. See the summary of features , and the review document for more information. However, for a closer look, refer to the online documentation . As a brief overview of the documentation, also refer to the hints recommending very important classes You can get this Java library and its documentation here: download Orbital library and documentation browse the documentation online browse the documentation online (alternative server, with less examples)

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