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21. Theorem Proving And Algebra
theorem Proving and algebra. Reading. Joseph Goguen, theorem Proving and algebra,draft textbook in preparation; Joseph Goguen, Proving and Rewriting;
http://www.cs.ucsd.edu/users/goguen/courses/thpro.html
##### Theorem Proving and Algebra
Optional course, 16 lectures in Michaelmas Term
Professor J A Goguen and Dr G Malcolm
##### Aims
This course of lectures treats algebraic proof techniques and their application to various problems in Computer Science. Exercises use the system for mechanical proofs in areas ranging from group theory to VLSI.
##### Synopsis
The following is an outline. Signature, algebra, equation and theory; homomorphism, initial (or word, or term) algebra, substitution; equational deduction, variety and completeness; term rewriting, interpretation and equivalence of theories, the theorem of constants; quotient algebras and rewriting modulo equations; induction; conditional equations; second order universal quantifiers. There is also an introduction to the OBJ system, and applications to group theory, abstract data types (including various number systems, lists and stacks), propositional calculus and digital hardware. There is a draft textbook, currently about two-thirds finished.
• Joseph Goguen

22. ABSTRACT ALGEBRA ON LINE: Contents
modules over a PID(10.7.5) First isomorphism theorem(7.1.1) Fitting's lemma for modules(10.4.5)Frattini's argument(7.8.5) Fundamental theorem of algebra(8.3.10
http://www.math.niu.edu/~beachy/aaol/contents.html
##### ABSTRACT ALGEBRA ON LINE
This site contains many of the definitions and theorems from the area of mathematics generally called abstract algebra. It is intended for undergraduate students taking an abstract algebra class at the junior/senior level, as well as for students taking their first graduate algebra course. It is based on the following books. Abstract Algebra Second Edition , by John A. Beachy and William D. Blair
Waveland Press
, P.O. Box 400, Prospect Heights, Illinois, 60070, Tel. 847 / 634-0081 Abstract Algebra II , by John A. Beachy
Interested students may also wish to refer to a closely related site that includes solved problems: the OnLine Study Guide for Abstract Algebra In addition to the Table of Contents, this page contains an index of definitions and theorems, so it can be searched for detailed references on subject area pages. Topics from the first volume are marked by the symbol and those from the second volume by the symbol Click here for the version with frames. The site is maintained by John Beachy as a service to students. email: beachy@math.niu.edu

23. ABSTRACT ALGEBRA ON LINE: Integers
INTEGERS. Excerpted from Beachy/Blair, Abstract algebra, 2nd Ed., © 1996 Chapter1. 1.1, 1.2 Divisors; prime factorization 1.3, 1.4 Congruences 1.1.3 theorem.
http://www.math.niu.edu/~beachy/aaol/integers.html
##### INTEGERS
Excerpted from Beachy/Blair, Abstract Algebra 2nd Ed.
##### Chapter 1
Divisors; prime factorization
Congruences
##### Divisors; prime factorization
integers , and will be denoted by Z 1.1.1. Definition. An integer a is called a multiple of an integer b if a=bq for some integer q. In this case we also say that b is a divisor In the above case we can also say that b is a factor of a, or that a is divisible by b. If b is not a divisor of a, meaning that a bq for all q Z , then we write b a. The set of all multiples of an integer a will be denoted by a Z Z Z 1.1.2. Axiom. [Well-Ordering Principle] Every nonempty set of natural numbers contains a smallest element. 1.1.3 Theorem. [Division Algorithm] quotient ) and r (the remainder ) such that a=bq+r, with 1.1.4. Theorem. Let I be a nonempty set of integers that is closed under addition and subtraction. Then I either consists of zero alone or else contains a smallest positive element, in which case I consists of all multiples of its smallest positive element. 1.1.5. Definition.

24. Complex Numbers: The Fundamental Theorem Of Algebra
Dave's Short Course on The Fundamental theorem of algebra. As remarkedbefore, in the 16th century Cardano noted that the sum of
http://www.clarku.edu/~djoyce/complex/fta.html
##### Dave's Short Course on The Fundamental Theorem of Algebra
As remarked before, in the 16th century Cardano noted that the sum of the three solutions to a cubic equation x bx cx d b , the negation of the coefficient of x . By the 17th century the theory of equations had developed so far as to allow Girard (1595-1632) to state a principle of algebra, what we call now "the fundamental theorem of algebra". His formulation, which he didn't prove, also gives a general relation between the n solutions to an n th degree equation and its n coefficients. An n th degree equation can be written in modern notation as x n a x n a n x a n x a n where the coefficients a a n a n , and a n are all constants. Girard said that an n th degree equation admits of n solutions, if you allow all roots and count roots with multiplicity. So, for example, the equation x x x + 1 = has the two solutions 1 and 1. Girard wasn't particularly clear what form his solutions were to have, just that there be n of them: x x x n , and x n Girard gave the relation between the n roots x x x n , and x n and the n coefficients a a n a n , and a n that extends Cardano's remark. First, the sum of the roots

25. Fundamental Theorem Of Algebra
Fundamental theorem of algebra. This is a very powerful algebraic tool.2.3 It says that given any polynomial. we can always rewrite it as.
http://ccrma-www.stanford.edu/~jos/complex/Fundamental_Theorem_Algebra.html
Complex Basics
Complex Roots

Complex Numbers
Contents ... Search
##### Fundamental Theorem of Algebra
This is a very powerful algebraic tool. It says that given any polynomial
we can always rewrite it as
where the points are the polynomial roots , and they may be real or complex. Complex Basics
Complex Roots

Complex Numbers
Contents ...
(How to cite this work)

by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),
Stanford University

26. Math Forum - Problems Library - Algebra, Pythagorean Theorem
algebra Pythagorean theorem Read about expectations for algebra inthe NCTM Standards algebra Standard for Grades 912. algebra
http://mathforum.org/library/problems/sets/alg_pythagorean.html
 TOPICS This page: Pythagorean theorem About Levels of Difficulty Algebra algebra/geometry Pythagorean theorem distance/rate/time exponents/roots/ ... PoW Library Algebra: Pythagorean Theorem Read about expectations for algebra in the NCTM Standards: Algebra Standard for Grades 9-12. Algebra problems that explore students' understanding of the Pythagorean theorem are listed below. For background information elsewhere on our site, explore High School Algebra and High School Geometry in the Ask Dr. Math archives; and see the Pythagorean theorem in the Dr. Math FAQ. For relevant sites on the Web, browse and search Basic Algebra and Equations in our Internet Mathematics Library; to find high-school sites, go to the bottom of the page, set the searcher for High School, and press the Search button. All Wet - Linda Benton Algebra, difficulty level 3. Solve simultaneous equations, find area, and use the Pythagorean theorem and percentages to find the spraying distance needed for a circular sprinkler in a rectangular yard. ... Ant Race - Terry Trotter Algebra, difficulty level 1. Two ants race to a piece of food. Your task is to figure out how far apart they were when the race started. [ ...

27. About "The Fundamental Theorem Of Algebra"
The Fundamental theorem ofalgebra states that any complex polynomial must have a complex root.......The Fundamental theorem of algebra.
http://mathforum.org/library/view/11467.html
 The Fundamental Theorem of Algebra Library Home Full Table of Contents Suggest a Link Library Help Visit this site: http://www.springer-ny.com/catalog/np/apr97np/DATA/0-387-94657-8.html Author: B. Fine, Fairfield Univ., CT; G. Rosenberger, Univ. of Dortmund, Germany Description: The Fundamental Theorem of Algebra states that any complex polynomial must have a complex root. This basic result, whose first accepted proof was given by Gauss, lies at the intersection of the theory of numbers and the theory of equations, and arises also in many other areas of mathematics. This book examines three pairs of proofs of the theorem from three different areas of mathematics: abstract algebra, complex analysis and topology. Levels: College Languages: English Resource Types: Textbooks Math Topics: Modern Algebra Complex Analysis Algebraic Number Theory Algebraic Topology ... Search http://mathforum.org/ webmaster@mathforum.org

28. Fundamental Theorem Of Algebra
Fundamental theorem of algebra. Preliminaries; Operations on Polynomials;Substitution in Polynomials; Fundamental theorem of algebra. Bibliography.
http://mizar.uwb.edu.pl/JFM/Vol12/polynom5.html
Journal of Formalized Mathematics
Volume 12, 2000

University of Bialystok

Association of Mizar Users
##### Fundamental Theorem of Algebra
Robert Milewski
University of Bialystok
This work has been partially supported by TYPES grant IST-1999-29001.
##### MML Identifier:
The terminology and notation used in this paper have been introduced in the following articles [
##### Contents (PDF format)
• Preliminaries
• Operations on Polynomials
• Substitution in Polynomials
• Fundamental Theorem of Algebra
##### Bibliography
1] Agnieszka Banachowicz and Anna Winnicka. Complex sequences Journal of Formalized Mathematics
2] Grzegorz Bancerek. The fundamental properties of natural numbers Journal of Formalized Mathematics
3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences Journal of Formalized Mathematics
4] Czeslaw Bylinski. Binary operations Journal of Formalized Mathematics
5] Czeslaw Bylinski. Functions and their basic properties Journal of Formalized Mathematics
6] Czeslaw Bylinski. Functions from a set to a set Journal of Formalized Mathematics
7] Czeslaw Bylinski. Partial functions Journal of Formalized Mathematics
8] Czeslaw Bylinski.
• 29. ThinkQuest Library Of Entries
Discrete algebra. Binomial theorem. The binomial theorem is a usefulformula for determining the algebraic expression that results
http://library.thinkquest.org/10030/11binoth.htm
##### Welcome to the ThinkQuest Internet Challenge of Entries
The web site you have requested, Seeing is Believing , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to Seeing is Believing click here Back to the Previous Page The Site you have Requested ...
##### Seeing is Believing
A ThinkQuest Internet Challenge 1997 Entry
Click image for the Site Languages : Site Desciption 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.
Students Peter Oakhill College, Castle Hill
Australia Suranthe H Oakhill College
Australia Coaches Tina Oakhill College, Castle Hill

30. ThinkQuest Library Of Entries
A listing of equations and definitions in introductory algebra.Category Kids and Teens School Time Math algebra...... Discrete algebra. Mathematical Induction Sequences and Series. Arithmetic ProgressionGeometric Progression Infinite Series. Binomial theorem. Back to Top.
http://library.thinkquest.org/10030/algecon.htm
##### Welcome to the ThinkQuest Internet Challenge of Entries
The web site you have requested, Seeing is Believing , is one of over 4000 student created entries in our Library. Before using our Library, please be sure that you have read and agreed to our To learn more about ThinkQuest. You can browse other ThinkQuest Library Entries To proceed to Seeing is Believing click here Back to the Previous Page The Site you have Requested ...
##### Seeing is Believing
A ThinkQuest Internet Challenge 1997 Entry
Click image for the Site Languages : Site Desciption 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.
Students Peter Oakhill College, Castle Hill
Australia Suranthe H Oakhill College
Australia Coaches Tina Oakhill College, Castle Hill

31. Abstract: The Fundamental Theorem Of Algebra
Abstract The Fundamental theorem of algebra. Freek Wiedijk. The Fundamental theoremof algebra states that every polynomial over the complex numbers has a root.
http://www.cee.hw.ac.uk/~fairouz/automath2002/abstracts/freekFTA.abst.html
##### Freek Wiedijk
The Fundamental Theorem of Algebra states that every polynomial over the complex numbers has a root. In Nijmegen we have formalised a constructive proof of this theorem in Coq. In this project, we wanted to also set up a library of results (about reals and complex numbers and polynomials) that could be re-used, by us and by others. We have therefore defined an algebraic hierarchy of monoids, groups, rings and so forth that allows to prove generic results and use them for concrete instantiations. In the talk I will briefly outline the FTA project. The main part will consist of an outline of the algebraic hierarchy and its use. This part will contain an explanation of the basic features of Coq.

32. Mathematics Archives - Topics In Mathematics - Computer Algebra
KEYWORDS Conference Proceedings, Distance learning, Multimedia, Numerical integrationand differentiation, Computer algebra, theorem proving, Applications of
http://archives.math.utk.edu/topics/computerAlgebra.html
 Topics in Mathematics Computer Algebra / Cryptology Genetic Algorithms Asian Technology Conference in Mathematics ADD. KEYWORDS: Conference Proceedings, Distance learning, Multimedia, Numerical integration and differentiation, Computer algebra, Theorem proving, Applications of Computer Algebra Systems (CAS), Graphing calculators Automatic Differentiation Bibliography Bibliography of the CATHODE (Computer Algebra Tools for Handling Ordinary Differential Equations) project A Bibliography of Computer Algebra Bibliography about the Maple Symbolic Algebra System ... Cryptologia ADD. KEYWORDS: Cryptology, Journal DOCUMENTA MATHEMATICA - Extra Volume ICM 1998 - Section: 14. Mathematical Aspects of Computer Science ADD. KEYWORDS: Articles SOURCE: DOCUMENTA MATHEMATICA, Proceedings of the International Congress of Mathematicians TECHNOLOGY: Postscript and DVI readers The Enigma Machine ADD. KEYWORDS: Software, History European Cryptography Resources Evolving Algebras ADD. KEYWORDS:

33. Mathematics Archives - Topics In Mathematics - Algebra
KEYWORDS Evolution of Algebraic Symbolism, Fundamental theorem ofAlgebra, Mathematical Induction, Weierstrass Product Inequality;
http://archives.math.utk.edu/topics/algebra.html
 Topics in Mathematics Algebra About - The Human Internet - College Algebra ADD. KEYWORDS: Tutorial, Inequalities, Absolute Values and Exponents, Fractional and Negative Exponents, Polynomials, Factoring Polynomials, Rational Functions, Compound Fractions, Solving Equations, Word Problems, Solving Quadratic Equations, Quadratic Formula, Complex Numbers, Inequalities, Quadratic Inequalities, Graphing Equations and Circles, Lines, Functions, Applications of Functions Algebra ADD. KEYWORDS: Algebra Postulates, Function Basics, Composite Functions, Even and Odd Functions, Inverse Functions, Linear, Quadratic, and Cubic Functions, Monotonic Functions, Periodic Functions Algebra ADD. KEYWORDS: Tutorial, Real Number System, Numerical Representations In Algebra, Algebraic Techniques, Quadratic Equations and Inequalities, Graphing, Functions, Polynomial Functions, Exponential and Logarithmic Functions, Linear Algebra, Discrete Algebra Algebra1: Graphing Linear Equations ADD. KEYWORDS:

 34. P06-Fundamental Theorem Of Algebra.html The Fundamental theorem of algebra. Exposition and application of the fundamentaltheorem of algebra. 2. The Fundamental theorem of algebra.http://www.mapleapps.com/powertools/precalc/html/P06-FundamentalTheoremofAlgebra

35. The Fundamental Theorem Of Algebra
The Fundamental theorem of algebra. theorem 1 Every nonconstant polynomialwith complex coefficients has a complex root. For example
http://www.shef.ac.uk/~puremath/theorems/ftalgebra.html
##### The Fundamental Theorem of Algebra
Theorem 1 Every nonconstant polynomial with complex coefficients has a complex root. For example, a nonconstant polynomial of degree 1 has the form f(z) = az+b with a 0, and this has a root z = -b/a. A polynomial of degree 2 has the form f(z) = az +bz+c, and this has roots given by the familiar quadratic formula z = (-b (b -4ac)])/2a. To use this we need to know how to take square roots of complex numbers, which is achieved by the formula
x+iy
= ((r+x)/2) + ((r-x)/2) i , where r = [ (x +y )]. (Note that the right hand side here only involves square root of positive real numbers.) Alternatively, we can use de Moivre's theorem: we have x+iy = re i q for some q , and then [ (x+iy)] = re i q The case of polynomials of degree 3 is more complicated. A typical cubic polynomial has the form f(z) = az +bz +cz+d. Consider the special case where a, b, c and d are real numbers and a 0, so we can think of f as a real-valued function of a real variable. When x is a large, positive real number the term ax will be much bigger than the other two terms and it follows that f(x) will be positive. Similarly, if x is a large negative real number then the term ax

36. Robbins Algebras Are Boolean
A web text by William McCune describing the solution of this problem by a theoremproving program, Category Science Math algebra...... Several, such as each of all x, n(n(x))=x exists 0 all x, x+0=x all x, x+x=x wereeasily shown by Argonne's theorem provers to make a Robbins algebra Boolean.
http://www-unix.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 February 5, 1998. 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 :

37. Mbox: JAR Special Issue CFP: Computer Algebra & Automated Theorem Proving
JAR special issue CFP Computer algebra Automated theorem Proving. vskip0.3cm {\Large\bf Computer algebra and Automated theorem Proving}.
http://www-unix.mcs.anl.gov/qed/mail-archive/volume-3/0079.html
 Dongming Wang Dongming.Wang@imag.fr Fri, 24 Nov 1995 15:29:40 +0100 Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Roman Matuszewski: "QED II - final report - available" Previous message: Geoff Sutcliffe: "CADE-14 Program Chair" % for LaTeX Computer algebra (CA) methods are among some of the most powerful ones developed for automated theorem proving (ATP) in several specialized mathematical domains. Typical examples are algebraic methods for ATP in geometry, which require complex and heavy polynomial operations. ATP techniques are being incorporated into the design and implementation of current CA systems. Research projects have been exploring the cooperation, combination and integration of CA and ATP systems. This special issue is devoted to reporting significant research developments and to motivating further investigations on the interaction of CA and ATP (systems) in research, education and industrial applications. Original research papers describing recent advances and new insights on all aspects of coupling CA and ATP are solicited. Specific topics of interest include (but

 38. The Prime Glossary: Partial Index: F fundamental theorem of algebra; Fundamental theorem of Arithmetic. (Entries withthe comments 'new' or 'modified' are new, or have been modified in the last 7http://primes.utm.edu/glossary/index.php?match=f

39. Fundamental Theorem Of Algebra
THE FUNDAMENTAL theorem OF algebra. Our object is to prove DeMoivre'sformula. Proof of the fundamental theorem of algebra. Let f(z
http://www.math.lsa.umich.edu/~hochster/419/fund.html
##### THE FUNDAMENTAL THEOREM OF ALGEBRA
Our object is to prove that every nonconstant polynomial f(z) in one variable z over the complex numbers C has a root, i.e. that there is a complex number r in C such that f(r) = 0. Suppose that The key point: one can get the absolute value of a nonconstant COMPLEX polynomial at a point where it does not vanish to decrease by moving along a line segment in a suitably chosen direction. We first review some relevant facts from calculus. Properties of real numbers and continuous functions Fact 1. Every sequence of real numbers has a monotone (nondecreasing or nonincreasing) subsequence. Proof. If the sequence has some term which occurs infinitely many times this is clear. Otherwise, we may choose a subsequence in which all the terms are distinct and work with that. Hence, assume that all terms are distinct. Call an element "good" if it is bigger than all the terms that follow it. If there are infinitely many good terms we are done: they will form a decreasing subsequence. If there are only finitely many pick any term beyond the last of them. It is not good, so pick a term after it that is bigger. That is not good, so pick a term after it that is bigger. Continuing in this way (officially, by mathematical induction) we get a strictly increasing subsequence. QED Fact 2. A bounded monotone sequence of real numbers converges.

 40. Complex Numbers : Fundamental Theorem Of Algebra metadata 1.8 Fundamental theorem of algebra, Fundamental theorem of algebra LetP (z) = be a polynomial of degree n (with real or complex coefficients).http://scholar.hw.ac.uk/site/maths/topic13.asp?outline=

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