21. Fund Theorem Of Algebra The Fundamental theorem of algebra (FTA) states. Every polynomial equation ofdegree n with complex coefficients has n roots in the complex numbers. 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 Version for printing 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.

22. 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 Version for printing 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. , 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

23. Computer Algebra, Theorem Proving, And Types Computer algebra, theorem Proving, and Types Concluding remarks on therelationship between types and theorem proving and a prospectus for what can (and http://www.cs.cornell.edu/Nuprl/PRLSeminar/PRLSeminar94_95/Wilson/Oct4.html

PRL Seminars Computer Algebra, Theorem Proving, and Types Todd Wilson October 4, 1994 Abstract Many computations a mathematician performs can be described in "algebraic" terms, that is, as dealing with various symbolic entities that are combined in restricted ways and are subject to laws (e.g., equations) specifying which combinations are equivalent. The term "computer algebra", as it appears in my title, has this general sense (as opposed to the more restrictive sense of "computational commutative algebra"), and my talk will discuss this subject and its relation to automatic theorem proving and type theory. In more detail, the talk will consist of the following: A survey of examples of computer algebra drawn from several areas of mathematics, including commutative algebra and algebraic geometry, invariant theory, (algebraic) number theory, group theory, Lie algebra, combinatorics, algebraic topology, and analysis (scientific computation). A discussion of the roles automatic theorem proving might have in these fields. A discussion of types, including

24. 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, and linear combinations. http://rachel5nj.tripod.com/

setAdGroup('67.18.104.18'); var cm_role = "live" var cm_host = "tripod.lycos.com" var cm_taxid = "/memberembedded" Search: Lycos Tripod Murderball Share This Page Report Abuse Edit your Site ... Next Nightlife on the Chalkboard So, what is this? If I ever get to updating it, it will be a place with a few tutorial things, nothing much, in the way of math. Since I 'm just in 7th grade math (pre-algebra) , it'll cover some of that (and a little other stuff I throw in) Oh yeah, and please sign my guestbook!! I've got to give some credit to my math book (Mathematics, Structure and Method by Mary Dolciani, Robert Sorgenfrey, and John Graham), and also my teacher (but not that much... some of the voc and the Balance thing in solving linear equations). A lot of this is mine though!!!! :) And I put it on the web. Links to My Pages: Combining Like Terms Solving Linear Equations Graphing Linear Equations Absolute Value Equations ... FOIL method Factoring Polynomials Contact Me

25. Fundamental Theorem Of Algebra - Wikipedia, The Free Encyclopedia In mathematics, the fundamental theorem of algebra states that every complexpolynomial of degree n has exactly n roots (zeroes), counted with multiplicity. http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra

26. 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/

Home About Computer Science Orbital library ... Site Map Orbital library Home Computer Science 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 summary of features review document Print Version Last modified: 2005-06-28

27. Fundamental Theorem Of Algebra - Wikipedia, The Free Encyclopedia More results from en.wikipedia.org PDF Current algebra, theorem and new super evolution equations =o http://en.wikipedia.org/wiki/Fundamental_Theorem_of_Algebra

Fundamental theorem of algebra From Wikipedia, the free encyclopedia. (Redirected from Fundamental Theorem of Algebra In mathematics , the fundamental theorem of algebra states that every complex polynomial of degree n has exactly n roots (zeroes), counted with multiplicity. More formally, if (where the coefficients a a n can be real or complex numbers), then there exist ( not necessarily distinct) complex numbers z z n such that This shows that the field of complex numbers , unlike the field of real numbers , is algebraically closed . An easy consequence is that the product of all the roots equals (−1) n a and the sum of all the roots equals - a n The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by Carl Friedrich Gauss in 1799. (An almost complete proof had been given earlier by d'Alembert .) Gauss produced several different proofs throughout his lifetime. All proofs of the fundamental theorem necessarily involve some analysis , or more precisely, the concept of continuity of real or complex polynomials. The main difficulty in the proof is to show that every non-constant polynomial has at least one zero. We mention approaches via

28. Fundamental Theorem Of Algebra Fundamental theorem of algebra. Complex numbers are in a sense perfect while The Fundamental theorem of algebra establishes this reason and is the topic http://www.cut-the-knot.org/do_you_know/fundamental.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents 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.

29. Fundamental Theorem Of Algebra The Fundamental theorem of algebra is a theorem about equation solving. It statesthat every polynomial equation over the field of complex numbers of degree http://www.cut-the-knot.org/do_you_know/fundamental2.shtml

Username: Password: Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents 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

30. Complex Numbers : Fundamental Theorem Of Algebra Fundamental theorem of algebra. Let P (z) = be a polynomial of degree n (withreal or complex coefficients). The fundamental theorem of algebra states http://scholar.hw.ac.uk/site/maths/topic13.asp?outline=

31. Elements Of Boolean Algebra Laws of Boolean algebra Laws of Boolean algebra. Commutative Law; Associative Law (b); T11 De Morgan s theorem (a) (b). Table 2 Boolean Laws http://www.ee.surrey.ac.uk/Projects/Labview/boolalgebra/

Boolean Algebra Introduction Laws of Boolean Algebra Commutative Law Associative Law ... On-line Quiz Introduction The most obvious way to simplify Boolean expressions is to manipulate them in the same way as normal algebraic expressions are manipulated. With regards to logic relations in digital forms, a set of rules for symbolic manipulation is needed in order to solve for the unknowns. A set of rules formulated by the English mathematician George Boole describe certain propositions whose outcome would be either true or false . With regard to digital logic, these rules are used to describe circuits whose state can be either, 1 (true) or (false) . In order to fully understand this, the relation between the AND gate OR gate and NOT gate operations should be appreciated. A number of rules can be derived from these relations as Table 1 demonstrates. P1: X = or X = 1 Table 1: Boolean Postulates Laws of Boolean Algebra Table 2 shows the basic Boolean laws. Note that every law has two expressions, (a) and (b). This is known as duality . These are obtained by changing every AND(.) to OR(+), every OR(+) to AND(.) and all 1's to 0's and vice-versa.

32. Parikh S Theorem In Commutative Kleene Algebra - Hopkins, Kozen Parikh s theorem says that the commutative image of every context free languageis the commutative image of some regular set. Pilling has shown that this http://citeseer.ist.psu.edu/hopkins99parikhs.html

33. Fundamental Theorem Of Algebra Fundamental theorem of algebra. Fundamental theorem of algebra.\fbox{\emph{Every $n$thorder polynomial possesses exactly http://ccrma-www.stanford.edu/~jos/complex/Fundamental_Theorem_Algebra.html

Complex Basics Complex Roots Complex Numbers Doc Top ... Index 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 Doc Top ... Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications '', by Julius O. Smith III W3K Publishing ISBN by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

34. Fundamental Theorem Of Algebra: Information From Answers.com fundamental theorem of algebra In mathematics , the fundamental theorem of algebrastates that every complex polynomial of degree n has exactly n. http://www.answers.com/topic/fundamental-theorem-of-algebra

showHide_TellMeAbout2('false'); Business Entertainment Games Health ... More... On this page: Wikipedia Best of Web Mentioned In Or search: - The Web - Images - News - Blogs - Shopping fundamental theorem of algebra Wikipedia fundamental theorem of algebra In mathematics , the fundamental theorem of algebra states that every complex polynomial of degree n has exactly n zeroes , counted with multiplicity. More formally, if (where the coefficients a a n can be real or complex numbers), then there exist ( not necessarily distinct) complex numbers z z n such that This shows that the field of complex numbers , unlike the field of real numbers , is algebraically closed n a and the sum of all the roots equals - a n The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by Carl Friedrich Gauss in 1799. (An almost complete proof had been given earlier by d'Alembert .) Gauss produced several different proofs throughout his lifetime. All proofs of the fundamental theorem necessarily involve some analysis , or more precisely, the concept of continuity of real or complex polynomials. The main difficulty in the proof is to show that every non-constant polynomial has at least one zero. We mention approaches via

35. PlanetMath: Fundamental Theorem Of Algebra proof of fundamental theorem of algebra (argument principle) (Proof) by rspuzioproof of fundamental theorem of algebra (Rouché s theorem) (Proof) by http://planetmath.org/encyclopedia/FundamentalTheoremOfAlgebra.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About fundamental theorem of algebra (Theorem) Let be a non-constant polynomial . Then there is with In other words, is algebraically closed "fundamental theorem of algebra" is owned by Evandar view preamble View style: HTML with images page images TeX source See Also: complex number complex Attachments: proof of the fundamental theorem of algebra (Liouville's theorem) (Proof) by Evandar proof of fundamental theorem of algebra (Proof) by scanez fundamental theorem of algebra result (Theorem) by rspuzio proof of fundamental theorem of algebra (due to D'Alembert) (Proof) by rspuzio proof of fundamental theorem of algebra (argument principle) (Proof) by rspuzio proof of fundamental theorem of algebra (Rouch©'s theorem) (Proof) by rspuzio Cross-references: algebraically closed polynomial There are 16 references to this object.

36. PlanetMath: Proof Of Fundamental Theorem Of Algebra (Rouché's Theorem) The fundamental theroem of algebra can be proven using Rouché s theorem. proof of fundamental theorem of algebra (Rouché s theorem) is owned by http://planetmath.org/encyclopedia/ProofOfFundamentalTheoremOfAlgebraRouchesTheo

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... Random Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About proof of fundamental theorem of algebra (Rouch©'s theorem) (Proof) The fundamental theroem of algebra can be proven using simple , it can be thought of as a ``toy model'' (see toy theorem ) for theorems on the zeros of analytic functions . For a variant of this proof in terms of the argument principle fundamental theorem of algebra (argument principle). Proof Denote the order of by Without loss of generality , we may assume that the leading coefficient of is and write Let . Suppose that . Since whenever . Hence, we have the string of inequalities Since polynomials in are analytic functions of in the whole complex plane , they are certainly analytic functions in the disk we conclude that and have the same number of zeros in the disk . Since has a single zero of order at , which counts as zeros according to multiplicity must also have zeros counted according to multiplicity. (By the way we chose

37. AoPS Math Forum :: View Topic - Fundamental Theorem Of Algebra All times are GMT 7 Hours, Fundamental theorem of algebra Post Posted TueApr 12, 2005 646 am Post subject Fundamental theorem of algebra http://www.artofproblemsolving.com/Forum/topic-33344.html

The AoPS Independent Study Programs begins Friday, Sept 9. Enrollments are limited. Enrollment deadline is Sept 19. Font Size: The time now is Fri Sep 16, 2005 4:09 pm All times are GMT - 7 Hours Fundamental theorem of algebra Mark the topic unread Math Forum College Playground ... Theorems, Formulas and Other Questions Moderators: amfulger fedja Kent Merryfield Moubinool Page of [2 Posts] Printer View Tell a Friend View previous topic View next topic Author Message liyi Navier-Stokes Equations Joined: 17 Jul 2003 Posts: 1534 Posted: Tue Apr 12, 2005 6:46 am Post subject: Fundamental theorem of algebra Theorem Every function has a real or complex root. Sketch of Proof Suppose ) such that where If there is a pair of such that , the theorem is proved. Consider the function then and where is a continuous function of Finally, let us consider and where is positive. If , then the integrand is continuous and it must holds that . Then we prove that the equality fails for large enough. Anyone who likes to complete the proof? Back to top liyi Navier-Stokes Equations Joined: 17 Jul 2003 Posts: 1534 Posted: Tue Apr 12, 2005 7:16 am

38. Robbins Algebras Are Boolean has been solved Every Robbins algebra is Boolean. This theorem was were easilyshown by Argonne s theorem provers to make a Robbins algebra Boolean. 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]:

39. Fundamental Theorem Of Algebra - Definition Of Fundamental Theorem Of Algebra In The fundamental theorem of algebra (now considered something of a misnomer bymany mathematicians) states that every complex polynomial of degree n has http://encyclopedia.laborlawtalk.com/Fundamental_theorem_of_algebra

Add to Favorites General Encyclopedia Legal ... Law forum Search Word: Visit our Law forums Labor Law Accidents Law Bankruptcy Law Business Law ... Tax Law The fundamental theorem of algebra (now considered something of a misnomer by many mathematicians) states that every complex polynomial of degree n has exactly n zeroes, counted with multiplicity. More formally, if (where the coefficients a a n can be real or complex numbers), then there exist ( not necessarily distinct) complex numbers z z n such that This shows that the field of complex numbers , unlike the field of real numbers , is algebraically closed n a and the sum of all the roots equals - a n The theorem had been conjectured in the 17th century but could not be proved since the complex numbers had not yet been firmly grounded. The first rigorous proof was given by Carl Friedrich Gauss in 1799. (An almost complete proof had been given earlier by d'Alembert .) Gauss produced several different proofs throughout his lifetime. All proofs of the fundamental theorem necessarily involve some analysis , or more precisely, the concept of continuity of real or complex polynomials. The main difficulty in the proof is to show that every non-constant polynomial has at least one zero. We mention approaches via

40. Fundamental Theorem Of Algebra Internet Resources for the Fundamental theorem of algebra Bibliography for theFundamental theorem of algebra short Bibliography for the Fundamental http://math.fullerton.edu/mathews/n2003/FunTheoremAlgebraBib.html

Research Experience for Undergraduates The Fundamental Theorem of Algebra Internet Resources for the Fundamental Theorem of Algebra Bibliography for the Fundamental Theorem of Algebra - short ... Return to Numerical Methods - Numerical Analysis (c) John H. Mathews 2004

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