Geometry.Net - Theorems_And_Conjectures: Algebra Theorem

The Fundamental Theorem Of Algebra.
How to think of a proof of the fundamental theorem of algebra. Prerequisites.A familiarity with polynomials and with basic real analysis. Statement.
http://www.dpmms.cam.ac.uk/~wtg10/ftalg.html

Extractions: A familiarity with polynomials and with basic real analysis. Every polynomial (with arbitrary complex coefficients) has a root in the complex plane. (Hence, by the factor theorem, the number of roots of a polynomial, up to multiplicity, equals its degree.) If you have heard of the impossibility of solving the quintic by radicals, or if you have simply tried and failed to solve such equations, then you will understand that it is unlikely that algebra alone will help us to find a solution of an arbitrary polynomial equation. In fact, what does it mean to solve a polynomial equation? When we `solve' quadratics, what we actually do is reduce the problem to solving quadratics of the particularly simple form x =C. In other words, our achievement is relative: if it is possible to find square roots, then it is possible to solve arbitrary quadratic equations. But is it possible to find square roots? Algebra cannot help us here. (What it can do is tell us that the existence of square roots does not lead to a contradiction of the field axioms. We simply "adjoin" square roots to the rational numbers and go ahead and do calculations with them - just as we adjoin i to the reals without worrying about its existence. See my
Theorem EQ-3: On Ternary Boolean Algebra
next up previous Next theorem EQ4 Group Theory Up Summary ofOtter Outputs Previous theorem EQ-2 Robbins algebra, theorem
http://www-fp.mcs.anl.gov/~lusk/papers/contest/node19.html
Theorem EQ-2: Robbins Algebra, Boolean
next up previous Next theorem EQ3 On Ternary Up Summary of Otter Outputs Previoustheorem EQ-1 The Commutator theorem EQ-2 Robbins algebra, 2 2 Boolean.
http://www-fp.mcs.anl.gov/~lusk/papers/contest/node18.html
Fundamental Theorem Of Algebra
Fundamental theorem of algebra Gauss' Proof of the Fundamental theorem ofalgebra Translated by Ernest Fandreyer, MS, Ed.D. Professor Emeritus
http://libraserv1.fsc.edu/proof/gauss.htm
Fundamental Theorem Of Algebra
Fundamental theorem of algebra. Keywords FTA, Fundamental, theorem, algebra, Constructive,Real, Complex, Polynomial. The README file of the contribution
http://coq.inria.fr/contribs/fta.html

Extractions: A constructive proof of the Fundamental Theorem of Algebra (every non-trivial polynomial equation P(z)=0 always has a solution in the complex plane) Download (archive compatible with Coq V7.4) Authors: Institution: Nijmegen university Keywords: FTA, Fundamental, Theorem, Algebra, Constructive, Real, Complex, Polynomial The README file of the contribution: This page was automatically generated from this description file
Fundamental Theorem Of Algebra
Fundamental theorem of algebra. The fundamental theorem of algebra (FTA)states Every polynomial of degree n with complex coefficients
http://www.und.edu/dept/math/history/fundalg.htm

Extractions: The fundamental theorem of algebra (FTA) states Every polynomial of degree n with complex coefficients has n roots in the complex numbers. There are many other equivalent versions of this, for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early work with equations only considered positive real roots so the FTA was not relevant. Cardan realized that one could work with numbers outside of the reals while studying a formula for the roots of a cubic equation. While solving x = 15x + 4 using the formula he got an answer involving the square root of -121. He manipulated this to obtain the correct answer, x = 4, even though he did not understand exactly what he was doing with these "complex numbers." In 1572 Bombelli created rules for these "complex numbers." In 1637 Descartes said that one can "imagine" for every equation of degree n n roots, but these imagined roots do not correspond to any real quantity. Albert Girard , a Flemish mathematiciam, was the first to claim that there are always n solutions to a polynomial of degree n in 1629 in . He does not say that the solutions are of the form a + b i , a, b real. Many mathematicians accepted Girard's claim that a polynomial equation must have
Fundamental Theorem Of Algebra
Fundamental theorem of algebra. The fundamental theorem of algebra (FTA)states Every polynomial of degree n with complex coefficients
http://www.und.edu/instruct/lgeller/fundalg.html

Extractions: The fundamental theorem of algebra (FTA) states: Every polynomial of degree n with complex coefficients has n roots in the complex numbers. There are many other equivalent versions of this, for example that every real polynomial can be expressed as the product of real linear and real quadratic factors. Early work with equations only considered positive real roots so the FTA was not relevant. Cardan realized that one could work with numbers outside of the reals while studying a formula for the roots of a cubic equation. While solving x = 15x + 4 using the formula he got an answer involving the square root of -121. He manipulated this to obtain the correct answer, x = 4, even though he did not understand exactly what he was doing with these "complex numbers." In 1572 Bombelli created rules for these "complex numbers." In 1637 Descartes said that one can "imagine" for every equation of degree n n roots, but these imagined roots do not correspond to any real quantity. Albert Girard , a Flemish mathematiciam, was the first to claim that there are always n solutions to a polynomial of degree n in 1629 in . He does not say that the solutions are of the form a + b i , a, b real. Many mathematicians accepted Girard's claim that a polynomial equation must have
The Binomial Theorem And Other Algebra
The Binomial theorem and other algebra. At its simplest, the binomial theoremgives an expansion of (1 + x) n for any positive integer n. We have
http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/node15.html

Extractions: There is a more general form: a b n a n na n-1 b a n-2 b a n-k b k b n with corresponding special cases. Formally this result is only valid for any positive integer n ; in fact it holds appropriately for more general exponents as we shall see in Chapter Another simple algebraic formula that can be useful concerns powers of differences: a b a b a b a b a b a ab b a b a b a a b ab b
The Fundamental Theorem Of Algebra
The Fundamental theorem of algebra. theorem 6.12 (Fundamental theorem of algebra)Every polynomial with complex coefficients has a root. Ran Levi 200003-13.
http://www.maths.abdn.ac.uk/~ran/mx4509/mx4509-notes/node18.html

Extractions: Next: Calculation of the Fundamental Up: Some First Applications Previous: The Degree of a S We now present one of the most classical theorems in the history of mathematics, which admits several proofs, some of which the reader might already be familiar with. Theorem 6.12 (Fundamental Theorem of Algebra) Every polynomial with complex coefficients has a root.
Fundamental Theorem Of Linear Algebra
Fundamental theorem of Linear algebra. Inner Products and Orthogonality.Thetheorem. An Example. Up to Linear algebra Part II
http://www.ma.iup.edu/projects/CalcDEMma/linalg2/linalg218.html
Leaving Cert. Higher Level Maths - Algebra - The Factor Theorem
You are here Home / Category Index / algebra / The Factor theorem.The Factor theorem By David Spollen. How to use this applet
http://www.netsoc.tcd.ie/~jgilbert/maths_site/applets/algebra/the_factor_theorem

Extractions: Search for: in Entire website Algebra Complex Numbers Matrices Sequences and series Differentiation Integration Circle Vectors Linear Transformations Line Geometry Trigonometry Probability Further Calculus and Series Website Home Algebra The Factor Theorem No Title ... Integration You are here: Home Category Index Algebra / The Factor Theorem The Factor Theorem: -Enter the co-efficients (a, b, c, d) of any given cubic polynomial in the boxes provided for them. -Enter the value of x in its box provided. -Click on the "Enter x" button. -An equation should appear containing your values and a report as to whether your x expression is a factor of your polynomial or not. Proof of the factor theorem: Some other key pointers:
NRICH | Secondary Topics | Algebra | Polynomials | Binomial Theorem
Top Level + 3D algebra + Applications + Equations Linear + Logical + Operations+ Percentages - Polynomials - Binomial theorem Binomial ( February 2001
http://www.nrich.maths.org.uk/topic_tree/Algebra/Polynomials/Binomial_Theorem/

Extractions: Bernard's Bag(P) - solutions(P) Penta Probs(P) - solutions(P) Let Me Try(P) - solutions(P) Kid's Mag(P) Play Games(P) Staff Room(P) 6 Problems - solutions 15+Challenges - solutions Articles Games LOGOland Editorial News Click on the folders to browse problem topics from the secondary site. You can then go directly to each of the problems. Top Level
Citations: Theorem Proving And Algebra - Goguen (ResearchIndex)
Retrieving documents Joseph Goguen. theorem Proving and algebra. MIT, to appear1996. Joseph Goguen. theorem Proving and algebra. MIT, to appear 1996.
http://citeseer.nj.nec.com/context/38836/0

Extractions: JAIST RESEARCH REPORT IS-RR-96-0024S 1 Logical Semantics.. - Razvan Diaconescu And (Correct) ....underlying logics. The following table shows the correspondence between specification programming paradigms and logics as they appear in the actual version of CafeOBJ, also pointing to some basic references. ABBREVIATION LOGIC SPEC PGM PARADIGM BASIC REF. MSA many sorted algebraic specification algebra OSA order sorted algebraic specification [13, 22, 15] algebra with subtypes HSA hidden sorted behavioural specification [16] algebra HOSA hidden order sorted behavioural specification [16, 1] algebra with subtypes RWL rewriting logic concurrent [27] algebraic specification OSRWL .
Citations: Automated Theorem Proving In Support Of Computer Algebra: Symbolic De
are several systems that study application like this AGLM99, FSS98, Fis00, DHHW98All of them use conventional theorem provers or computer algebra systems in
http://citeseer.nj.nec.com/context/1301627/273622

Extractions: This paper is cited in the following contexts: Towards an Knowledge-Centered Infrastructure for Web-Based.. - Kohlhase (2001) (Correct) ....base is that for a helpful answer, we have to determine, which cases t our situation, or at least which cases can be eliminated, a task which requires proving (in) consistency of the case descriptions with the situation at hand. There are several systems that study application like this [ , FSS98, Fis00, DHHW98] All of them use conventional theorem provers or computer algebra systems in the background, and could bene t from MathWeb, as well as contribute to it, if used as mathematical services. Finally, the user query can often contain variables (wildcards) itself.
No Match For Fundamental Theorem Of Algebra
No match for Fundamental theorem of algebra. Sorry, the term Fundamentaltheorem of algebra is not in the dictionary. Check the spelling
http://wombat.doc.ic.ac.uk/foldoc/foldoc.cgi?Fundamental Theorem of Algebra
PinkMonkey.com Algebra Study Guide 4.1 Theorem
CHAPTER 4 QUADRATIC EQUATIONS. 4.1 theorem. If a, b Ï R and ab= 0 then either a = 0 or b = 0. Corollary 1 If a . b = 0 but a
http://www.pinkmonkey.com/studyguides/subjects/algebra/chap4/a0404101.asp
Fundamental Theorem Of Algebra
Fundamental theorem Of algebra. Start your search on Fundamental theoremOf algebra. Other educational search engines Ask Jeeves
http://www.virtualology.com/virtualpubliclibrary/hallofeducation/Mathematics/Fun

Extractions: Unauthorized Site: This site and its contents are not affiliated, connected, associated with or authorized by the individual, family, friends, or trademarked entities utilizing any part or the subjects entire name. Any official or affiliated sites that are related to this subject will be hyper linked below upon submission and Virtualology's review. 2000 by Virtualology TM . All rights reserved.
Theorem 4.1.8: Algebra On Series
theorem 4.1.8 algebra on Series. Let and be two absolutely convergent series.Then The sum of the two series is again absolutely convergent.
http://www.shu.edu/projects/reals/numser/proofs/sumalgbr.html

Extractions: Theorem 4.1.8: Algebra on Series Let and be two absolutely convergent series. Then: The sum of the two series is again absolutely convergent. Its limit is the sum of the limit of the two series. The difference of the two series is again absolutely convergent. Its limit is the difference of the limit of the two series. The product of the two series is again absolutely convergent. Its limit is the product of the limit of the two series ( Cauchy Product Context The proof of the first statement is a simple application of the triangle inequality. Let A n n and B n n Assume that A n converges to A and B n converges to B . Then S n + b + b n + b n if n is big enough. Therefore the sequence n converges to A + B , as required. The proof for the difference of sums is similar. The proof for the Cauchy product, on the other hand, is much more complicated and will be given in the statement on the Cauchy Product Interactive Real Analysis , ver. 1.9.3
Theorem 6.5.7: Algebra With Derivatives
theorem 6.5.7 algebra with Derivatives. Addition Rule If f and gare differentiable at x = c then f(x) + g(x) is differentiable
http://www.shu.edu/projects/reals/cont/proofs/diffalg.html

Extractions: Theorem 6.5.7: Algebra with Derivatives Addition Rule : If f and g are differentiable at x = c then f(x) + g(x) is differentiable at x = c and (f(x) + g(x)) = f'(x) + g'(x) Product Rule : If f and g are differentiable at x = c then f(x) g(x) is differentiable at x = c and (f(x) g(x)) = f'(x) g(x) + f(x) g'(x) Quotient Rule : If f and g are differentiable at x = c , and g(c) # then f(x) / g(x) is differentiable at x = c , and ( f(x) / g(x) ) = Chain Rule : If g is differentiable at x = c , and f is differentiable at x = g(c) then f(g(x)) is differentiable at x = c , and f(g(x)) = f'(g(x)) g'(x) Context These proofs, except for the chain rule, consist of adding and subtracting the same terms and rearranging the result. We will prove the product and chain rule, and leave the others as an exercise. Product rule: Chain Rule: A 'quick and dirty' proof would go as follows: Since g is differentiable, g is also continuous at x = c. Therefore, as x approaches c we know that g(x) approaches g(c). The first factor, by a simple substitution, converges to f'(u), where u = g(c). The second factor converges to g'(c). Hence, by our rule on product of limits we see that the final limit is going to be f'(u) g'(c) = f'(g(c)) g'(c), as required. But this 'simple substitution' may not be mathematically precise. Here is a better proof of the chain rule. Define the function h(t) as follows, for a fixed s = g(c):
Gausss 1799 Proof Of The Fundamental Theorem Of Algebra
EDITORIAL. Teach Gausss 1799 Proof Of the FundamentalTheorem of algebra. From Spring 2002 21st Century issue.
http://www.21stcenturysciencetech.com/articles/Spring02/Gauss_02.html

Extractions: EDITORIAL From Spring 2002 21st Century issue. An Induced Mental Block A New Curriculum We have all heard the frequent laments among our co-thinkers and professional colleagues at the sadly reduced state of science and mathematics education in our nation. As in all such matters, after the righteous indignation and hand-wringing, is over, one must ask oneself the realistic question: Are you part of the problem, or part of the solution? If you are not sure, we have a proposal for you. To introduce it, I ask you to perform the following experiment. STEP 1: As a suitable subject, locate any person who has attended high school within the last 50 or so years. You may include yourself. Now, politely ask that person, if he or she would please construct for you a square root. Among the technically educated, it is very common, next, to see the diagonal of the square appear, often with the label 2 attached. As this has nothing whatsoever to do with the solution, I have found it most effective to point out in such cases, that the problem is really much simpler than that. No knowledge of the Pythagorean Theorem, nor any higher mathematics, is required. An Induced Mental Block What is the problem? No student of the classical method of education, which has been around for at least the past 2,500 years, could ever have any problem with this simple exercise. The mental block which arises here is the perfectly lawful result of the absurd and prevalent modern-day teaching that number can exist independent of any physically determining principle. This is the ivory-tower view of mathematics, which holds sway from grade school to university, and reaches up like a hand from the grave, even into the peer review process governing what can be reported as the results of experimental physics.

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