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Topic: Quadratic irrational

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PlanetMath: e is not a quadratic irrational This is version 8 of e is not a quadratic irrational, born on 2003-11-21, modified 2005-03-18. "e is not a quadratic irrational" is owned by mathcam. To do this, we show that it can not be the root of any quadratic polynomial with integer coefficients. planetmath.org /encyclopedia/EIsIrrational.html   (109 words)

Math Forum - Ask Dr. Math The reason it can't have just one irrational root is that an irrational multiplied by a rational is still irrational, so the constant at the end of the quadratic would be irrational. Date: 4/10/96 at 14:23:37 From: kelli rostkowski Subject: Algebra: quadratic equations Would you explain to me why a quadratic equation cannot have one irrational or one imaginary root. Date: 4/27/96 at 19:42:54 From: Doctor Steven Subject: Re: Algebra: quadratic equations I'm assuming you mean a quadratic equation with rational coefficients. mathforum.org /dr.math/problems/rostkowski4.10.96.html   (280 words)

Thesis Abstracts As ?(x) maps quadratic irrationals to rational numbers, it is shown that both generalizations send natural classes of pairs of cubic irrational numbers in the same cubic number field to pairs of rational numbers. Continued fractions are closely tied to distinguishing quadratic irrationals and determining properties of the algebraic fields that they determine. Using the theory of continued fractions, we produce a new sharp Diophantine inequality involving an irrational number and a rational approximation to that number, such that the only solutions are precisely all the best rational approximates to the given irrational number; that is, the complete list of its convergents. www.williams.edu /mathematics/sabstracts.html   (4563 words)

irrational.htm After developing the quadratic formula, students are ready to talk about how solutions that have a discriminant that is not a perfect square, the answers do not work out evenly. Since they have already proven that SQR(2) is irrational, they now know the limit of what their calculator can do in terms of approximating values of irrational numbers. One example is to study how to solve quadratic equations by looking for x-intercepts (zeros) of the function on a graphing calculator. personal.bgsu.edu /~brahier/irrational.htm   (639 words)

Secondary Education in Canada: A Student Transfer Guide, 1998 - Mathematics The contents include radicals and irrational numbers, quadratic functions and equations, quadratic relations and systems, exponential functions and logarithms, trigonometry, trigonometric identities and formulas, and circular functions. Besides simplifying rational expressions, this course extends the study of functions and relations with particular emphasis on the linear function; furthermore, the distinction between quadratic and linear functions is emphasized. 30 sub-topics are considered within the topics of rational numbers, irrational numbers, organizing and interpreting data, similarity and trigonometry, reasoning, analytic geometry, expressions and equations. www.cmec.ca /tguide/1998/english/13.stm   (7812 words)

Irrational Numbers Proof that the square root of 2 is irrational number... irrational number definition of irrational number in computing dictionary - by t... Fascinating irrational numbers: Pi and square roots - lesson from Homeschool Mat... www.scienceoxygen.com /math/56.html   (135 words)

Calculating the simple continued fraction of a quadratic irrational This program finds the continued fraction expansion of a quadratic irrational (u+tsqrt(d))/v, where d,t,u,v are integers, d >1 and nonsquare, t and v nonzero. Calculating the simple continued fraction of a quadratic irrational We use the continued fraction algorithm as described in K. Rosen, Elementary Number theory and its applications, 379-381 and Knuth's The art of computer programming, Vol. www.numbertheory.org /php/surd.html   (95 words)

Continued Fractions and Characteristic Recurrences Lagrange proved the simple continued fraction of an irrational number is periodic if and only if the number is a quadratic irrational, i.e., the root of a quadratic equation, such as x^2 - 2 = 0. As a result, the numerators and denominators of successive convergents of the continued fraction for sqrt(N) can always be generated by a simple second order linear recurrence. However, it's not so easy to define the analagous sequence for CUBE root of 2. www.mathpages.com /home/kmath434.htm   (365 words)

mp_arc 02-148 For a specific class of maps, we prove that if the trace is a quadratic irrational (the simplest nontrivial case, comprising 8 maps), then the periodic orbits are organized into finitely many renormalizable families, with exponentially increasing period, plus a finite number of exceptional families. 02-148 Kouptsov K. L., Lowenstein J. H, Vivaldi F. Quadratic rational rotations of the torus and dual lattice maps (2556K, zipped postscript) Mar 26, 02 We develop a general formalism for computed-assisted proofs concerning the orbit structure of certain non ergodic piecewise affine maps of the torus, whose eigenvalues are roots of unity. www.ma.utexas.edu /mp_arc-bin/mpa?yn=02-148   (147 words)

mp_arc 97-589 For badly approximated t (e.g., when t is a quadratic irrational, or, more generally, when t has bounded quotients in its continued fraction expansion), E(t,x) is a "1/2-derivative" more regular than E(0,x). For irrational t, the regularity of E(t,x) is better and depends on how well t is approximated by rationals. We describe different thin classes of irrationals which prescribe their particular regularity to the fundamental solution. www.ma.utexas.edu /mp_arc-bin/mpa?yn=97-589   (173 words)

KUMON PHILIPPINES: Math & Reading Programs Level J aims to develop the students' ability to work with algebraic expressions, factorization, irrational numbers, quadratic equations, the remainder theorem, the factor theorem, and the proof of identities and inequalities, in preparation for higher-level math studies. Level K aims to develop the students' ability to work with a variety of functions including quadratic, fractional, irrational, exponential, logarithmic and trigonometric functions. Level I aims for students to further develop their algebraic skills gained up to Level H to master operations mainly with quadratic polynomials, equations and functions, e.g. www.kumon.com.ph /math.htm   (1082 words)

Kolmogorov-Arnold-Moser On the other hand, it is known that quadratic irrationals, i.e., solutions of quadratic equations with integer coefficients, have periodic continued fraction expansions which implies that the corresponding approximations behave asymptotically as (Richter and Scholz 1987) In this sense, the quadratic irrationals are the ``most irrational'' numbers because they have the smallest possible The theorem also gives a hint as to which irrational tori are the most robust. www-nonlinear.physik.uni-bremen.de /nlp/publications/ChaosHTML/r14richter/node7.html   (425 words)

Periodic and Fixed Points of the Gauss Map Corollary: The periodic points of the Gauss map are the reciprocals of the reduced quadratic irrationals. To prove the corollary, we note that every rational initial point is ``attracted'' to the artificial fixed point at 0, while every quadratic irrational is ultimately ``attracted'' to a periodic orbit. is a root of a quadratic with integer coefficients). www.cecm.sfu.ca /organics/papers/corless/confrac/html/node6-an.shtml   (543 words)

Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations It is shown that grain boundaries are quasiperiodic tillings made of inflatable sequences of structural units.^A general grain boundary is a quasiperiodic grid of dislocations with the same inflation multiplier (a quadratic irrational) in every direction.^12 refs., 7 figs. www.osti.gov /energycitations/product.biblio.jsp?osti_id=6096971   (135 words)

research2.html It is well-known that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet L-function which is very close to 1, a weak consequence of the Generalized Riemann Hypothesis). One might ask whether it is possible to find quadratic polynomials with arbitrarily long strings of consecutive prime values; that is whether, for any given N can we find A for which n^2+n+A is prime for n=0,1,2... In this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than ``all'' as in Rabinowitsch's result). www.math.ucalgary.ca /~ramollin/research2.html   (1412 words)

Mathematics 10, 20: Aids for Planning (Mathematics 20) To identify an irrational number and to demonstrate the ability to add, subtract, multiply, and divide square root radicals. To demonstrate the ability to solve quadratic equations by factoring, and by taking the square root of both sides of an equation. To solve quadratic equations by a) factoring, and b) by taking the square roots of both sides of an equation. www.sasked.gov.sk.ca /docs/secmath/m20ap.html   (1068 words)

Physics Help and Math Help - Physics Forums - Factoring irrational equations If you really are intent on solving it, if factoring into quadratic factors doesn't work, you'll either have to resort to the quartic formula (very ugly, though it might be less ugly if you work through the derivation, sort of like completing the square instead of using the quadratic formula directly) or numerical approximation. The quadratic equation will probably be required, but I’m still not sure how to get the final answer. However, if you plug in these real numbers, none of them work, therefore meaning that x is an irrational number and is complex/imaginary. www.physicsforums.com /printthread.php?t=58980   (998 words)

PlanetMath: Davenport-Schmidt theorem which is not rational or quadratic irrational, there are infinitely many rational or real quadratic irrational The height of the rational or quadratic irrational number Davenport, H. Schmidt, M. Wolfgang: Approximation to real numbers by quadratic irrationals. planetmath.org /encyclopedia/DavenportSchmidt.html   (88 words)

Danville Area Community College - Catalog - Course Descriptions Instruction covers ratios, fractions and percents, two variable and linear equations, slopes, functions, graphs, linear and quadratic equations, inequalities and absolute values, rational and irrational numbers, and quadratic equations. Demonstrate the ability to work with slopes, functions graphing, linear and quadratic equations, and direct and inverse variations. Demonstrate the ability to work with quadratic equations. www.dacc.cc.il.us /catalogs/0405/courses/index.php?course=ASEM031   (638 words)

Irrationality of Quadratic Sums then the overall sum is irrational (and this condition is clearly met by my "quadratic" unit fraction sequences.) To prove this irrationality criterion, Dean observes that if the infinite sum 1/d[1] + 1/d[2] +... Is the sum of and infinite quadratic sequence necessarily irrational? If each denominator is greater than or equal to the SQUARE of the preceeding denominator, then we will say the sequence is "quadratic". www.mathpages.com /home/kmath455.htm   (293 words)

Untitled Document Determine from the discriminant of a quadratic equation whether the roots are rational or irrational. Determine from the discriminant of a quadratic equation whether the roots are imaginary, rational, or irrational. Graph quadratic equations and observe where the graph crosses the x-axis, or note that it does not. www.angelfire.com /moon/bronsieboy/PAGE3.HTM   (1732 words)

Underdamped Frenkel-Kontorova Dynamics on Quasiperiodic Substrates However, when the length scales are quadratic irrationals, or have some commensurability with each other, the static friction will be nonzero for all choices of interaction parameters. When the length scales are related by the golden mean (a quadratic irrational) the static friction is always nonzero. From considerations based on the connection of this problem to standard map theory for dynamical systems with three incommensurate frequencies [3], we postulate that zero static friction is generally possible for incommensurate ratios of the length scales involved. www.cond-mat.physik.uni-mainz.de /cecam_workshop/vanossi.html   (322 words)

quadratic equations 2 The discriminant of a quadratic equation is the value under the square root sign in the quadratic formula. In fact, there will be 2 different irrational solutions because 37 is not a perfect square number. Using the Discriminant to predict the roots of a quadratic equation www.equation-solver.com /quadratic-equations-2.htm   (329 words)

Convergence Of Random Walks On The Circle Generated By An Irrational Rotation - Su (ResearchIndex) We obtain bounds for rates when # is any irrational, and a sharp rate when # is a quadratic irrational. Su obtained discrepancy bounds for this walk, showing that it converges most quickly when ff is a quadratic irrational. 0.1: Irrational Numbers of Constant Type --- A New Characterization - Manash Mukherjee (1998) citeseer.ist.psu.edu /su98convergence.html   (637 words)

Mathematics Course Descriptions Basic language of algebra, addition, multiplication of real numbers, equations and problems, variables, solving equations, polynomials and their factors, rational expressions in open sentences, functions and their graphs, irrational numbers, quadratic equations in three variables, basic statistics. Sets in algebra, open sentences in one variable, linear equations, problem solving, special products and factoring, rational numbers, relations and functions, irrational numbers and quadratic equation, exponent functions and logarithms, progression, and polynomial functions. Areas of study include integers, vectors, scientific notation, using formulas, work with linear equations, statistics, probability, inequalities, systems of equations, polynomials, factors, quadratic functions and right triangle relationships. www.iola.k12.wi.us /ismath/coursdesc.htm   (572 words)

THE FAMILY OF METALLIC MEANS The members of the MMF are the only positive quadratic irrational numbers that originate GSFS (with additive properties), which are, simultaneously, geometric progressions. All the members of this family are positive quadratic irrational numbers that are the positive solutions of quadratic equations of the type are also quadratic Pisot numbers with purely periodic continued fraction expansions, where the condition that the terms of the continued fraction have to be positive, has been relaxed [28]. www.mi.sanu.ac.yu /vismath/spinadel   (3062 words)

Brownian friezes The Euler-Lagrange theorem claims that a continued fraction expansion of x is periodic, if and only if x is irrational of quadratic type. An important part of the set of irrational numbers is the set of transcendental numbers. It is therefore the biggest solution of the quadratic equation x = [ www.mi.sanu.ac.yu /vismath/kocic/ch2.htm   (894 words)

University of Lethbridge - Mathematics & Computer Science - Colloquium Archive 2001 For the uninitiated, we give an overview of the topics in the title, beginning with the notion of a quadratic irrational, moving through the use of continued fraction to solve Pell's equations, to the continued fraction algorithm, using ideal theory, and this author's recent method for finding solutions of dual norm form quadratic Diophantine equations. An explanation is given for this prime production by this author's 1996 discovery of the reasons in terms of the class group structure of complex quadratic number fields. We explore how sums of two integral squares play a role in determining solutions of Diophantine equations, and examples thereof motivate a discussion of prime producing quadratic polynomials. www.uleth.ca /fas/mcs/extra_content/colloquium/colArchive2001.html   (1092 words)

10th Algebra 2 The student will recognize multiple representations of functions (linear, quadratic, and absolute value) and convert between a graph, a table, and symbolic form. The student will select, justify, and apply a technique to solve a quadratic equation over the set of complex numbers. - graph quadratic equations of the form y = (x – h) www.asfg.mx /highschool/Math/curriculum/10thalgebra.htm   (1496 words)

Index quadratic irrational numbers (which are also called quadratic surds). Check if a quadratic surd belong to the same field as the instance. Check if the instance is equal to another quadratic surd. www.spaceroots.org /rkcheck-doc/index-all.html   (470 words)

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