61. Algebra In The Renaissance, Part 2 I found this part of the lecture particularly interesting and thought he did a goodjob presenting it. The next mathematician discussed was rafael bombelli. http://public.csusm.edu/DJBarskyWebs/330CollageOct17.html

Algebra in the Renaissance, Part 2 The discussion was started by talking about art in the Renaissance. The idea of perspective in a painting began to be used in the Renaissance. To achieve realism, objects further away must be made to appear smaller. The painter Leon Battista Alberti (1404-1472) wrote a text on the subject of geometry as it relates to perspective in painting. The main topic centered around solving the "cubic" problem. Several mathematicians of the fifteenth and sixteenth century built upon the work of the Islamic mathematicians. We discussed Scipione del Ferro (1465-1526) who discovered an algebraic method for solving the cubic equation x ^3 + cx = d. Del Ferro taught Antonio Fiore. Niccolo Tartaglia (1499-1557) claimed that he discovered the solution to the cubic equations of the form x^3 + bx^2 = d. Tartaglia told Gerolamo Cardano his secret, however Cardano published the work when he discovered that it had earlier been discovered by del Ferro. It is interesting to follow the long history of one problem. After Dr. Barsky's commentary on the lack of a Nobel prize for mathematics and the mathematician of the day (Vickery), David Trigg began to talk about how the third dimension was represented in the art of this time period. The topics covered consisted of Copernicus and Kepler in Astronomy, the addition of perspective to make two dimensional art appear as three dimensional, Scipione Del Ferro, Antonio Fiore, Niccolo Tartaglia, Gerolamo Cardano and the "Artis Magnae", Libre de Ludo Aleae, Raphael Bombelli, and Simon Stevin.

62. A Look To The Past In 1572 rafael bombelli (15261573) published his treatise, Algebra, in which hegave one more step in the solution of cubic equations, expressing solutions in http://ued.uniandes.edu.co/servidor/em/recinf/tg18/Vizmanos/Vizmanos-2.html

Will elementary algebra disappear with the use of new graphing calculators?. A look to the past What do we understand elementary algebra to be? Elementary algebra is the language with which we communicate the majority of mathematics. Thanks to algebra we can work with concepts at an abstract level and then apply them. Elementary algebra begins as a generalization of arithmetic and then focuses on its own structure and greater logical coherence. From there comes the importance of the various uses of algebraic symbols. When we write A + B, we can be indicating the sum of two natural numbers, the sum of two algebraic expressions, or even the sum of two matrices. Thus there is, at first, representations and symbolism, and later the development of algorithms and procedures to work formally with algebraic expressions. But what we today understand to be algebra has been the fruit of the efforts of many generations that have been contributing their grains of sand in constructing this magnificent building. It seems that the Egyptians already knew methods for solving first degree equations. In the

63. WebQuest:  Resources http//wwwgap.dcs.st-and.ac.uk/~history/Mathematicians/Cardan.html. rafael bombelli.http//www-gap.dcs.st-and.ac.uk/~history/Mathematicians/bombelli.html. http://michelle_sinclair.tripod.com/resources.htm

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 Back to WebQuest (The teacher is not responsible for any inappropriate material the student may find on the internet while researching for this project) MacTutor History of Mathematics Archive http://www-groups.dcs.st-and.ac.uk/~history/index.html Mathematicians in Richard S. Westfall's Archive http://www-groups.dcs.st-and.ac.uk/~history/External/Westfall_list.html Who was Fibonacci? http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibBio.html Leonardo Pisano Fibonacci http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html Jordanus Nemorarius http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Jordanus.html Nicole of Oresme http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Oresme.html Regiomontanus http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Regiomontanus.html Ludovico Ferrari http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Ferrari.html Francois Viete http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Viete.html

64. Historia Matematica Mailing List Archive: Re: [HM] Synthetic Division rafael bombelli da Bologna, L Algebra. Prima edizione integrale. Introduzionedi U. Forti Prefazione di E. Bortolotti. Feltrinelli editore, Milano 1966. http://sunsite.utk.edu/math_archives/.http/hypermail/historia/nov98/0262.html

Re: [HM] Synthetic Division Prof. Lueneburg luene@mathematik.uni-kl.de Mon, 30 Nov 1998 17:35:59 +0100 (MEZ) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Next message: Ed Sandifer: "Re: [HM] Synthetic Division" Previous message: Judith Grabiner: "[HM] Ball's view of Maclaurin" This is exactly what I was looking for. I got the message when I was about going to my lecture. After lecturing, I took my xerocopies from the 1966 edition of all five books in one and, indeed, I found in Book II what you described. The 1966 edition contains the three books according to the first edition. Books IV und V were published for the first time in 1929. Rafael Bombelli da Bologna, L'Algebra. Prima edizione integrale. Introduzione di U. Forti - Prefazione di E. Bortolotti. Feltrinelli editore, Milano 1966 I know what I have to do this evening. Thanks a lot, Heinz Lueneburg sandifer@wcsu.ctstateu.edu Next message: Ed Sandifer: "Re: [HM] Synthetic Division" Previous message: Judith Grabiner: "[HM] Ball's view of Maclaurin"

65. Cardano Y Tartaglia Translate this page Mucho más que un triángulo Gerolamo Cardano. Renacentista tenaz Ludovico deFerrari. La idea en un destello rafael bombelli. El valor de la claridad. http://www.nivola.com/cardanoindex.htm

De la Edad Media al Renacimiento El siglo XV Las matemáticas del ábaco Regiomontano y la trigonometría Alberto Durero y la geometría El calendario gregoriano. Un problema de astronomía Luca Pacioli y la Summa de Arithmetica Dos problemas de Maestro Biaggio comentados por Maestro Benedetto Los versos con los que Tartaglia comunicó la solución a Cardano La demostración de Cardano de la regla de la ecuación cúbica Análisis de la demostración de Cardano La resolución de la ecuación general de tercer grado La demostración de Bombelli El caso irreducible y los números complejos La resolución de la ecuación de cuarto grado Los protagonistas de esta historia Scipione del Ferro. El álgebra en silencio Niccoló Tartaglia. Mucho más que un triángulo Gerolamo Cardano. Renacentista tenaz Ludovico de Ferrari. La idea en un destello Rafael Bombelli. El valor de la claridad Puntos suspensivos Panorama de los siglos XV y XVI Bibliografía Portada Contraportada Otras obras

66. LookSmart Australia Archimedes, Arnauld, Antoine, Barrow, Isaac. Bernoullis, bombelli, rafael,Buergi, Joost. Carcavi, Pierre de, Cardano, Girolamo, Cavalieri, B. http://explore.looksmart.com.au/synd-oz/explore/index.jsp?catPath=302562;317836;

67. KYMAA Newsletter, March 1996 rafael bombelli of Bologna Renaissance Algebraist by Daniel J.Curtin, Northern Kentucky University. Following the work of http://web.centre.edu/mat/kymaa/past/1996S.html

Kentucky Section Newsletter Spring Issue March, 1996 Please allow this entire file to load into your Web-browser to avoid errors. Table of Contents From the Chair ANNUAL MEETING KY SECTION MAA MEETING FACULTY REGISTRATION FORM KY SECTION MAA MEETING STUDENT REGISTRATION FORM ... KENTUCKY SECTION OFFICERS 1995-96 From the Chair Murray Meeting Our annual meeting is March 29-30, at Murray State University. Even if you have not yet registered to attend the meeting, it is not too late. Reading the enclosed meeting program should persuade you that the event will be lively, interesting, and entertaining. Another valuable dimension of meetings is to renew contact with colleagues and meet new ones. Invite a colleague who hasn't been attending our meetings. Invite a student. Are You Giving Us a Line? The context here is "on line," and the answer is Yes. KYMAA now has a presence on the World Wide Web, due to the efforts of Webmeister Lyn Miller (WKU). The page eagerly awaits your perusal. The URL is: http://www4.wku.edu/~miller/KYMAA.homepage.html

68. Week99 eg 4) Luca bombelli, Joohan Lee, David Meyer and rafael D. Sorkin,Spacetime as a causal set, Phys. Rev. Lett. 59 (1987), 521. http://math.ucr.edu/home/baez/week99.html

March 15, 1997 This Week's Finds in Mathematical Physics (Week 99) John Baez Life here at the Center for Gravitational Physics and Geometry is tremendously exciting. In two weeks I have to return to U. C. Riverside and my mundane life as a teacher of calculus, but right now I'm still living it up. I'm working with Ashtekar, Corichi, and Krasnov on computing the entropy of black holes using the loop representation of quantum gravity, and also I'm talking to lots of people about an interesting 4-dimensional formulation of the loop representation in terms of "spin foams" - roughly speaking, soap-bubble-like structures with faces labelled by spins. Here are some papers I've come across while here: 1) Lee Smolin, The future of spin networks, in The Geometric Universe: Science, Geometry, and the Work of Roger Penrose, eds. S. Hugget, Paul Tod, and L.J. Mason, Oxford University Press, 1998. Also available as gr-qc/9702030 I've spoken a lot about spin networks here on This Week's Finds. They were first invented by Penrose as a radical alternative to the usual way of thinking of space as a smooth manifold. For him, they were purely discrete, purely combinatorial structures: graphs with edges labelled by "spins" j = 0, 1/2, 1, 3/2, etc., and with three edges meeting at each vertex. He showed how when these spin networks become sufficiently large and complicated, they begin in certain ways to mimic ordinary 3-dimensional Euclidean space. Interestingly, he never got around to publishing his original paper on the subject, so it remains available only if you know someone who knows someone who has it:

69. Also Available At A Href= A Href= Http//math.ucr.edu/home/baez This ties their work to the work of rafael Sorkin on causal sets, eg 4) Luca bombelli,Joohan Lee, David Meyer and rafael D. Sorkin, Spacetime as a causal set http://math.ucr.edu/home/baez/twf.ascii/week99

Similarly, given two objects v and w in a category, the hom functor gives a *set* hom(x,y) namely the set of morphisms from x to y. Note that the inner product is linear in w and conjugate-linear in y, and similarly, the hom functor hom(x,y) is covariant in y and contravariant in x. This hints at the category theory secretly underlying quantum mechanics. In quantum theory the inner product represents the *amplitude* to pass from v to w, while in category theory hom(x,y) is the *set* of ways to get from x to y. In Feynman path integrals, we do an integral over the set of ways to get from here to there, and get a number, the amplitude to get from here to there. So when physicists do Feynman path integration - just like a shepherd counting sheep - they are engaged in a process of decategorification! To understand this analogy better, note that any morphism f: x -> y in Hilb can be turned around or "dualized" to obtain a morphism f*: y -> x. This is usually called the "adjoint" of f, and it satisfies

70. Resultados Clínicos Y Radiológicos De La Técnica De Osteotomía En Translate this page rafael Alberto Galindo González* *Trabajo de grado presentado como requisito b cabezaelipsoide, normotrófica y móvil según la clasificación de bombelli. http://www.encolombia.com/ortopedia7293resultados.htm

RESULTADOS CLÍNICOS Y RADIOLÓGICOS DE LA TÉCNICA DE OSTEOTOMÍA EN VALGO-EXTENSIÓN EN PACIENTES CON OSTEOARTROSIS DE LA CADERA RESUMEN INTRODUCCIÓN OBJETIVOS Rafael Alberto Galindo González* *Trabajo de grado presentado como requisito parcial para optar al título de Ortopedista y Traumatólogo. Universidad Pontificia Bolivariana, Facultad d Medicina, Medellín. RESUMEN Desde octubre de 1984 hasta abril de 1992, un total de 32 caderas tuvieron una osteotomía intertrocantérea en valgo extensión por osteoartrosis, en el Instituto de Seguros Sociales, Clínica León XIII, Medellín. Se incluyeron en el presente estudio, 22 caderas (dos pacientes con intervención bilateral). No hubo predominio según el sexo. El promedio de edad al momento de la intervención fue de 39.4 años (Rango 25 a 58). En el 59.1% correspondieron a caderas derechas. El tiempo promedio de seguimiento fue de 54.2 meses (Rango 10 a 90 meses). 31.8% de las caderas tuvieron algún tipo de cirugía previa a la osteotomía, sin afectar en forma estadísticamente significativa el resultado final.

71. Scuola ER | Archivio | La Matematica A DIDAMATICA 2000 Translate this page game, arricchito da un ampio apparato storico didattico e biografico, che attraversouna fantastica ricostruzione della vita di rafael bombelli permette al http://www.scuolaer.it/page.asp?IDCategoria=116&IDSezione=0&ID=11116

72. Olympiades Academiques Sujets 2002 Translate this page RENNES Voir la Correction Le mathématicien italien rafael bombelli (1526-1572) aproposé dans un traité d’algèbre publié en 1572 une méthode permettant http://www.maths-express.com/olympiades_lycee/2002/sujets2002.htm

73. Polski Dom Aukcyjny 'SZTUKA' In Basilic Vatic. Anno 1740 Petrus bombelli Incid. Et Vendit Rom Anno Dom.1785. 200. 47. Sadeler, rafael jr. ( 1584-1632 ) Swieta Rodzina , ok. http://www.sztuka.com.pl/index.php?ac=103&id=6

74. A History Of Hypercomplex Numbers 1572, rafael bombelli (15301590) publishes Algebra, making use of his wild idea that one could use these square roots of negative numbers to get to the real http://history.hyperjeff.net/hypercomplex_slim.html

Back to the Full version Jeff Biggus Part of the HyperJeff Network Sketching the History of Hypercomplex Numbers last updated Monday, 14-Jan-2002 00:20:51 CST Brahmagupta (598-670) writes Khandakhadyaka which solves quadratic equations and allows for the possibility of negative solutions. pre Abraham bar Hiyya Ha-Nasi writes the work Hibbur ha-Meshihah ve-ha-Tishboret , translated in 1145 into Latin as Liber embadorum , which presents the first complete solution to the quadratic equation. Nicolas Chuquet (1445-1500) writes Triparty en la sciences des nombres . The fourth part of which contains the "Regle des premiers," or the rule of the unknown, what we would today call an algebra. He introduced an exponential notation, allowing positive, negative, and zero powers. In solving general equations he showed that some equations lead to imaginary solutions, but dismisses them ("Tel nombre est ineperible"). pre Nicolo Fontana (Tartaglia) finds the general method for solving all types of cubic equations and tells Cardano, under the promise that Cardano tell no one until he publishes first. Cardno tells everyone in 1545. Geronimo Cardano (1501-1576) writes Ars magna on the solutions of cubic and quartic equations. In it, solutions to polynomials which lead to square roots of negative quantities occur, but Cardano calls them "sophistic" and concludes that it is "as subtle as it is useless."

75. Raffael Bombelli Translate this page bombelli, Raffael. Raffael bombelli wurde am 20. Januar 1526 in derKathedrale San Pietro in Bologna getauft. In den Jahren vor 1560 http://www.mathe.tu-freiberg.de/~hebisch/cafe/bombelli.html

Bombelli, Raffael L'Algebra

76. Cerme 1 - Proceedings: Contents Vol. II REFLECTIONS AND EXAMPLES Giorgio T. Bagni. bombelli s Algebra (1572)and Imaginary Numbers; Educational Problems The Focus of Our Work; http://www.fmd.uni-osnabrueck.de/ebooks/erme/cerme1-proceedings/cerme1_contents2

E uropean Society for R esearch in M athematics E ducation Institute for Cognitive Mathematics Contents Vol. II European Research in Mathematics Education I.I + I.II Proceedings of the First Conference of the European Society for Research in Mathematics Education Vol. I + II Editor: Inge Schwank, 1999 Publishing House: Forschungsinstitut fuer Mathematikdidaktik, Osnabrueck. Internet-Versions Vol. I: ISBN 3-925386-50-5 pdf-file 2.706 kB, V1.01 Vol. II: ISBN 3-925386-51-3 pdf-file 1.321 kB, V1.0 Paper-Versions Vol. I: ISBN: 3-925386-53-X how to order Vol. II: ISBN: 3-925386-54-8 how to order Overview including direct link to complete single contributions in pdf-format Vol. I: Table of contents I html-file Abstracts I html-file Vol. II:

77. Complex Numbers And Geometry A short time later, in 1572, another mathematician, rafael Bombellihelped to shape the nature of algebra for the next 400 years. http://campus.northpark.edu/math/PreCalculus/Transcendental/Trigonometric/Comple

Section 5.1: Complex Numbers and Geometry While the quadratic formula , has been known to give solutions to the quadratic equation, ax bx c = , since the time of the ancient Babylonian civilization (around 2000 BC), the simple looking equation, x + 1 = 0, was an enigma until relatively recently. That is because our number concept has historically been limited to those numbers which can be graphed on the real number line. In this section, we will see how the real number system is only a part of a larger number system, call the "complex" numbers. Moreover, we will see how the nice geometric interpretations of addition, multiplication, and negation of real numbers generalize to the complex numbers . We will also learn about a new operation, which applies to complex numbers, called conjugation , and discuss its geometric significance. The Origins of Complex Numbers For thousands of years, mathematicians considered the equation x + 1 = to be insolvable. From a functional point of view, we know that the range of the square function f x x , contains only positive numbers, so that

78. Un Poquito De La Historia Del álgebra Translate this page Entre 1545 y 1560, los matemáticos italianos Girolamo Cardano y rafael Bombellise dieron cuenta de que el uso de los números imaginarios era indispensable http://redescolar.ilce.edu.mx/redescolar/act_permanentes/mate/mate3a/mate3a.htm

S A D E cosechas y de materiales. Ya para entonces tenían un método para resolver ecuaciones de primer grado que se llamaba el "método de la falsa posición". No tenían notación simbólica pero utilizaron el jeroglífico hau A E E arithmos , que significa número. Los problemas de álgebra que propuso prepararon el terreno de lo que siglos más tarde sería "la teoría de ecuaciones". A pesar de lo rudimentario de su notación simbólica y de lo poco elegantes que eran los métodos que usaba, se le puede considerar como uno de los precursores del álgebra moderna. E S algoritmo que, usada primero para referirse a los métodos de cálculos numéricos en oposición a los métodos de cálculo con ábaco, adquirió finalmente su sentido actual de "procedimiento sistemático de cálculo". En cuanto a la palabra Al-jabr wal muqabala.

79. Den Italienske Matematiker Rafaello Bombelli Var Interesseret I At Fuldstændigg Den italienske matematiker Rafaello bombelli var interesseret i at fuldstændiggørekvadratrødderne. bombelli kaldte dem ”udspekulerede tal”. Tallinie. http://fp.worldonline.dk/fpeneven/Bombelli.htm

Den italienske matematiker Rafaello Bombelli var interesseret i at fuldstændiggøre kvadratrødderne. Han stillede spørgsmålet: Hvad er kvadratroden af minus 1, dvs. Man skal altså specificere det tal, som ganget med sig selv giver -1. Løsningen kan ikke være 1, fordi I - I = 1, men den er heller ikke -1, fordi (-1) - (-1) = I. Da kvadratet på et vilkårligt tal, positivt eller negativt, altid er positivt, er opgaven uløselig. Og dog forlanger fuldstændigheden, at der skal være en løsning. Bombefli indførte derfor et nyt tal, i, som pr. definition gav -1, når det blev ganget med sig selv. Dette tal blev prototypen på samtlige imaginære tal: tal, der ganget med sig selv giver et negativt tal. Dette kunne virke som en fej måde at løse problemer på, men den svarede helt til indførelsen af de negative tal. Stillet over for et i modsat fald ubesvarligt spørgsmål definerede hinduerne ganske enkelt -1 som svaret på spørgsmålet: Hvad er minus 1? Det er lettere at acceptere forestillingen om -1, fordi vi fx sammenligner med det velkendte fænomen gæld, hvorimod vi i den virkelige verden ikke råder over noget, som kan til understøtte begrebet et imaginært tal og dernæst det vi i dag kalder for komplekse tal.

80. La Radice Quadrata http://digilander.libero.it/basecinque/numeri/radiquad.htm

BASE Cinque Appunti di Matematica ricreativa BASE Cinque Collezione La radice quadrata Rilassatevi, questo algoritmo è una ciliegina! La stima iniziale L'algoritmo Un altro esempio Spiegazione dell'algoritmo ... I consigli di Enrico Delfini Sorpresina! Digita un numero nella casella e clicca sul pulsante. Leggi qui il risultato. Esiste un metodo semplice per calcolare "a mano" la radice quadrata di un numero? Ne esistono diversi, ma non si può dire che siano semplicissimi. Il procedimento che viene ancora oggi insegnato nella scuola media è lo stesso che Rafael Bombelli presentò nella sua Opera su Algebra del 1550. Questo algoritmo è difficile da ricordare soprattutto se viene imparato meccanicamente, senza capirne le motivazioni. Gli strumenti per capirlo si acquisiscono nel primo anno della scuola superiore, con lo studio del calcolo letterale e dei cosiddetti prodotti notevoli. Gli antichi hanno faticato a lungo per costruire le tavole delle radici quadrate e i moderni hanno inventato le calcolatrici tascabili. Se il vostro obiettivo è risolvere dei problemi allora è meglio utilizzare le tavole o la calcolatrice.

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