The "Greatest European Mathematician of the Middle Ages"
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On K-Fibonacci Numbers of Arithmetic Indexes
Applied Mathematics and Computation 208 (2009) 180–185 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc On k-Fibonacci numbers of arithmetic indexes Sergio Falcon *, Angel Plaza Department of Mathematics, University of Las Palmas de Gran Canaria (ULPGC), Campus de Tafira, 35017 Las Palmas de Gran Canaria, Spain article info abstract Keywords: In this paper, we study the sums of k-Fibonacci numbers with indexes in an arithmetic k-Fibonacci numbers sequence, say an þ r for fixed integers a and r. This enables us to give in a straightforward Sequences of partial sums way several formulas for the sums of such numbers. Ó 2008 Elsevier Inc. All rights reserved. 1. Introduction One of the more studied sequences is the Fibonacci sequence [1–3], and it has been generalized in many ways [4–10]. Here, we use the following one-parameter generalization of the Fibonacci sequence. Definition 1. For any integer number k P 1, the kth Fibonacci sequence, say fFk;ngn2N is defined recurrently by Fk;0 ¼ 0; Fk;1 ¼ 1; and Fk;nþ1 ¼ kFk;n þ Fk;nÀ1 for n P 1: Note that for k ¼ 1 the classical Fibonacci sequence is obtained while for k ¼ 2 we obtain the Pell sequence. Some of the properties that the k-Fibonacci numbers verify and that we will need later are summarized below [11–15]: pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi n n 2 2 r1Àr2 kþ k þ4 kÀ k þ4 [Binet’s formula] Fk;n ¼ r Àr , where r1 ¼ 2 and r2 ¼ 2 . These roots verify r1 þ r2 ¼ k, and r1 Á r2 ¼1 1 2 2 nþ1Àr 2 [Catalan’s identity] Fk;nÀrFk;nþr À Fk;n ¼ðÀ1Þ Fk;r 2 n [Simson’s identity] Fk;nÀ1Fk;nþ1 À Fk;n ¼ðÀ1Þ n [D’Ocagne’s identity] Fk;mFk;nþ1 À Fk;mþ1Fk;n ¼ðÀ1Þ Fk;mÀn [Convolution Product] Fk;nþm ¼ Fk;nþ1Fk;m þ Fk;nFk;mÀ1 In this paper, we study different sums of k-Fibonacci numbers. -
From Al-Khwarizmi to Algorithm (71-74)
Olympiads in Informatics, 2017, Vol. 11, Special Issue, 71–74 71 © 2017 IOI, Vilnius University DOI: 10.15388/ioi.2017.special.11 From Al-Khwarizmi to Algorithm Bahman MEHRI Sharif University of Technology, Iran e-mail: [email protected] Mohammad ibn Musa al-Khwarizmi (780–850), Latinized as Algoritmi, was a Persian mathematician, astronomer, and geographer during the Abbasid Caliphate, a scholar in the House of Wisdom in Baghdad. In the 12th century, Latin translations of his work on the Indian numerals introduced the decimal number system to the Western world. Al-Khwarizmi’s The Compendious Book on Calculation by Completion and Balancing resented the first systematic solu- tion of linear and quadratic equations in Arabic. He is often considered one of the fathers of algebra. Some words reflect the importance of al-Khwarizmi’s contributions to mathematics. “Algebra” (Fig. 1) is derived from al-jabr, one of the two operations he used to solve quadratic equations. Algorism and algorithm stem from Algoritmi, the Latin form of his name. Fig. 1. A page from al-Khwarizmi “Algebra”. 72 B. Mehri Few details of al-Khwarizmi’s life are known with certainty. He was born in a Per- sian family and Ibn al-Nadim gives his birthplace as Khwarazm in Greater Khorasan. Ibn al-Nadim’s Kitāb al-Fihrist includes a short biography on al-Khwarizmi together with a list of the books he wrote. Al-Khwārizmī accomplished most of his work in the period between 813 and 833 at the House of Wisdom in Baghdad. Al-Khwarizmi contributions to mathematics, geography, astronomy, and cartogra- phy established the basis for innovation in algebra and trigonometry. -
On the Prime Number Subset of the Fibonacci Numbers
Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach On the Prime Number Subset of the Fibonacci Numbers Lacey Fish1 Brandon Reid2 Argen West3 1Department of Mathematics Louisiana State University Baton Rouge, LA 2Department of Mathematics University of Alabama Tuscaloosa, AL 3Department of Mathematics University of Louisiana Lafayette Lafayette, LA Lacey Fish, Brandon Reid, Argen West SMILE Presentations, 2010 LSU, UA, ULL Sieve Theory Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions What is a sieve? What is a sieve? A sieve is a method to count or estimate the size of “sifted sets” of integers. Well, what is a sifted set? A sifted set is made of the remaining numbers after filtering. Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions What is a sieve? What is a sieve? A sieve is a method to count or estimate the size of “sifted sets” of integers. Well, what is a sifted set? A sifted set is made of the remaining numbers after filtering. Lacey Fish, Brandon Reid, Argen West LSU, UA, ULL Sieve Theory Introduction to Sieves Fibonacci Numbers Fibonacci sequence mod a prime Probability of Fibonacci Primes Matrix Approach Basic Definitions History Two Famous and Useful Sieves Sieve of Eratosthenes Brun’s Sieve Lacey Fish, Brandon Reid, Argen West -
A MATHEMATICIAN's SURVIVAL GUIDE 1. an Algebra Teacher I
A MATHEMATICIAN’S SURVIVAL GUIDE PETER G. CASAZZA 1. An Algebra Teacher I could Understand Emmy award-winning journalist and bestselling author Cokie Roberts once said: As long as algebra is taught in school, there will be prayer in school. 1.1. An Object of Pride. Mathematician’s relationship with the general public most closely resembles “bipolar” disorder - at the same time they admire us and hate us. Almost everyone has had at least one bad experience with mathematics during some part of their education. Get into any taxi and tell the driver you are a mathematician and the response is predictable. First, there is silence while the driver relives his greatest nightmare - taking algebra. Next, you will hear the immortal words: “I was never any good at mathematics.” My response is: “I was never any good at being a taxi driver so I went into mathematics.” You can learn a lot from taxi drivers if you just don’t tell them you are a mathematician. Why get started on the wrong foot? The mathematician David Mumford put it: “I am accustomed, as a professional mathematician, to living in a sort of vacuum, surrounded by people who declare with an odd sort of pride that they are mathematically illiterate.” 1.2. A Balancing Act. The other most common response we get from the public is: “I can’t even balance my checkbook.” This reflects the fact that the public thinks that mathematics is basically just adding numbers. They have no idea what we really do. Because of the textbooks they studied, they think that all needed mathematics has already been discovered. -
Positional Notation Or Trigonometry [2, 13]
The Greatest Mathematical Discovery? David H. Bailey∗ Jonathan M. Borweiny April 24, 2011 1 Introduction Question: What mathematical discovery more than 1500 years ago: • Is one of the greatest, if not the greatest, single discovery in the field of mathematics? • Involved three subtle ideas that eluded the greatest minds of antiquity, even geniuses such as Archimedes? • Was fiercely resisted in Europe for hundreds of years after its discovery? • Even today, in historical treatments of mathematics, is often dismissed with scant mention, or else is ascribed to the wrong source? Answer: Our modern system of positional decimal notation with zero, to- gether with the basic arithmetic computational schemes, which were discov- ered in India prior to 500 CE. ∗Bailey: Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. Email: [email protected]. This work was supported by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DE-AC02-05CH11231. yCentre for Computer Assisted Research Mathematics and its Applications (CARMA), University of Newcastle, Callaghan, NSW 2308, Australia. Email: [email protected]. 1 2 Why? As the 19th century mathematician Pierre-Simon Laplace explained: It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very sim- plicity and the great ease which it has lent to all computations put our arithmetic in the first rank of useful inventions; and we shall appre- ciate the grandeur of this achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity. -
Fibonacci Number
Fibonacci number From Wikipedia, the free encyclopedia • Have questions? Find out how to ask questions and get answers. • • Learn more about citing Wikipedia • Jump to: navigation, search A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, … , are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, ... (Sometimes this sequence is considered to start at F1 = 1, but in this article it is regarded as beginning with F0=0.) The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India. [1] [2] • [edit] Origins The Fibonacci numbers first appeared, under the name mātrāmeru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chandah-shāstra, the Art of Prosody, 450 or 200 BC). Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well-known text on these. -
Twelve Simple Algorithms to Compute Fibonacci Numbers Arxiv
Twelve Simple Algorithms to Compute Fibonacci Numbers Ali Dasdan KD Consulting Saratoga, CA, USA [email protected] April 16, 2018 Abstract The Fibonacci numbers are a sequence of integers in which every number after the first two, 0 and 1, is the sum of the two preceding numbers. These numbers are well known and algorithms to compute them are so easy that they are often used in introductory algorithms courses. In this paper, we present twelve of these well-known algo- rithms and some of their properties. These algorithms, though very simple, illustrate multiple concepts from the algorithms field, so we highlight them. We also present the results of a small-scale experi- mental comparison of their runtimes on a personal laptop. Finally, we provide a list of homework questions for the students. We hope that this paper can serve as a useful resource for the students learning the basics of algorithms. arXiv:1803.07199v2 [cs.DS] 13 Apr 2018 1 Introduction The Fibonacci numbers are a sequence Fn of integers in which every num- ber after the first two, 0 and 1, is the sum of the two preceding num- bers: 0; 1; 1; 2; 3; 5; 8; 13; 21; ::. More formally, they are defined by the re- currence relation Fn = Fn−1 + Fn−2, n ≥ 2 with the base values F0 = 0 and F1 = 1 [1, 5, 7, 8]. 1 The formal definition of this sequence directly maps to an algorithm to compute the nth Fibonacci number Fn. However, there are many other ways of computing the nth Fibonacci number. -
The Astronomers Tycho Brahe and Johannes Kepler
Ice Core Records – From Volcanoes to Supernovas The Astronomers Tycho Brahe and Johannes Kepler Tycho Brahe (1546-1601, shown at left) was a nobleman from Denmark who made astronomy his life's work because he was so impressed when, as a boy, he saw an eclipse of the Sun take place at exactly the time it was predicted. Tycho's life's work in astronomy consisted of measuring the positions of the stars, planets, Moon, and Sun, every night and day possible, and carefully recording these measurements, year after year. Johannes Kepler (1571-1630, below right) came from a poor German family. He did not have it easy growing Tycho Brahe up. His father was a soldier, who was killed in a war, and his mother (who was once accused of witchcraft) did not treat him well. Kepler was taken out of school when he was a boy so that he could make money for the family by working as a waiter in an inn. As a young man Kepler studied theology and science, and discovered that he liked science better. He became an accomplished mathematician and a persistent and determined calculator. He was driven to find an explanation for order in the universe. He was convinced that the order of the planets and their movement through the sky could be explained through mathematical calculation and careful thinking. Johannes Kepler Tycho wanted to study science so that he could learn how to predict eclipses. He studied mathematics and astronomy in Germany. Then, in 1571, when he was 25, Tycho built his own observatory on an island (the King of Denmark gave him the island and some additional money just for that purpose). -
Newton.Indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag
omslag Newton.indd | Sander Pinkse Boekproductie | 16-11-12 / 14:45 | Pag. 1 e Dutch Republic proved ‘A new light on several to be extremely receptive to major gures involved in the groundbreaking ideas of Newton Isaac Newton (–). the reception of Newton’s Dutch scholars such as Willem work.’ and the Netherlands Jacob ’s Gravesande and Petrus Prof. Bert Theunissen, Newton the Netherlands and van Musschenbroek played a Utrecht University crucial role in the adaption and How Isaac Newton was Fashioned dissemination of Newton’s work, ‘is book provides an in the Dutch Republic not only in the Netherlands important contribution to but also in the rest of Europe. EDITED BY ERIC JORINK In the course of the eighteenth the study of the European AND AD MAAS century, Newton’s ideas (in Enlightenment with new dierent guises and interpre- insights in the circulation tations) became a veritable hype in Dutch society. In Newton of knowledge.’ and the Netherlands Newton’s Prof. Frans van Lunteren, sudden success is analyzed in Leiden University great depth and put into a new perspective. Ad Maas is curator at the Museum Boerhaave, Leiden, the Netherlands. Eric Jorink is researcher at the Huygens Institute for Netherlands History (Royal Dutch Academy of Arts and Sciences). / www.lup.nl LUP Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 1 Newton and the Netherlands Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. 2 Newton and the Netherlands.indd | Sander Pinkse Boekproductie | 16-11-12 / 16:47 | Pag. -
On Popularization of Scientific Education in Italy Between 12Th and 16Th Century
PROBLEMS OF EDUCATION IN THE 21st CENTURY Volume 57, 2013 90 ON POPULARIZATION OF SCIENTIFIC EDUCATION IN ITALY BETWEEN 12TH AND 16TH CENTURY Raffaele Pisano University of Lille1, France E–mail: [email protected] Paolo Bussotti University of West Bohemia, Czech Republic E–mail: [email protected] Abstract Mathematics education is also a social phenomenon because it is influenced both by the needs of the labour market and by the basic knowledge of mathematics necessary for every person to be able to face some operations indispensable in the social and economic daily life. Therefore the way in which mathe- matics education is framed changes according to modifications of the social environment and know–how. For example, until the end of the 20th century, in the Italian faculties of engineering the teaching of math- ematical analysis was profound: there were two complex examinations in which the theory was as impor- tant as the ability in solving exercises. Now the situation is different. In some universities there is only a proof of mathematical analysis; in others there are two proves, but they are sixth–month and not annual proves. The theoretical requirements have been drastically reduced and the exercises themselves are often far easier than those proposed in the recent past. With some modifications, the situation is similar for the teaching of other modern mathematical disciplines: many operations needing of calculations and math- ematical reasoning are developed by the computers or other intelligent machines and hence an engineer needs less theoretical mathematics than in the past. The problem has historical roots. In this research an analysis of the phenomenon of “scientific education” (teaching geometry, arithmetic, mathematics only) with respect the methods used from the late Middle Ages by “maestri d’abaco” to the Renaissance hu- manists, and with respect to mathematics education nowadays is discussed. -
Famous Mathematicians Key
Infinite Secrets AIRING SEPTEMBER 30, 2003 Learning More Dzielska, Maria. Famous Mathematicians Hypatia of Alexandria. Cambridge, MA: Harvard University The history of mathematics spans thousands of years and touches all parts of the Press, 1996. world. Likewise, the notable mathematicians of the past and present are equally Provides a biography of Hypatia based diverse.The following are brief biographies of three mathematicians who stand out on several sources,including the letters for their contributions to the fields of geometry and calculus. of Hypatia’s student Synesius. Hypatia of Alexandria Bhaskara wrote numerous papers and Biographies of Women Mathematicians books on such topics as plane and spherical (c. A.D. 370–415 ) www.agnesscott.edu/lriddle/women/ Born in Alexandria, trigonometry, algebra, and the mathematics women.htm of planetary motion. His most famous work, Contains biographical essays and Egypt, around A.D. 370, Siddhanta Siromani, was written in A.D. comments on woman mathematicians, Hypatia was the first 1150. It is divided into four parts: “Lilavati” including Hypatia. Also includes documented female (arithmetic), “Bijaganita” (algebra), resources for further study. mathematician. She was ✷ ✷ ✷ the daughter of Theon, “Goladhyaya” (celestial globe), and a mathematician who “Grahaganita” (mathematics of the planets). Patwardhan, K. S., S. A. Naimpally, taught at the school at the Alexandrine Like Archimedes, Bhaskara discovered and S. L. Singh. Library. She studied astronomy, astrology, several principles of what is now calculus Lilavati of Bhaskaracharya. centuries before it was invented. Also like Delhi, India:Motilal Banarsidass, 2001. and mathematics under the guidance of her Archimedes, Bhaskara was fascinated by the Explains the definitions, formulae, father.She became head of the Platonist concepts of infinity and square roots. -
Fibonacci Is Not a Pizza Topping Bevel Cut April, 2020 Andrew Davis Last Month I Saw an Amazing Advertisement Appearing in Our Woodworking Universe
Fibonacci is Not a Pizza Topping Bevel Cut April, 2020 Andrew Davis Last month I saw an amazing advertisement appearing in our woodworking universe. You might say that about all of the Woodpeckers advertisements – promoting good looking, even sexy tools and accessories for the woodworker who has money to burn or who does some odd task many times per day. I don’t fall into either category, though I often click on the ads to see the details and watch a video or two. These days I seem to have time to burn, if not money. Anyway, as someone who took too many math courses in my formative years, this ad absolutely caught my attention. I was left guessing two things: what is a Fibonacci and why would anyone need such a tool? Mr. Fibonacci and his Numbers Mathematician Leonardo Fibonacci was born in Pisa – now Italy – in 1170 and is credited with the first CNC machine and plunge router. Fibonacci popularized the Hindu–Arabic numeral system in the Western World (thereby replacing Roman numerals) primarily through his publication in 1202 called the Book of Calculation. Think how hard it would be to do any woodworking today if dimensions were called out as XVI by XXIV (16 x 24). But for this bevel cut article, the point is that he introduced Europe to the sequence of Fibonacci numbers. Fibonacci numbers are a sequence in which each number is the sum of the two preceding numbers. Here are the first Fibonacci numbers. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418.