On Popularization of Scientific Education in Italy Between 12Th and 16Th Century
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"Computers" Abacus—The First Calculator
Component 4: Introduction to Information and Computer Science Unit 1: Basic Computing Concepts, Including History Lecture 4 BMI540/640 Week 1 This material was developed by Oregon Health & Science University, funded by the Department of Health and Human Services, Office of the National Coordinator for Health Information Technology under Award Number IU24OC000015. The First "Computers" • The word "computer" was first recorded in 1613 • Referred to a person who performed calculations • Evidence of counting is traced to at least 35,000 BC Ishango Bone Tally Stick: Science Museum of Brussels Component 4/Unit 1-4 Health IT Workforce Curriculum 2 Version 2.0/Spring 2011 Abacus—The First Calculator • Invented by Babylonians in 2400 BC — many subsequent versions • Used for counting before there were written numbers • Still used today The Chinese Lee Abacus http://www.ee.ryerson.ca/~elf/abacus/ Component 4/Unit 1-4 Health IT Workforce Curriculum 3 Version 2.0/Spring 2011 1 Slide Rules John Napier William Oughtred • By the Middle Ages, number systems were developed • John Napier discovered/developed logarithms at the turn of the 17 th century • William Oughtred used logarithms to invent the slide rude in 1621 in England • Used for multiplication, division, logarithms, roots, trigonometric functions • Used until early 70s when electronic calculators became available Component 4/Unit 1-4 Health IT Workforce Curriculum 4 Version 2.0/Spring 2011 Mechanical Computers • Use mechanical parts to automate calculations • Limited operations • First one was the ancient Antikythera computer from 150 BC Used gears to calculate position of sun and moon Fragment of Antikythera mechanism Component 4/Unit 1-4 Health IT Workforce Curriculum 5 Version 2.0/Spring 2011 Leonardo da Vinci 1452-1519, Italy Leonardo da Vinci • Two notebooks discovered in 1967 showed drawings for a mechanical calculator • A replica was built soon after Leonardo da Vinci's notes and the replica The Controversial Replica of Leonardo da Vinci's Adding Machine . -
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. -
The "Greatest European Mathematician of the Middle Ages"
Who was Fibonacci? The "greatest European mathematician of the middle ages", his full name was Leonardo of Pisa, or Leonardo Pisano in Italian since he was born in Pisa (Italy), the city with the famous Leaning Tower, about 1175 AD. Pisa was an important commercial town in its day and had links with many Mediterranean ports. Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They are still called "bougies" in French, but the town is a ruin today says D E Smith (see below). So Leonardo grew up with a North African education under the Moors and later travelled extensively around the Mediterranean coast. He would have met with many merchants and learned of their systems of doing arithmetic. He soon realised the many advantages of the "Hindu-Arabic" system over all the others. D E Smith points out that another famous Italian - St Francis of Assisi (a nearby Italian town) - was also alive at the same time as Fibonacci: St Francis was born about 1182 (after Fibonacci's around 1175) and died in 1226 (before Fibonacci's death commonly assumed to be around 1250). By the way, don't confuse Leonardo of Pisa with Leonardo da Vinci! Vinci was just a few miles from Pisa on the way to Florence, but Leonardo da Vinci was born in Vinci in 1452, about 200 years after the death of Leonardo of Pisa (Fibonacci). His names Fibonacci Leonardo of Pisa is now known as Fibonacci [pronounced fib-on-arch-ee] short for filius Bonacci. -
Suanpan” in Chinese)
Math Exercise on the Abacus (“Suanpan” in Chinese) • Teachers’ Introduction • Student Materials Introduction Cards 1-7 Practicing Basics Cards 8-11 Exercises Cards 12, 14, 16 Answer keys Cards 13, 15, 17 Learning: Card 18 “Up,” “Down,” “Rid,” “Advance” Exercises: Addition (the numbers 1-9) Cards 18-28 Advanced Addition Cards 29-30 Exercises: Subtraction Cards 31-39 (the numbers 1-9) Acknowledgment: This unit is adapted from A Children’s Palace, by Michele Shoresman and Roberta Gumport, with illustrations by Elizabeth Chang (University of Illinois Urbana-Champagne, Center for Asian Studies, Outreach Office, 3rd ed., 1986. Print edition, now out of print.) 1 Teachers’ Introduction: Level: This unit is designed for students who understand p1ace value and know the basic addition and subtraction facts. Goals: 1. The students will learn to manipulate one form of ca1cu1ator used in many Asian countries. 2. The concept of p1ace value will be reinforced. 3. The students will learn another method of adding and subtracting. Instructions • The following student sheets may be copied so that your students have individual sets. • Individual suanpan for your students can be ordered from China Sprout: http://www.chinasprout.com/shop/ Product # A948 or ATG022 Evaluation The students will be able to manipulate a suanpan to set numbers, and to do simple addition and subtraction problems. Vocabulary suanpan set beam rod c1ear ones rod tens rod hundreds rod 2 Card 1 Suanpan – Abacus The abacus is an ancient calculator still used in China and other Asian countries. In Chinese it is called a “Suanpan.” It is a frame divided into an upper and lower section by a bar called the “beam.” The abacus can be used for addition, subtraction, multiplication, and division. -
An EM Construal of the Abacus
An EM Construal of the Abacus 1019358 Abstract The abacus is a simple yet powerful tool for calculations, only simple rules are used and yet the same outcome can be derived through various processes depending on the human operator. This paper explores how a construal of the abacus can be used as an educational tool but also how ‘human computing’ is illustrated. The user can gain understanding regarding how to operate the abacus for addition through a constructivist approach but can also use the more informative material elaborated in the EMPE context. 1 Introduction encapsulates all of these components together. The abacus can, not only be used for addition and 1.1 The Abacus subtraction, but also for multiplication, division, and even for calculating the square and cube roots of The abacus is a tool engineered to assist with numbers [3]. calculations, to improve the accuracy and the speed The Japanese also have a variation of the to complete such calculations at the same time to abacus known as the Soroban where the only minimise the mental calculations involved. In a way, difference from the Suanpan is each column an abacus can be thought of as one’s pen and paper contains one less Heaven and Earth bead [4]. used to note down the relevant numerals for a calculation or the modern day electronic calculator we are familiar with. However, the latter analogy may be less accurate as an electronic calculator would compute an answer when a series of buttons are pressed and requires less human input to perform the calculation compared to an abacus. -
Cracking the Cubic: Cardano, Controversy, and Creasing Alissa S
Cracking the Cubic: Cardano, Controversy, and Creasing Alissa S. Crans Loyola Marymount University MAA MD-DC-VA Spring Meeting Stevenson University April 14, 2012 These images are from the Wikipedia articles on Niccolò Fontana Tartaglia and Gerolamo Cardano. Both images belong to the public domain. 1 Quadratic Equation A brief history... • 400 BC Babylonians • 300 BC Euclid 323 - 283 BC This image is from the website entry for Euclid from the MacTutor History of Mathematics. It belongs to the public domain. 3 Quadratic Equation • 600 AD Brahmagupta To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. Brahmasphutasiddhanta Colebook translation, 1817, pg 346 598 - 668 BC This image is from the website entry for Brahmagupta from the The Story of Mathematics. It belongs to the public domain. 4 Quadratic Equation • 600 AD Brahmagupta To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. Brahmasphutasiddhanta Colebook translation, 1817, pg 346 598 - 668 BC ax2 + bx = c This image is from the website entry for Brahmagupta from the The Story of Mathematics. It belongs to the public domain. 4 Quadratic Equation • 600 AD Brahmagupta To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value. -
History of Algebra and Its Implications for Teaching
Maggio: History of Algebra and its Implications for Teaching History of Algebra and its Implications for Teaching Jaime Maggio Fall 2020 MA 398 Senior Seminar Mentor: Dr.Loth Published by DigitalCommons@SHU, 2021 1 Academic Festival, Event 31 [2021] Abstract Algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities (letters and other general sym- bols) defined by the equations that they satisfy. Algebraic problems have survived in mathematical writings of the Egyptians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts. In this paper, we will discuss historically famous mathematicians from all over the world along with their key mathematical contributions. Mathe- matical proofs of ancient and modern discoveries will be presented. We will then consider the impacts of incorporating history into the teaching of mathematics courses as an educational technique. 1 https://digitalcommons.sacredheart.edu/acadfest/2021/all/31 2 Maggio: History of Algebra and its Implications for Teaching 1 Introduction In order to understand the way algebra is the way it is today, it is important to understand how it came about starting with its ancient origins. In a mod- ern sense, algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities defined by the equations that they sat- isfy. Algebraic problems have survived in mathematical writings of the Egyp- tians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts, but these concepts had a heavier focus on geometry [1]. The combination of all of the discoveries of these great mathematicians shaped the way algebra is taught today. -
The Abacus: Instruction by Teachers of Students with Visual Impairments Sheila Amato, Sunggye Hong, and L
CEU Article The Abacus: Instruction by Teachers of Students with Visual Impairments Sheila Amato, Sunggye Hong, and L. Penny Rosenblum Structured abstract: Introduction: This article, based on a study of 196 teachers of students with visual impairments, reports on the experiences with and opin ions related to their decisions about instructing their students who are blind or have low vision in the abacus. Methods: The participants completed an online survey on how they decide which students should be taught abacus computation skills and which skills they teach. Data were also gathered on those who reported that they did not teach computation with the abacus. Results: The participants resided in the United States and Canada and had various numbers of years of teaching experience. More than two-thirds of those who reported that they taught abacus computation skills indicated that they began instruction when their students were between preschool and the second grade. When students were provided with instruction in abacus computation, the most frequently taught skills were the operations of addition and subtraction. More than two-thirds of the participants reported that students were allowed to use an abacus on high- stakes tests in their state or province. Discussion: Teachers of students with visual impairments are teaching students to compute using the Cranmer abacus. A small number of participants reported they did not teach computation with an abacus to their students because of their own lack of knowledge. Implications for practitioners: The abacus has a role in the toolbox of today’s students with visual impairments. Among other implications for educational practice, further studies are needed to examine more closely how teachers of students with visual impairments are instructing their students in computation with an abacus. -
The History of the Primality of One—A Selection of Sources
THE HISTORY OF THE PRIMALITY OF ONE|A SELECTION OF SOURCES ANGELA REDDICK, YENG XIONG, AND CHRIS K. CALDWELLy This document, in an altered and updated form, is available online: https:// cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.html. Please use that document instead. The table below is a selection of sources which address the question \is the number one a prime number?" Whether or not one is prime is simply a matter of definition, but definitions follow use, context and tradition. Modern usage dictates that the number one be called a unit and not a prime. We choose sources which made the author's view clear. This is often difficult because of language and typographical barriers (which, when possible, we tried to reproduce for the primary sources below so that the reader could better understand the context). It is also difficult because few addressed the question explicitly. For example, Gauss does not even define prime in his pivotal Disquisitiones Arithmeticae [45], but his statement of the fundamental theorem of arithmetic makes his stand clear. Some (see, for example, V. A. Lebesgue and G. H. Hardy below) seemed ambivalent (or allowed it to depend on the context?)1 The first column, titled `prime,' is yes when the author defined one to be prime. This is just a raw list of sources; for an evaluation of the history see our articles [17] and [110]. Any date before 1200 is an approximation. We would be glad to hear of significant additions or corrections to this list. Date: May 17, 2016. 2010 Mathematics Subject Classification. -
The Reception of Gerolamo Cardano's Liber De Ludo Aleae
READING IN CONTEXT: THE RECEPTION OF GEROLAMO CARDANO’S LIBER DE LUDO ALEAE A RESEARCH PAPER SUBMITTED TO DR. SLOAN DESPEAUX DEPARTMENT OF MATHEMATICS WESTERN CAROLINA UNIVERSITY BY NATHAN BOWMAN 1 Any person who enjoys gambling games would immensely desire to master the theory of probability and find a fool-proof way of winning at these games. Many have made valiant attempts at creating a successful system. One man, in the 16th century, genuinely believed that he had cracked the code. This man was Gerolamo Cardano. He detailed his discovery in his book Liber de Ludo Aleae, also known as The Book on Games of Chance. This book, a discussion of probability and gambling, was frowned upon due to numerous reasons, both moral and practical. Historians of mathematics have often discussed the work by Pascal, Fermat and Bernoulli; however, it seems that Cardano’s Liber de Ludo Aleae has often been left on the back of the shelf to collect dust. This poses the question of why Cardano’s book has been underappreciated. In this paper, we will seek to understand the reception of Liber de Ludo Aleae by looking deeper into Cardano as a gambler and mathematician, considering how he perceived his own book, and finally, by evaluating contemporary opinions and scholarly critiques. Much of what we know about Cardano comes from his own personal writings.1 He wrote many works including his autobiography entitled Da Propia Vita. In it, Cardano wrote that he was born in Pavia on September 24, 1501. He was born outside of wedlock, a situation usually disdained by the general public of the time. -
Algorithms, Recursion and Induction: Euclid and Fibonacci 1 Introduction
Algorithms, Recursion and Induction: Euclid and Fibonacci Desh Ranjan Department of Computer Science Old Dominion University, Norfolk, VA 23529 1 Introduction The central task in computing is designing of efficient computational meth- ods, or algorithms, for solving problems. An algorithm is a precise description of steps to be performed to accomplish a task or solve a problem. Recursion or recursive thinking is a key concept in solving problems and designing efficient algorithms to do so.Often when trying to solve a problem, one encounters a situation where the solution to a larger instance of the problem can be obtained if an appropriate smaller instance (or several smaller instances) of the same problem could be solved. Similarly, often one can define functions where values at “larger” arguments of the function are based on its own value at “smaller” argument values. In the same vein, sometimes it is possible to describe how to construct larger objects from smaller objects of similar type. However, the obvious question/difficulty here is how does one obtain so- lution to smaller instances. If one were to think as before (recursively!) this would require solving further smaller instances (sub-instances) of the same problem. Solving these sub-instances could lead to requiring solution of sub- sub-instances, etc. This could all become very confusing and the insight that the larger problem could have been solved using the solutions to the smaller problems would be useless if indeed one could not in some systematic way take advantage of it. Use of proper recursive definitions and notation allows us to avoid this confusion and harness the power of recursive thinking. -
Europe Starts to Wake Up: Leonardo of Pisa
Europe Starts to Wake up: Leonardo of Pisa Leonardo of Pisa, also known as Fibbonaci (from filius Bonaccia, “son of Bonnaccio”) was the greatest mathematician of the middle ages. He lived from 1175 to 1250, and traveled widely in the Mediterranean while young. He recognized the superiority of the Hindu-Arabic number system, and wrote LIber Abaci, the Book of Counting, to explain its merits (and to set down most of the arithmetic knowledge of the time). Hindu – Arabic Numerals: Fibbonacci also wrote the Liber Quadratorum or “Book of Squares,” devoted entirely to second-degree Diophantine equations. Example: Find rational numbers x, u, v satisfying: x22+ x= u x22− xv= Fibonacci’s solution involved finding three squares that form an arithmetic sequence, say ab22= −=d, c22b+d, 2 so d is the common difference. Then let x = b . One such solution is given by d ab22==1, 25, c2=49 so that x = 25 . 24 In Liber Quadratorum Fibonacci also gave examples of cubic equations whose solutions could not be rational numbers. Finally, Fibonacci proved that all Pythagorean triples (i.e. positive integers a, b, c satisfying ab22+=c2, can be obtained by letting as==2 tbs22−tc=s2+t2, where s and t are any positive integers. It was known in Euclid’s time that this method would always general Pythagorean triples; Fibonacci showed that it in fact produced all Pythagorean triples. Here are a few Pythagorean triples to amaze your friends and confuse your enemies: s t a b c 1 2435 1 4 8 15 17 1 5102426 1 6123537 2 3 12 5 13 2 4161220 2 5202129 2 6243240 3 4 24 7 25 3 5301634 3 6362745 3 7424058 4 5 40 9 41 4 6482052 18 174 6264 29952 30600 The Fibonacci Sequence Of course, Fibonacci is best known for his sequence, 1, 1, 2, 3, 5, 8, 13, .