The History of Mathematics, Part 8. Chuck Garner, Ph.D. February 3, 2014

Size: px
Start display at page:

Download "The History of Mathematics, Part 8. Chuck Garner, Ph.D. February 3, 2014"

Transcription

1 History of Mathematics, Part 8 Chuck, Ph.D. Department of Mathematics Rockdale Magnet School for Science and Technology February 3, 2014

2 Outline

3 Outline

4 of Alexandria 325 bc-265 bc laws of nature are but the mathematical thoughts of God.

5 Not much is known about his life Possibly of the first faculty in Alexandria Assembled the Elements in 13 books Wrote other books Data; supplement to books I-VI On Divisions; constructions of figures into 2 parts Phaenomena; astronomy Optics; perspective Many no longer exist: Elements of Music, Pseudoria, Porisms, Conics, Surface-Loci

6 and the Elements took previous works and studied them He assembled these into a logical framework, c. 300 bc All of geometry can be deduced from basic axioms, postulates, and definitions Much of Elements is not s, nor does he claim it is Influential due to its organization Rendered all other geometry works obsolete

7 Outline

8 of the Elements Book I Axioms, postulates, definitions, and 48 fundamental propositions Book II Geometric algebra Book III Circles Book IV Constructions of polygons Book V Eudoxian theory of proportion Book VI Application of Eudoxian proportion to similiar figures

9 of the Elements Book VII Definitions and propositions in number theory Book VIII Continued proportions and geometric progressions Book IX Number theory (FTA and Infinitude of Primes) Book X Definitions and results on incommensurables Book XI Solid geometry Book XII Measurement of solids Book XIII Platonic solids

10 Through Time Additions to Book XIV Continuation of Book XIII (Hypsicles, c. 125 bc) Book XV Construction of solids (Isidorus, c. 525 ad) 888 ad edition of the Elements First printed edition in 1482 Other editions...

11 1482 Edition First printed edition

12 1543 Edition First interactive edition

13 1574 Edition Published by mathematician and astronomer Christopher Clavius

14 1661 Edition First English edition to include Data

15 1776 Edition Published in Glasgow by Foulis Press

16 Through Time Byrne s 1847 color edition Heath s 1909 translation Green Lion Press edition David Joyce s dynamic Elements

17 Outline

18 of the Elements Compass and straightedge are ean tools Revolutionized mathematical thought Became a template for presenting mathematical thought Geometry became the standard notion of what mathematics is Arithmetic and algebra was for the common; geometry for the learned Geometry was the gateway to all other learning d thinking for millenia Attempting to fix gaps in led to much more mathematics

19 Declaration of Independence We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.

20 Abraham Lincoln and From Lincoln s law partner, Billy Henderson: He studied and nearly mastered the six books of since he was a member of Congress. He began a course of rigid mental discipline with the intent to improve his faculties, especially his powers of logic and language. Hence his fondness for, which he carried with him on the circuit till he could demonstrate with ease all the propositions in the six books; often studying far into the night, with a candle near his pillow, while his fellow-lawyers, half a dozen in a room, filled the air with interminable snoring.

21 Abraham Lincoln and Lincoln on why he studied : In the course of my law reading I constantly came upon the word demonstrate. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?...i consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man. At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father s house, and stayed there till I could give any proposition in the six books of at sight. I then found out what demonstrate means, and went back to my law studies.

22 Abraham Lincoln and Lincoln, from the fourth Lincoln-Douglas debate: If you have ever studied geometry, you remember that by a course of reasoning, proves that all the angles in a triangle are equal to two right angles. has shown you how to work it out. Now, if you undertake to disprove that proposition, and to show that it is erroneous, would you prove it to be false by calling a liar?

23 Outline

24 Article on a particular proposition of Article on the regular polyhedra Parts of Book I of the Elements with commentaries; Reader, 3.B Discovering the Elements; Reader, 3.F Debating geometrical algebra; Reader, 3.G Origins and impact of the Elements; Math Through the Ages, Sketch 14 Next: Thinker and the Thug

Springfield, went home to my father s house, and stayed there till I could give any proposition in the

Springfield, went home to my father s house, and stayed there till I could give any proposition in the Euclid Euclid (c. 325 BCE c. 265 CE) was of the best selling mathematics textbook of all time. The Elements has introduced millions to the powers of logic and language. As a struggling lawyer at age forty,

More information

Math 0095: Developmental Mathematics Emporium

Math 0095: Developmental Mathematics Emporium Math 0095: Developmental Mathematics Emporium Course Titles: Credit hours: Prerequisites: Math 0099: Early Foundations of College Mathematics Math 0100: Foundations of College Mathematics Math 0101: Foundations

More information

Math 0095: Developmental Emporium Mathematics

Math 0095: Developmental Emporium Mathematics Math 0095: Developmental Emporium Mathematics Course Titles: Credit hours: Prerequisites: Math 0099: Early Foundations of College Mathematics Math 0100: Foundations of College Mathematics Math 0101: Foundations

More information

Euclidean Geometry. The Elements of Mathematics

Euclidean Geometry. The Elements of Mathematics Euclidean Geometry The Elements of Mathematics Euclid, We Hardly Knew Ye Born around 300 BCE in Alexandria, Egypt We really know almost nothing else about his personal life Taught students in mathematics

More information

Greece. Chapter 5: Euclid of Alexandria

Greece. Chapter 5: Euclid of Alexandria Greece Chapter 5: Euclid of Alexandria The Library at Alexandria What do we know about it? Well, a little history Alexander the Great In about 352 BC, the Macedonian King Philip II began to unify the numerous

More information

In Defense of Euclid. The Ancient Greek Theory of Numbers

In Defense of Euclid. The Ancient Greek Theory of Numbers In Defense of Euclid The Ancient Greek Theory of Numbers The Poetry of Euclid A unit is that by virtue of which each of the things that exist is called one.» The Elements, book VII, definition 1. Our Goal:

More information

Item 8. Constructing the Square Area of Two Proving No Irrationals. 6 Total Pages

Item 8. Constructing the Square Area of Two Proving No Irrationals. 6 Total Pages Item 8 Constructing the Square Area of Two Proving No Irrationals 6 Total Pages 1 2 We want to start with Pi. How Geometry Proves No Irrations They call Pi the ratio of the circumference of a circle to

More information

Dr Prya Mathew SJCE Mysore

Dr Prya Mathew SJCE Mysore 1 2 3 The word Mathematics derived from two Greek words Manthanein means learning Techne means an art or technique So Mathematics means the art of learning related to disciplines or faculties disciplines

More information

Lecture 8. Eudoxus and the Avoidance of a Fundamental Conflict

Lecture 8. Eudoxus and the Avoidance of a Fundamental Conflict Lecture 8. Eudoxus and the Avoidance of a Fundamental Conflict Eudoxus of Cnidus Eudoxus, 480 BC - 355 BC, was a Greek philosopher, mathematician and astronomer who contributed to Euclid s Elements. His

More information

Euclidean Geometry Proofs

Euclidean Geometry Proofs Euclidean Geometry Proofs History Thales (600 BC) First to turn geometry into a logical discipline. Described as the first Greek philosopher and the father of geometry as a deductive study. Relied on rational

More information

http://radicalart.info/physics/vacuum/index.html The Scientific Revolution In the 1500s and 1600s the Scientific Revolution changed the way Europeans looked at the world. People began to make conclusions

More information

p, p or its negation is true, and the other false

p, p or its negation is true, and the other false Logic and Proof In logic (and mathematics) one often has to prove the truthness of a statement made. A proposition is a (declarative) sentence that is either true or false. Example: An odd number is prime.

More information

Lecture 1: Axioms and Models

Lecture 1: Axioms and Models Lecture 1: Axioms and Models 1.1 Geometry Although the study of geometry dates back at least to the early Babylonian and Egyptian societies, our modern systematic approach to the subject originates in

More information

Credited with formulating the method of exhaustion for approximating a circle by polygons

Credited with formulating the method of exhaustion for approximating a circle by polygons MATH 300 History of Mathematics Figures in Greek Mathematics Sixth Century BCE Thales of Miletus May have formulated earliest theorems in geometry (e.g., ASA) Predicted an eclipse in 585 BCE Pythagoras

More information

What is a Revolution? A Revolution is a complete change, or an overthrow of a government, a social system, etc.

What is a Revolution? A Revolution is a complete change, or an overthrow of a government, a social system, etc. CW10 p374 Vocab What is a Revolution? A Revolution is a complete change, or an overthrow of a government, a social system, etc. The Scientific Revolution In the 1500s and 1600s the Scientific Revolution

More information

The Scientific Revolution

The Scientific Revolution The Scientific Revolution What is a Revolution? A Revolution is a complete change, or an overthrow of a government, a social system, etc. The Scientific Revolution In the 1500s and 1600s the Scientific

More information

Algebra I. Course Outline

Algebra I. Course Outline Algebra I Course Outline I. The Language of Algebra A. Variables and Expressions B. Order of Operations C. Open Sentences D. Identity and Equality Properties E. The Distributive Property F. Commutative

More information

MATH10040: Chapter 0 Mathematics, Logic and Reasoning

MATH10040: Chapter 0 Mathematics, Logic and Reasoning MATH10040: Chapter 0 Mathematics, Logic and Reasoning 1. What is Mathematics? There is no definitive answer to this question. 1 Indeed, the answer given by a 21st-century mathematician would differ greatly

More information

Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions

Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Solutions Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. A prime number

More information

Geometry I (CM122A, 5CCM122B, 4CCM122A)

Geometry I (CM122A, 5CCM122B, 4CCM122A) Geometry I (CM122A, 5CCM122B, 4CCM122A) Lecturer: Giuseppe Tinaglia Office: S5.31 Office Hours: Wed 1-3 or by appointment. E-mail: giuseppe.tinaglia@kcl.ac.uk Course webpage: http://www.mth.kcl.ac.uk/

More information

Why write proofs? Why not just test and repeat enough examples to confirm a theory?

Why write proofs? Why not just test and repeat enough examples to confirm a theory? P R E F A C E T O T H E S T U D E N T Welcome to the study of mathematical reasoning. The authors know that many students approach this material with some apprehension and uncertainty. Some students feel

More information

Pell s Equation Claire Larkin

Pell s Equation Claire Larkin Pell s Equation is a Diophantine equation in the form: Pell s Equation Claire Larkin The Equation x 2 dy 2 = where x and y are both integer solutions and n is a positive nonsquare integer. A diophantine

More information

Plate 1. Portrait of Simon Stevin by an unknown artist. Library of Leyden University.

Plate 1. Portrait of Simon Stevin by an unknown artist. Library of Leyden University. The renovation of the arts and sciences by the Renaissance has been especially glorious in Italy and the Netherlands. Among the outstanding scholars of that period Simon Stevin, born at Bruges and living

More information

DISCOVERING GEOMETRY Over 6000 years ago, geometry consisted primarily of practical rules for measuring land and for

DISCOVERING GEOMETRY Over 6000 years ago, geometry consisted primarily of practical rules for measuring land and for Name Period GEOMETRY Chapter One BASICS OF GEOMETRY Geometry, like much of mathematics and science, developed when people began recognizing and describing patterns. In this course, you will study many

More information

Proof of a theorem of Fermat that every prime number of the form 4n + 1 is a sum of two squares

Proof of a theorem of Fermat that every prime number of the form 4n + 1 is a sum of two squares Proof of a theorem of Fermat that every prime number of the form 4n + 1 is a sum of two squares Leonhard Euler 1. When I recently 1 considered these numbers which arise from the addition of two squares,

More information

FACTORIZATION AND THE PRIMES

FACTORIZATION AND THE PRIMES I FACTORIZATION AND THE PRIMES 1. The laws of arithmetic The object of the higher arithmetic is to discover and to establish general propositions concerning the natural numbers 1, 2, 3,... of ordinary

More information

Euclid and The Elements

Euclid and The Elements Euclid and The Elements C R Pranesachar When a crime is committed and the culprit responsible for the crime is nabbed, the job of the investigators is not over. To convince the judge, they have to prove

More information

Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers

Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Math Circle Beginners Group February 28, 2016 Euclid and Prime Numbers Warm-up Problems 1. What is a prime number? Give an example of an even prime number and an odd prime number. (a) Circle the prime

More information

Grade 7/8 Math Circles. Mathematical Thinking

Grade 7/8 Math Circles. Mathematical Thinking Faculty of Mathematics Waterloo, Ontario N2L 3G1 Centre for Education in Mathematics and Computing Grade 7/8 Math Circles March 22 & 23 2016 Mathematical Thinking Today we will take a look at some of the

More information

Chapter 12: Ruler and compass constructions

Chapter 12: Ruler and compass constructions Chapter 12: Ruler and compass constructions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4120, Spring 2014 M. Macauley (Clemson) Chapter

More information

MATH 115 Concepts in Mathematics

MATH 115 Concepts in Mathematics South Central College MATH 115 Concepts in Mathematics Course Outcome Summary Course Information Description Total Credits 4.00 Total Hours 64.00 Concepts in Mathematics is a general education survey course

More information

Grade 6 Math Circles. Ancient Mathematics

Grade 6 Math Circles. Ancient Mathematics Faculty of Mathematics Waterloo, Ontario N2L 3G1 Introduction Grade 6 Math Circles October 17/18, 2017 Ancient Mathematics Centre for Education in Mathematics and Computing Have you ever wondered where

More information

SSWH13 The student will examine the intellectual, political, social, and economic factors that changed the world view of Europeans.

SSWH13 The student will examine the intellectual, political, social, and economic factors that changed the world view of Europeans. SSWH13 The student will examine the intellectual, political, social, and economic factors that changed the world view of Europeans. a. Explain the scientific contributions of Copernicus, Galileo, Kepler,

More information

Abstract. History of the problem

Abstract. History of the problem HOW THE GREEKS MIGHT HAVE DISCOVERED AND APPROXIMATE IRRATIONAL NUMBERS László Filep, PhD Institute of Mathematics and Computer Science, College of Nyíregyháza, Nyíregyháza, Hungary Web: www.nyf.hu E-mail:

More information

3. Euclid and the Elements

3. Euclid and the Elements 3. Euclid and the Elements (Burton, 4.1 4.3) Alexander the Great s political empire fragmented shortly after his death in 323 B. C. E., but the cultural effects of his conquests were irreversible and defined

More information

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that:

Scott Taylor 1. EQUIVALENCE RELATIONS. Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: Equivalence MA Relations 274 and Partitions Scott Taylor 1. EQUIVALENCE RELATIONS Definition 1.1. Let A be a set. An equivalence relation on A is a relation such that: (1) is reflexive. That is, (2) is

More information

The WhatPower Function à An Introduction to Logarithms

The WhatPower Function à An Introduction to Logarithms Classwork Work with your partner or group to solve each of the following equations for x. a. 2 # = 2 % b. 2 # = 2 c. 2 # = 6 d. 2 # 64 = 0 e. 2 # = 0 f. 2 %# = 64 Exploring the WhatPower Function with

More information

Math Project 1 We haven t always been a world where proof based mathematics existed and in fact, the

Math Project 1 We haven t always been a world where proof based mathematics existed and in fact, the 336 Math Project 1 We haven t always been a world where proof based mathematics existed and in fact, the 1 use of proofs emerged in ancient Greek mathematics sometime around 300 BC. It was essentially

More information

The Limit of Humanly Knowable Mathematical Truth

The Limit of Humanly Knowable Mathematical Truth The Limit of Humanly Knowable Mathematical Truth Gödel s Incompleteness Theorems, and Artificial Intelligence Santa Rosa Junior College December 12, 2015 Another title for this talk could be... An Argument

More information

The Arithmetic of Reasoning. Chessa Horomanski & Matt Corson

The Arithmetic of Reasoning. Chessa Horomanski & Matt Corson The Arithmetic of Reasoning LOGIC AND BOOLEAN ALGEBRA Chessa Horomanski & Matt Corson Computers Ask us questions, correct our grammar, calculate our taxes But Misunderstand what we re sure we told them,

More information

MthEd/Math 300 Williams Fall 2011 Midterm Exam 2

MthEd/Math 300 Williams Fall 2011 Midterm Exam 2 Name: MthEd/Math 300 Williams Fall 2011 Midterm Exam 2 Closed Book / Closed Note. Answer all problems. You may use a calculator for numerical computations. Section 1: For each event listed in the first

More information

Scientific Revolution

Scientific Revolution Age of Revolutions Scientific Revolution Scientific Revolution Period of time in which a new way of thinking came about. The beliefs held by many for so long were now being questioned. Use logic and reason

More information

Glossary of Logical Terms

Glossary of Logical Terms Math 304 Spring 2007 Glossary of Logical Terms The following glossary briefly describes some of the major technical logical terms used in this course. The glossary should be read through at the beginning

More information

The Scientific Revolution

The Scientific Revolution Chapter 18, Section 2 The Scientific Revolution (Pages 670 679) Setting a Purpose for Reading Think about these questions as you read: How did the Scientific Revolution change life in the 1600s? What is

More information

Euclid Geometry And Non-Euclid Geometry. Have you ever asked yourself why is it that if you walk to a specific place from

Euclid Geometry And Non-Euclid Geometry. Have you ever asked yourself why is it that if you walk to a specific place from Hu1 Haotian Hu Dr. Boman Math 475W 9 November 2016 Euclid Geometry And Non-Euclid Geometry Have you ever asked yourself why is it that if you walk to a specific place from somewhere, you will always find

More information

Workshop 1- Building on the Axioms. The First Proofs

Workshop 1- Building on the Axioms. The First Proofs Boston University Summer I 2009 Workshop 1- Building on the Axioms. The First Proofs MA341 Number Theory Kalin Kostadinov The goal of this workshop was to organize our experience with the common integers

More information

Diodorus s Master Argument Nino B. Cocchiarella For Classes IC and IIC Students of Professor Giuseppe Addona

Diodorus s Master Argument Nino B. Cocchiarella For Classes IC and IIC Students of Professor Giuseppe Addona Diodorus s Master Argument Nino B. Cocchiarella For Classes IC and IIC Students of Professor Giuseppe Addona In my Remarks on Stoic Logic that I wrote for you last year, I mentioned Diodorus Cronus s trilemma,

More information

Math Number 842 Professor R. Roybal MATH History of Mathematics 24th October, Project 1 - Proofs

Math Number 842 Professor R. Roybal MATH History of Mathematics 24th October, Project 1 - Proofs Math Number 842 Professor R. Roybal MATH 331 - History of Mathematics 24th October, 2017 Project 1 - Proofs Mathematical proofs are an important concept that was integral to the development of modern mathematics.

More information

NOTE The book 'Eight Essays on Geometry for an Oral Tradition' will hopefully be available later in 2016, free online

NOTE The book 'Eight Essays on Geometry for an Oral Tradition' will hopefully be available later in 2016, free online THREE BOOKS AND A PAPER: A GEOMETRY PROJECT Andrew Nicholas ABSTRACT The aim of the project was to produce a sort of miniature Vedic mathematics version of Euclid's 'Elements', covering much less ground

More information

Introduction to Logic

Introduction to Logic Introduction to Logic L. Marizza A. Bailey June 21, 2014 The beginning of Modern Mathematics Before Euclid, there were many mathematicians that made great progress in the knowledge of numbers, algebra

More information

CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC

CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC CHAPTER 2 INTRODUCTION TO CLASSICAL PROPOSITIONAL LOGIC 1 Motivation and History The origins of the classical propositional logic, classical propositional calculus, as it was, and still often is called,

More information

{ }. The dots mean they continue in that pattern.

{ }. The dots mean they continue in that pattern. INTEGERS Integers are positive and negative whole numbers, that is they are;... 3, 2, 1,0,1,2,3... { }. The dots mean they continue in that pattern. Like all number sets, integers were invented to describe

More information

INTRODUCTION TO LOGIC

INTRODUCTION TO LOGIC INTRODUCTION TO LOGIC L. MARIZZA A. BAILEY 1. The beginning of Modern Mathematics Before Euclid, there were many mathematicians that made great progress in the knowledge of numbers, algebra and geometry.

More information

Pierre de Fermat ( )

Pierre de Fermat ( ) Section 04 Mathematical Induction 987 8 Find the sum of the first ten terms of the sequence: 9 Find the sum of the first 50 terms of the sequence: 0 Find the sum of the first ten terms of the sequence:

More information

Mathematics for Computer Scientists

Mathematics for Computer Scientists Mathematics for Computer Scientists Lecture notes for the module G51MCS Venanzio Capretta University of Nottingham School of Computer Science Chapter 6 Modular Arithmetic 6.1 Pascal s Triangle One easy

More information

THE INTRODUCTION OF COMPLEX NUMBERS*

THE INTRODUCTION OF COMPLEX NUMBERS* THE INTRODUCTION OF COMPLEX NUMBERS* John N. Crossley Monash University, Melbourne, Australia Any keen mathematics student will tell you that complex numbers come in when you want to solve a quadratic

More information

Assignment 3 Logic and Reasoning KEY

Assignment 3 Logic and Reasoning KEY Assignment 3 Logic and Reasoning KEY Print this sheet and fill in your answers. Please staple the sheets together. Turn in at the beginning of class on Friday, September 8. Recall this about logic: Suppose

More information

Lesson 14: An Axiom System for Geometry

Lesson 14: An Axiom System for Geometry 219 Lesson 14: n xiom System for Geometry We are now ready to present an axiomatic development of geometry. This means that we will list a set of axioms for geometry. These axioms will be simple fundamental

More information

Critical Thinking: Sir Isaac Newton

Critical Thinking: Sir Isaac Newton Critical Thinking: Sir Isaac Name: Date: Watch this NOVA program on while finding the answers for the following questions: https://www.youtube.com/watch?v=yprv1h3cgqk 1.In 19 a British Economist named

More information

Gödel s Incompleteness Theorem. Behrad Taghavi Department of Physics & Astronomy, Stony Brook University.

Gödel s Incompleteness Theorem. Behrad Taghavi Department of Physics & Astronomy, Stony Brook University. Gödel s Incompleteness Theorem Behrad Taghavi Department of Physics & Astronomy, Stony Brook University. Gödel s Incompleteness Theorem or Is there any guaranty that there would always be a job for physicists

More information

Name (please print) Mathematics Final Examination December 14, 2005 I. (4)

Name (please print) Mathematics Final Examination December 14, 2005 I. (4) Mathematics 513-00 Final Examination December 14, 005 I Use a direct argument to prove the following implication: The product of two odd integers is odd Let m and n be two odd integers Since they are odd,

More information

Grade 6 Math Circles November 1 st /2 nd. Egyptian Mathematics

Grade 6 Math Circles November 1 st /2 nd. Egyptian Mathematics Faculty of Mathematics Waterloo, Ontario N2L 3G Centre for Education in Mathematics and Computing Grade 6 Math Circles November st /2 nd Egyptian Mathematics Ancient Egypt One of the greatest achievements

More information

STATION #1: NICOLAUS COPERNICUS

STATION #1: NICOLAUS COPERNICUS STATION #1: NICOLAUS COPERNICUS Nicolaus Copernicus was a Polish astronomer who is best known for the astronomical theory that the Sun was near the center of the universe and that the Earth and other planets

More information

Section 2.1: Introduction to the Logic of Quantified Statements

Section 2.1: Introduction to the Logic of Quantified Statements Section 2.1: Introduction to the Logic of Quantified Statements In the previous chapter, we studied a branch of logic called propositional logic or propositional calculus. Loosely speaking, propositional

More information

THE MATHEMATICS OF EULER. Introduction: The Master of Us All. (Dunham, Euler the Master xv). This quote by twentieth-century mathematician André Weil

THE MATHEMATICS OF EULER. Introduction: The Master of Us All. (Dunham, Euler the Master xv). This quote by twentieth-century mathematician André Weil THE MATHEMATICS OF EULER Introduction: The Master of Us All All his life he seems to have carried in his head the whole of the mathematics of his day (Dunham, Euler the Master xv). This quote by twentieth-century

More information

0. Introduction. Math 407: Modern Algebra I. Robert Campbell. January 29, 2008 UMBC. Robert Campbell (UMBC) 0. Introduction January 29, / 22

0. Introduction. Math 407: Modern Algebra I. Robert Campbell. January 29, 2008 UMBC. Robert Campbell (UMBC) 0. Introduction January 29, / 22 0. Introduction Math 407: Modern Algebra I Robert Campbell UMBC January 29, 2008 Robert Campbell (UMBC) 0. Introduction January 29, 2008 1 / 22 Outline 1 Math 407: Abstract Algebra 2 Sources 3 Cast of

More information

Direct Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24

Direct Proof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Direct Proof Fall / 24 Direct Proof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Direct Proof Fall 2014 1 / 24 Outline 1 Overview of Proof 2 Theorems 3 Definitions 4 Direct Proof 5 Using

More information

From Hipparchus to a Helix Ruler

From Hipparchus to a Helix Ruler From Hipparchus to a Helix Ruler Jerold Mathews This paper gives a brief history of the chord function { a precursor of the sine function { from the mathematics/astronomy of Hipparchus and Ptolemy to the

More information

STRATEGIES OF PROBLEM SOLVING

STRATEGIES OF PROBLEM SOLVING STRATEGIES OF PROBLEM SOLVING Second Edition Maria Nogin Department of Mathematics College of Science and Mathematics California State University, Fresno 2014 2 Chapter 1 Introduction Solving mathematical

More information

Logical Form, Mathematical Practice, and Frege s Begriffsschrift. Danielle Macbeth Haverford College

Logical Form, Mathematical Practice, and Frege s Begriffsschrift. Danielle Macbeth Haverford College Logical Form, Mathematical Practice, and Frege s Begriffsschrift Danielle Macbeth Haverford College Logic Colloquium 2015, Helsinki, August 3 8, 2015 Outline written mathematics report of reasoning display

More information

Size of the Earth and the Distances to the Moon and the Sun

Size of the Earth and the Distances to the Moon and the Sun Size of the Earth and the Distances to the Moon and the Sun Objectives Using observations of the Earth-Moon-Sun system and elementary geometry and trigonometry, we will duplicate the methods of the ancient

More information

Is There Any Evidence for a Creator in the Universe?

Is There Any Evidence for a Creator in the Universe? Is There Any Evidence for a Creator in the Universe? By Claude LeBlanc, M.A., Magis Center, 2016 Opening Prayer Father, you give us the ability to learn about the world you created. Through our senses

More information

All great designs are driven by a motivator. A single or series of entities prompt the

All great designs are driven by a motivator. A single or series of entities prompt the The Driving Force: Mathematics or the Universe? All great designs are driven by a motivator. A single or series of entities prompt the development of the design, shaping and influencing the end product.

More information

Equivalent Forms of the Axiom of Infinity

Equivalent Forms of the Axiom of Infinity Equivalent Forms of the Axiom of Infinity Axiom of Infinity 1. There is a set that contains each finite ordinal as an element. The Axiom of Infinity is the axiom of Set Theory that explicitly asserts that

More information

WUCT121. Discrete Mathematics. Logic. Tutorial Exercises

WUCT121. Discrete Mathematics. Logic. Tutorial Exercises WUCT11 Discrete Mathematics Logic Tutorial Exercises 1 Logic Predicate Logic 3 Proofs 4 Set Theory 5 Relations and Functions WUCT11 Logic Tutorial Exercises 1 Section 1: Logic Question1 For each of the

More information

Is mathematics discovery or invention?

Is mathematics discovery or invention? Is mathematics discovery or invention? From the problems of everyday life to the mystery of existence Part One By Marco Dal Prà Venice - Italy May 2013 1 Foreword This document is intended as a starting

More information

Supplementary Material for MTH 299 Online Edition

Supplementary Material for MTH 299 Online Edition Supplementary Material for MTH 299 Online Edition Abstract This document contains supplementary material, such as definitions, explanations, examples, etc., to complement that of the text, How to Think

More information

6. MESH ANALYSIS 6.1 INTRODUCTION

6. MESH ANALYSIS 6.1 INTRODUCTION 6. MESH ANALYSIS INTRODUCTION PASSIVE SIGN CONVENTION PLANAR CIRCUITS FORMATION OF MESHES ANALYSIS OF A SIMPLE CIRCUIT DETERMINANT OF A MATRIX CRAMER S RULE GAUSSIAN ELIMINATION METHOD EXAMPLES FOR MESH

More information

CHAPTER 5 INTRODUCTION TO EUCLID S GEOMETRY. 5.1 Introduction

CHAPTER 5 INTRODUCTION TO EUCLID S GEOMETRY. 5.1 Introduction 78 MATHEMATICS INTRODUCTION TO EUCLID S GEOMETRY CHAPTER 5 5.1 Introduction The word geometry comes form the Greek words geo, meaning the earth, and metrein, meaning to measure. Geometry appears to have

More information

PRINCIPLE OF MATHEMATICAL INDUCTION

PRINCIPLE OF MATHEMATICAL INDUCTION Chapter 4 PRINCIPLE OF MATHEMATICAL INDUCTION Analysis and natural philosopy owe their most important discoveries to this fruitful means, which is called induction Newton was indebted to it for his theorem

More information

Arab Mathematics Bridges the Dark Ages. early fourth century and the European Giants in the seventeenth and eighteenth

Arab Mathematics Bridges the Dark Ages. early fourth century and the European Giants in the seventeenth and eighteenth John Griffith Arab Mathematics Bridges the Dark Ages When most people think of Mathematics, they tend to think of people like Plato, Aristotle, Newton, Leibniz, and a plethora of Greek Mathematicians.

More information

Invention of Algebra by Arab Mathematicians. Alex Gearty, Shane Becker, Lauren Ferris

Invention of Algebra by Arab Mathematicians. Alex Gearty, Shane Becker, Lauren Ferris Invention of Algebra by Arab Mathematicians Alex Gearty, Shane Becker, Lauren Ferris The Algebra of Squares and Roots The Hindu-Arabic Numeral System - Here we see the evolution of the Brahmi system as

More information

Incoming Magnet Precalculus / Functions Summer Review Assignment

Incoming Magnet Precalculus / Functions Summer Review Assignment Incoming Magnet recalculus / Functions Summer Review ssignment Students, This assignment should serve as a review of the lgebra and Geometry skills necessary for success in recalculus. These skills were

More information

Introducing Proof 1. hsn.uk.net. Contents

Introducing Proof 1. hsn.uk.net. Contents Contents 1 1 Introduction 1 What is proof? 1 Statements, Definitions and Euler Diagrams 1 Statements 1 Definitions Our first proof Euler diagrams 4 3 Logical Connectives 5 Negation 6 Conjunction 7 Disjunction

More information

S2 (2.2) Equations.notebook November 24, 2015

S2 (2.2) Equations.notebook November 24, 2015 Daily Practice 7.10.2015 Q1. Multiply out and simplify 7(x - 3) + 2(x + 1) Q2. Simplify the ratio 14:21 Q3. Find 17% of 5000 Today we will be learning to solve equations. Homework Due tomorrow. Q4. Given

More information

Complex Numbers: A Brief Introduction. By: Neal Dempsey. History of Mathematics. Prof. Jennifer McCarthy. July 16, 2010

Complex Numbers: A Brief Introduction. By: Neal Dempsey. History of Mathematics. Prof. Jennifer McCarthy. July 16, 2010 1 Complex Numbers: A Brief Introduction. By: Neal Dempsey History of Mathematics Prof. Jennifer McCarthy July 16, 2010 2 Abstract Complex numbers, although confusing at times, are one of the most elegant

More information

Finding an Equation of a Line

Finding an Equation of a Line Lesson 3-4 Finding an Equation of a Line Vocabulary point-slope form piecewise linear function BIG IDEA Postulates and theorems of geometry about lines tell when exactly one line is determined from given

More information

Summer Work Packet for MPH Math Classes

Summer Work Packet for MPH Math Classes Summer Work Packet for MPH Math Classes Students going into Geometry AC Sept. 2017 Name: This packet is designed to help students stay current with their math skills. Each math class expects a certain

More information

P1-763.PDF Why Proofs?

P1-763.PDF Why Proofs? P1-763.PDF Why Proofs? During the Iron Age men finally started questioning mathematics which eventually lead to the creating of proofs. People wanted to know how and why is math true, rather than just

More information

Chapter One. The Real Number System

Chapter One. The Real Number System Chapter One. The Real Number System We shall give a quick introduction to the real number system. It is imperative that we know how the set of real numbers behaves in the way that its completeness and

More information

Introduction to Geometry

Introduction to Geometry Introduction to Geometry What is Geometry Why do we use Geometry What is Geometry? Geometry is a branch of mathematics that concerns itself with the questions of shape, size, position of figures, and the

More information

Algebra SEMESTER ONE. K12.com { Pg. 1 } Course Overview. Unit 1: Algebra Basics. Unit 2: Properties of Real Numbers

Algebra SEMESTER ONE. K12.com { Pg. 1 } Course Overview. Unit 1: Algebra Basics. Unit 2: Properties of Real Numbers Algebra Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform manipulations with numbers, variables, equations, and inequalities. They also learn

More information

Math 202 notes. Jason Riedy. 27 August, Review 1. 2 Ruling out possibilities Logic puzzles The pigeonhole principle 2

Math 202 notes. Jason Riedy. 27 August, Review 1. 2 Ruling out possibilities Logic puzzles The pigeonhole principle 2 Math 202 notes Jason Riedy 27 August, 2008 Contents 1 Review 1 2 Ruling out possibilities 1 2.1 Logic puzzles............................. 2 3 The pigeonhole principle 2 4 Mathematical reasoning 3 5 Next

More information

Logic CHAPTER. 3.1 A Little Dash of Logic Two Methods of Logical Reasoning p. 101

Logic CHAPTER. 3.1 A Little Dash of Logic Two Methods of Logical Reasoning p. 101 CHAPTER Logic Riding a bicycle is a skill which, once learned, is rarely forgotten. What s more, bicycles are enough alike that if you can ride one bike, you can pretty much ride them all. This is an example

More information

ALGEBRA AND GEOMETRY. Cambridge University Press Algebra and Geometry Alan F. Beardon Frontmatter More information

ALGEBRA AND GEOMETRY. Cambridge University Press Algebra and Geometry Alan F. Beardon Frontmatter More information ALGEBRA AND GEOMETRY This text gives a basic introduction and a unified approach to algebra and geometry. It covers the ideas of complex numbers, scalar and vector products, determinants, linear algebra,

More information

Preface to the First Edition. xxvii 0.1 Set-theoretic Notation xxvii 0.2 Proof by Induction xxix 0.3 Equivalence Relations and Equivalence Classes xxx

Preface to the First Edition. xxvii 0.1 Set-theoretic Notation xxvii 0.2 Proof by Induction xxix 0.3 Equivalence Relations and Equivalence Classes xxx Table of Preface to the First Edition Preface to the Second Edition page xvii xxi Mathematical Prolegomenon xxvii 0.1 Set-theoretic Notation xxvii 0.2 Proof by Induction xxix 0.3 Equivalence Relations

More information

Table of Contents. Number and Operation. Geometry. Measurement. Lesson 1 Goldbach s Conjecture Lesson 2 Micro Mites... 11

Table of Contents. Number and Operation. Geometry. Measurement. Lesson 1 Goldbach s Conjecture Lesson 2 Micro Mites... 11 Table of Contents Number and Operation Lesson 1 Goldbach s Conjecture........................ 5 Prime Factorization Lesson 2 Micro Mites.................................... 11 Division with Decimals Lesson

More information

Variable/Equality Concept

Variable/Equality Concept Congruences and equations Mathematics Unit 8: A transition to Algebra Variable/Equality Concept What is algebra? How would you characterize when a student moves from studying arithmetic to algebra? What

More information

35 Chapter CHAPTER 4: Mathematical Proof

35 Chapter CHAPTER 4: Mathematical Proof 35 Chapter 4 35 CHAPTER 4: Mathematical Proof Faith is different from proof; the one is human, the other is a gift of God. Justus ex fide vivit. It is this faith that God Himself puts into the heart. 21

More information

ALGEBRA. COPYRIGHT 1996 Mark Twain Media, Inc. ISBN Printing No EB

ALGEBRA. COPYRIGHT 1996 Mark Twain Media, Inc. ISBN Printing No EB ALGEBRA By Don Blattner and Myrl Shireman COPYRIGHT 1996 Mark Twain Media, Inc. ISBN 978-1-58037-826-0 Printing No. 1874-EB Mark Twain Media, Inc., Publishers Distributed by Carson-Dellosa Publishing Company,

More information