Whenever the topic of mathematics is mentioned, people tend to indicate their weakness in the subject as a result of not having enjoyed its instruction during their school experience. Many students unfortunately do not have very positive experiences when learning mathematics, which can result from teachers who have a tendency "to teach to the test". This is truly unfortunate for several reasons. First, basic algebra and geometry, which are taken by almost all students, are not difficult subjects, and all students should be able to master them with the proper motivational instruction. Second, we live in a technical age, and being comfortable with basic mathematics can certainly help you deal with life's daily challenges. Other, less tangible reasons, are the pleasure one can experience from understanding the many intricacies of mathematics and its relation to the real world, experiencing the satisfaction of solving a mathematical problem, and discovering the intrinsic beauty and historical development of many mathematical expressions and relationships. These are some of the experiences that this book is designed to deliver to the reader. The book offers 101 mathematical gems, some of which may require a modicum of high school mathematics and others, just a desire to carefully apply oneself to the ideas. Many folks have spent years encountering mathematical terms, symbols, relationships and other esoteric expressions. Their origins and their meanings may never have been revealed, such as the symbols +, -, =, π. ꝏ, √, ∑, and many others. This book provides a delightful insight into the origin of mathematical symbols and popular theorems such as the Pythagorean Theorem and the Fibonacci Sequence, common mathematical mistakes and curiosities, intriguing number relationships, and some of the different mathematical procedures in various countries. The book uses a historical and cultural approach to the topics, which enhances the subject matter and greatly adds to its appeal. The mathematical material can, therefore, be more fully appreciated and understood by anyone who has a curiosity and interest in mathematics, especially if in their past experience they were expected to simply accept ideas and concepts without a clear understanding of their origins and meaning. It is hoped that this will cast a new and positive picture of mathematics and provide a more favorable impression of this most important subject and be a different experience than what many may have previously encountered. It is also our wish that some of the fascination and beauty of mathematics shines through in these presentations.
Author(s): Arthur D Kramer, Alfred S Posamentier
Series: Problem Solving In Mathematics And Beyond, 27
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 402
City: Singapore
Contents
About the Authors
Introduction
Chapter 1 All About Numbers and Symbols
UNIT 1 Some Ancient Number Systems
UNIT 2 The Origin of Our Numerals
UNIT 3 The History of Zero
UNIT 4 The Origin of the Equal Sign
UNIT 5 The Birth of Multiplication and its Symbol
UNIT 6 The Origin of the Division Symbols
UNIT 7 The Origins of Fractions
UNIT 8 The Emergence of Exponent Notation
UNIT 9 The Origin of the Square-Root Symbol
UNIT 10 The Origin of Percent %, Number #, and Dollar $ Symbols
Percent Symbol %
Number Symbol #
Dollar Sign $
UNIT 11 The Ubiquitous Number π
The Area of a Circle Formula
The Value of π
UNIT 12 One of the Earliest Approximations of π
UNIT 13 The Origin of the Basic Trigonometric Names: Sine, Tangent, and Secant
UNIT 14 How Numbers are Categorized
UNIT 15 Imaginary Numbers are Not Really Imaginary
UNIT 16 The Birth of Algebra
UNIT 17 The Infinity Concept and Its Symbol ∞
UNIT 18 The “Quad” in Quadratic
UNIT 19 Prime Numbers: Background and Properties
UNIT 20 Perfect Numbers
UNIT 21 Triangular Numbers and Their Relation to Square Numbers
UNIT 22 Numerology
UNIT 23 Very Small and Very Large Numbers
UNIT 24 The Googol is Much Older Than Google
UNIT 25 Armstrong or Narcissistic Numbers
UNIT 26 Friendly Numbers or Amicable Numbers
UNIT 27 Happy and Unhappy Numbers
UNIT 28 Repunits, Numbers Consisting Only of 1’s
UNIT 29 Palindromic Numbers
UNIT 30 The Ubiquitous Number 1089
UNIT 31 Kaprekar Numbers
The Kaprekar Constant
UNIT 32 The Transcendency of Transcendental Numbers
Chapter 2 Arithmetic Curiosities
UNIT 33 How the Romans did Multiplication
UNIT 34 How Complex Calculations were Done Before Electronics
UNIT 35 American and European Subtraction Algorithms
UNIT 36 Extracting a Square Root
UNIT 37 Divisibility by Prime Numbers
Divisibility by 3 (or 9)
Divisibility by 7
Divisibility by 11
Divisibility by 13
Divisibility by 17
UNIT 38 Successive Percentages
UNIT 39 Casting Out Nines
UNIT 40 Division by Zero
UNIT 41 Mistaken Assumptions
UNIT 42 The Ulam–Collatz Loop
UNIT 43 A Cyclic Number Loop
UNIT 44 Curious Number Properties
Curious numbers with unit-digit 9
The Curious Number 11
The Curious Number 110
The Sum of Squares Surprise
Squares That Can Be Partitioned into Squares
Splitting Numbers
Combining Powers
Powers of Digits
Five Unique Numbers
More Number Curiosities
The Devil Number
Number coincidences
UNIT 45 Magic Squares
Chapter 3 Aspects of Measurement
UNIT 46 The Origin of Measurement
UNIT 47 The Metric System Versus the U.S. Customary System
UNIT 48 The Origins of Today’s Calendar
UNIT 49 How Eratosthenes Measured the Earth
UNIT 50 The Origin of the Cartesian Plane
UNIT 51 Why the Normal Bell-Shaped Curve is Called Normal
UNIT 52 Measures of Central Tendency
The Arithmetic Mean
The Geometric Mean
The Harmonic Mean
UNIT 53 Comparing The Three Means Algebraically
UNIT 54 Comparing the Three Means Geometrically
UNIT 55 The Rule of 72
Chapter 4 Geometric Novelties
UNIT 56 The Platonic Solids
Unit 57 Euler’s Theorem Revisited
UNIT 58 The Beginnings of Geometry
UNIT 59 The Origin of the American Secondary School Geometry Course
UNIT 60 The Pythagorean Theorem and Its Origins
Egypt
Mesopotamia
India
China
UNIT 61 President James A. Garfield’s Contribution to Mathematics
UNIT 62 The Golden Ratio — and the Golden Rectangle and Triangle
UNIT 63 The Center of Gravity of Triangles and Quadrilaterals
UNIT 64 Napoleon’s Geometry Theorem
UNIT 65 Simson’s Line
UNIT 66 The Nine-Point Circle
UNIT 67 A Famous Problem of Antiquity — Angle Trisection
UNIT 68 The Ever-Present Equilateral Triangle
UNIT 69 The Four-Color Map Problem
UNIT 70 Optical Illusions
UNIT 71 The World of Non-Euclidean Geometries
UNIT 72 Introduction to Spherical Geometry
UNIT 73 Cartography is Mathematically Challenging
UNIT 74 A Look at Fractals
Chapter 5 Probability
UNIT 75 The History of Probability
UNIT 76 The Famous Birthday Problem
UNIT 77 The Remarkable Probability in the Game of Craps
UNIT 78 Some Statements are Neither True nor False
UNIT 79 The Pigeonhole Principle
UNIT 80 The Heads and Tails Conundrum
UNIT 81 The Monty Hall Problem
UNIT 82 The False Positive Paradox
UNIT 83 Fair Division
Chapter 6 A Potpourri of Mathematical Topics
UNIT 84 Logical Thinking
UNIT 85 More Mathematical Conundrums
UNIT 86 Mistakes in Mathematics
Geometric Mistake
Algebraic Mistake
Geometric Mistake
Arithmetic Mistake
UNIT 87 The Hands of an Analog Clock
UNIT 88 Pythagorean Triples
UNIT 89 Fermat’s Last Theorem
UNIT 90 Pure Mathematics and Prime Numbers
UNIT 91 The Fibonacci Numbers
UNIT 92 Diophantine Equations
UNIT 93 The Goldbach Conjecture
UNIT 94 Triskaidekaphobia
UNIT 95 The Parabola — A Very Special Curve
UNIT 96 Kepler’s Three Famous Laws
Kepler’s First Law
Kepler’s Second Law
Kepler’s Third Law
UNIT 97 Beyond the Quadratic Equation
UNIT 98 Évariste Galois — A Short Brilliant Life
UNIT 99 A Look at Einstein’s Special Relativity
UNIT 100 A Clear Look at Calculus
Differential Calculus
Integral Calculus
Unit 101 Euler’s Remarkable Relationship
Index
