The Mathematics of Everyday Life

by Alfred S. Posamentier, Christian Spreitzer

Overview

Two experienced math educators help the average reader discover not only the everyday usefulness of math but the fun that comes from mastering the basics of arithmetic, algebra, geometry, and more. If you think of mathematics as a series of pointless classroom exercises without much relevance to real life, this book will change your mind. As the authors show, math is deeply embedded in almost every aspect of daily life--from managing your personal finances, making consumer purchases, and sharpening your computational skills, to learning to apply mathematical concepts that will give you a better grasp of both ordinary and extraordinary events and help you better appreciate the world we live in. With some basic geometry under your belt, you'll discover that there is an optimal point on a soccer field from which to shoot a goal. And you'll be more clever with the gears of a bike. If you like to play cards or go to the casino, knowing something about probability will give you an edge. You'll also have an enhanced understanding of the "whispering effect" inside the Capitol rotunda, why a car's headlights are so bright, and even why sewer covers are round. After reading this entertaining and instructive book, you'll come away with a whole new awareness of how elegantly mathematics explains everyday experiences and observations--from present day items to classical art and architecture.

Product Details

ISBN-13:	9781633883888
Publisher:	Rowman & Littlefield Publishers, Inc.
Publication date:	08/21/2018
Sold by:	Barnes & Noble
Format:	eBook
Pages:	424
File size:	13 MB
Note:	This product may take a few minutes to download.

About the Author

Alfred S. Posamentier has published over sixty books in the area of mathematics and mathematics education, including, most recently, The Joy of Mathematics (with with Robert Geretschlaeger, Charles Li, and Christian Spreitzer) and The Circle: A Mathematical Exploration beyond the Line (with Robert Geretschlaeger). After having been on the faculty of the City College of the City University of New York for forty years, where he was a professor of mathematics education and dean of the School of Education, he subsequently held the same positions at Mercy College, New York, for five years. He was also executive director for internationalization and sponsored programs at Long Island University, New York.

Christian Spreitzer is a coauthor of The Joy of Mathematics (with Alfred Posamentier, Robert Geretschlaeger, and Charles Li). He teaches mathematics and physics at the University College of Teacher Education in Lower Austria and at the University of Vienna. He was educated as a mathematician and a physicist and received his master's degree from the University of Vienna. He has published research articles in peer-reviewed journals on mathematics as well as mathematics education.

Read an Excerpt

Introduction

We hear so often that mathematics is important for us to better appreciatethe world around us. This is the book that will guide you througha wide variety of aspects of our world that are accompanied by mathematical explanations. Unfortunately, because of restricted curriculum guidelines, your teachers and school probably did not have the time to navigate the many excursions that we will be presenting here. Had they been able to enrich mathematics instruction through these many applications, your math instruction would have been much more enjoyable. Teachers are strictly directed by curriculum guidelines to cover topics considered essential foundation blocks for further study in mathematics as well as the other STEM fields. This is compounded even more so in many schools where teachers are rated by the scores that their students achieve on standardized tests. Hence, there is a lot of “teaching to the test” and very little beyond that. In this book, we plan to dispel the very popular notion that mathematics is tedious and boring. We will be presenting a host of topics and ideas you are unlikely to have encountered during your school instruction, and they will give you a true feeling of how mathematics is a part of our everyday lives and how it can be used to explain many of the concepts we experience, the things we see, the decisions we make, and the overall understanding we have of the world around us.

Most people experience mathematics in a formal sense during their school years. All too often, mathematics is presented as a collection of mechanical techniques that supposedly allow the learner to apply these methods and concepts to a variety of encounters in everyday life. But the lifestyles and interests of the general populace vary broadly. Compound that with the lack of applications in the curriculum, and the average person is left with the feeling that he or she learns mathematics because it had to be learned and not because it is useful. In this book, we hope to convince you not only that mathematics is useful, but also that it can help us explain many things in everyday life that we tend to take for granted—and in unexpected ways.

There are people who feel that they can explain the world through mathematics. Oftentimes there may be a bit of a “stretch” in such explanations. Yet there are numerous parts of our experiential world that are based on mathematical concepts and principles. These can be appreciated in a variety of forms of art, architecture, nature, and finance; further, the daily aspects of our culture and lifestyles have mathematical explanations, or even mathematical origins. Most people are unaware as they confront these things that simple mathematics is being applied or can be used to better explain what is being observed.

As mentioned above, mathematical applications can be found in art and architecture, where the proportions of pictures, their subjects, and the nature of the structures themselves can be explained with well-established mathematical relationships. The golden ratio, which shows its tested beauty on the dimensions of the golden rectangle, is used throughout art and architecture. The structures that have for centuries become icons in our society—such as famous cathedrals, the Parthenon, the pyramids in Egypt—and other notable structures, even those built in modern times—such as the Headquarters of the United Nations in New York—all exhibit the beautiful golden ratio.

Additionally, for paintings to convey proper depth perception, lines of perspectivity also must come into play; we can see this in Leonardo da Vinci’s famous The Last Supper mural in Milan, Italy, for example. Through their drawings and paintings, well-known artists have demonstrated their awareness of mathematical concepts. Notably among these is the famous German artist Albrecht Dürer, who studied Leonardo’s techniques and used them in some of his drawings. He also showed a keen awareness of mathematics in his 1514 etching Melencolia I, in which he included an incredibly rich magic square that has many prop-erties beyond those of normal magic squares. We hope to give you the tools you need to appreciate art and architecture from a mathematical standpoint, as well as an aesthetic one.

Perhaps the most ubiquitous numbers in mathematics are the Fibonacci numbers, which continuously arise in just about every aspect of our lives. For example, in nature the numbers of spirals on pine cones and pineapples are always Fibonacci numbers, as is the arrangement of the branches on a pear tree. You’ll have to read on to learn about even more amazing appearances and applications of the Fibonacci numbers. (By the way, the Fibonacci numbers can also generate the golden ratio!)

Beyond art and the natural world, there are also unexpected, curious mathematical explanations for understanding excellence in sports. For example, you can use very simple high-school geometry to determine the optimal point along the sideline of a soccer field from which to shoot a goal. You can also use simple geometry in billiards to easily determine the point at which to hit a ball with the cue so that it hits a cushion and ricochets onto another cushion or ball. You might not think about this when playing a game of billiards for fun with your friends, but this is just another example of how math is all around you.

As you know, we all do lots of calculating and estimating in the course of our everyday lives. But you will be surprised, and perhaps entertained, by the many shortcuts and unusual relationships that can be used to make these tasks almost trivial, such as when converting miles to and from kilometers. We will present a variety of useful shopping shortcuts, investment insights, even how best to wrap a present!

Mathematics can help you navigate the globe and even appreciate and understand rainbows and the other curves that we encounter. For instance, when you travel along a road with timed traffic lights, mathematics can explain how this is done. Have you ever wondered why all sewer covers are round? That, too, will be explained. There are curves that enable us to have whispering galleries, and curves that hold up bridges. All of these are special properties that can be explained very easily through elementary mathematics.

The field of probability allows us insight into some unusual aspects of reality, too. It is clearly to a gambler’s advantage to understand concepts of probability, for oftentimes a correct assessment of a wagering situation can be quite counterintuitive. There are game shows on television, most notably Let’s Make a Deal, that have been a hot topic of controversy regarding how to determine the best strategy to win the game. Probability can also affect your worldview, particularly when you are reading a newspaper and journalists enthusiastically offer statistical evidence to support a position; at this point, knowledge of probability concepts can be helpful not only to understand the presented material but also to criticize it intelligently. We hope to enlighten you in this regard. We will also provide some curious insights into the card game of poker.

As you will see, mathematical problem-solving strategies are often used in everyday life. For example, using extremes to solve certain problems can be very effective not only in mathematics, but also with issues we face regularly. When confronted with a decision to be made, we say to ourselves, “Well, in the worst-case scenario, such and such would be the case.” This allows us to move ahead with a sensible procedure for dealing with the situation at hand. Mathematical problem-solving strategies can guide us in the way we think about common, everyday decisions to be made.

These are just a few of the plethora of mathematical applications in our everyday lives. Often, we are not even aware that mathematics can explain and facilitate an understanding of what we see and how we can best deal with these situations. The mathematical concepts we consider in this book will require nothing more than a recollection of what you had been taught up to the tenth grade in high school. Join us now as we begin our journey through an investigation of a wealth of topics that either depend on mathematics or can use mathematics to explain their functioning, or, perhaps, even allow us to appreciate the world around us in an enhanced, sophisticated fashion because now many things we may have just accepted without question will become more meaningful.

Table of Contents

Introduction 11

Chapter 1 Historical High Points In The Development of Mathematical Applications 15

The Origin of Our Number Symbols 15

The Most Important Number in Mathematics 17

The Famous Fibonacci Numbers 20

Arithmetic in Ancient Egypt 21

Where the Terms Related to Our Clock Evolve 28

A Minute History of Timekeeping 28

Babylonian Mathematics and the Sexagesimal System 29

Babylonian Minutes and Seconds Have Survived to This Day 33

Roman Numerals Are Everywhere around Us 34

Mathematics on the Calendar 39

How We Overlook Our Calendar 48

Chapter 2 Mathematics In Our Everyday Lives-Arithmetic Shortcuts and Thinking Mathematically 53

Arithmetic with the Numbers 9 and 11 54

How 9s Can Check Your Arithmetic 60

Rules for Divisibility 62

A Quick Method to Multiply by Factors of Powers of 10 68

Arithmetic with Numbers of Terminal Digit 5 69

Multiplying Two-Digit Numbers Less Than 20 71

Mental Arithmetic Can Be More Challenging-but Useful! 73

Arithmetic with Logical Thinking 74

Using the Fibonacci Numbers to Convert Kilometers to and from Miles 75

Thinking "Outside the Box" 79

Solving Problems by Considering Extremes 82

The Working-Backward Strategy in Problem Solving 86

Chapter 3 Mathematical Appearances and Applications In Everyday-Life Problems 93

Shopping with Mathematical Support 93

Successive Percentages 94

Raising Interest! 100

The Rule of 72 104

Paper Sizes and the Root of All ISO 106

Comparing Areas and Perimeters 110

Mathematics in Home Construction 113

The Perfect Manhole Cover 117

Design Your Own Coffee-Cup Sleeve! 124

How to Optimally Wrap a Present 128

Chapter 4 Probability, Games, and Gambling 135

Friday the Thirteenth! 135

Unexpected Birthday Matches 137

Selecting Clothes 141

Playing Cards, a Counterintuitive Probability 142

Mathematics in Poker 144

Mathematical Logic of Tic-Tac-Toe 150

The Monty Hall Problem 154

Business Applications 158

Mathematics of Life Insurance 163

The Most Misunderstood Average 168

What We Need to Know about Averages 170

Comparing Measures of Central Tendency 172

Chapter 5 Sports and Games-Explained Mathematically 181

The Best Angle to Throw a Ball 181

Optimizing Your Shot at Soccer 187

A Game of Angles 192

Playing Billiards Cleverly 201

Mathematics on a Bicycle 206

The Spirograph Toy 212

Chapter 6 The World and Its Nature 225

Measures of and on the Earth 225

Navigating the Globe 229

What Is Relativity? 232

Coloring a Map 233

Crossing Bridges 237

Mathematics in Nature 243

The Male Bee's Family Tree 244

Fibonacci Numbers in the Plant World 246

The Pine Cone and Others 247

Leaf Arrangement-Phyllotaxis 251

The Fibonacci Numbers on the Human Body 255

The Geometry of Rainbows 258

Chapter 7 Appearances of Mathematics In Art and Architecture 275

Golden Ratio Sightings 276

Displaying a Watch 281

Applications in Art 284

Perspectivity in Art 296

Numbers in Art 305

Viewing a Statue Optimally 309

The Most Overlooked Curve 312

The One-Sided Belt-the Möbius Strip 315

Chapter 8 The Technology Around Us-From A Mathematical Perspective 319

A Fascination with the Clock 319

The Mathematics of Paper Folding 322

Building a Skewed Tower 328

Whispering Galleries 334

Looking inside a Flashlight 345

Coffee with Caustics 351

Green Traffic Lights All the Way 356

Safety in Numbers 365

The ISBN System 376

How the Global Positioning System (GPS) Works 381

Acknowledgments 391

Notes 393

Index 401

Editorial Reviews

"A delightfully entertaining, instructive, and useful book—one that reminds us that mathematics can be both stimulating and extremely handy. From understanding designs in the natural world to the reason manhole covers are round; from probability in poker to the perplexing Monty Hall Problem; and from the geometry of the Spirograph to the geometrical aesthetics of Raphael, Da Vinci, and Albrecht Dürer, The Mathematics of Everyday Life unveils seeming mysteries while teaching us shortcuts to problem solving and how to see things from a new perspective—in sum, how to think logically and mathematically.”—Scott Krawczyk, PhD, dean, Richard L. Conolly College of Liberal Arts and Sciences, LIU Brooklyn“Spectacular! Once again, Alfred S. Posamentier and Christian Spreitzer have reached into their rich reservoir of numerical knowledge to explain the beauty and eloquence of mathematics. They take us on a vigorous voyage, helping us to understand how math can help us solve problems of everyday life. Even those of us who are ‘not good at math' will appreciate how important the basic principles they describe so well are to our daily experiences.”—Allen I. Hyman, MD, FCCM, Professor Emeritus of Anesthesiology, Columbia University in the City of New York“This book is fun! Even for those of us whose study of mathematics was some time ago, the examples and puzzles are easy to understand. We are offered really useful ways of using math in shopping (what is a genuine discount?), converting miles to kilometers (useful when traveling abroad), and completing a host of other everyday tasks. I particularly liked the authors' insistence that using our own ‘built-in computer'—our brain—is more fun than using our phone or other devices for everyday calculations. I hope many people read the book and agree with that advice.”—Baroness Pauline Perry, formerly Her Majesty's Chief Inspector of Education“Almost everything is math! That's what Posamentier and Spreitzer would like to reveal to us. Wherever you look—in paintings, sports, nature, shopping, or gambling—all of it works with math.Surprising, astonishing, entertaining...This new book tells you all about your unknown life with math. Whether or not you like math, you will like this book.” —Gerhard Ackermann, PhD, professor and retired president, Beuth University of Applied Sciences, Berlin

From the Publisher