Excerpts

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. Excerpted from The Mathematics of Everyday Life by Alfred S. Posamentier, Christian Spreitzer All rights reserved by the original copyright owners. Excerpts are provided for display purposes only and may not be reproduced, reprinted or distributed without the written permission of the publisher.