Math Bytes

Math Bytes: Google Bombs, Chocolate-Covered Pi, and Other Cool Bits in Computing

Copyright Date: 2014
Pages: 152
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  • Book Info
    Math Bytes
    Book Description:

    This book provides a fun, hands-on approach to learning how mathematics and computing relate to the world around us and help us to better understand it. How can reposting on Twitter kill a movie's opening weekend? How can you use mathematics to find your celebrity look-alike? What is Homer Simpson's method for disproving Fermat's Last Theorem? Each topic in this refreshingly inviting book illustrates a famous mathematical algorithm or result--such as Google's PageRank and the traveling salesman problem--and the applications grow more challenging as you progress through the chapters. But don't worry, helpful solutions are provided each step of the way.

    Math Bytesshows you how to do calculus using a bag of chocolate chips, and how to prove the Euler characteristic simply by doodling. Generously illustrated in color throughout, this lively and entertaining book also explains how to create fractal landscapes with a roll of the dice, pick a competitive bracket for March Madness, decipher the math that makes it possible to resize a computer font or launch an Angry Bird--and much, much more. All of the applications are presented in an accessible and engaging way, enabling beginners and advanced readers alike to learn and explore at their own pace--a bit and a byte at a time.

    eISBN: 978-1-4008-5142-3
    Subjects: Mathematics, Technology

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
    (pp. ix-xiv)
  4. 1 Your First Byte
    (pp. 1-4)

    Mathematics helps create the landscapes on distant planets in the movies. The web pages listed by a search engine are possible through mathematical computation. The fonts we use in word processors result from graphs of functions. These applications use a computer although each is possible to perform by hand, at least theoretically. Modern computers have allowed applications of math to become a seamless part of everyday life.

    Before the advent of the digital computer, the only computer we had to rely on was a person’s mind. Few people possessed the skill to perform complex computations quickly. Johan Zacharias Dase (1824–...

  5. 2 Deceiving Arithmetic
    (pp. 5-10)

    Addition and subtraction can seem easy enough. In fact, that is largely what a computer does, and a computer only adds 1s and 0s. Yet, even this most foundational mathematical operation can be tricky and require attention.

    In the 7th season ofThe Simpsons, Homer has a nightmare in which he travels into 3D space. In this episode entitled “Treehouse of Horror VI” as Homer struggles with this newfound reality, an equation flies past 3D Homer that reads:\[{{1782}^{12}}+{{1841}^{12}}={{1972}^{12}}.\]

    Simple enough, unless you happen to remember some fateful words written in 1637.

    It may have been a dark and dreary...

  6. 3 Two by Two
    (pp. 11-20)

    Suppose you could take a million dollars today or be given one cent today and twice as much as the day before for the next 30 days. So, if you took the second option, you’d take home a penny today, two pennies tomorrow and so forth. Which would you choose? The more lucrative answer will give insight on how a movie might bomb soon into its opening weekend.

    Let’s step back into Roman times. General Terentius returns home after one of his many victorious battles. Soon after entering the city, he requests an audience with the emperor, and he is...

  7. 4 Infinite Detail
    (pp. 21-31)

    Let’s start with a game.

    1. Place a dot halfway between square 1 and 2.

    2. Roll a die and place a new dot halfway between your last dot and:

    the middle of square 1 if you roll 1 or 2

    the middle of square 2 if you roll 3 or 4

    the middle of square 3 if you roll 5 or 6

    3. Return to Step 2.

    Play for a while! What shape emerges?

    The image that would emerge from the game that started this chapter, if you played the game on the previous page long enough, is the image in Figure...

  8. 5 Plot the Course
    (pp. 32-41)

    In this chapter, we will see how functions likey= 3x+ 1 andy= 5−x2enable a computer to create a font like the one that comprises these words or plot the path of an Angry Bird through the air.

    In days of old, calligraphers sat carefully lettering books with their artful creations. These masterpieces of script appear from cultures such as Indian, Tibetan, Persian, Islamic, Chinese, Japanese, and Western.

    Let’s learn the fundamental ideas behind creating fonts on a computer. Later, we will see a font that’s a puzzle and could even be used as a...

  9. 6 Doodling into a Labyrinth
    (pp. 42-53)

    In this chapter, we begin with doodling as inspired by [8] and, by the end, use a math theorem to create a maze.

    We begin with doodling. Place a pencil on a piece of paper and maybe even close your eyes. Create your doodle by drawing in big sweeping motions, keeping the pencil on the paper at all times. Remember that the longer you doodle the harder the exercise will be. On the left in Figure 6.1 is my doodle.

    Let the math begin by placing a dot at every crossing point in your doodle. Also place a dot at...

  10. 7 Obama-cize Yourself
    (pp. 54-60)

    During the 2008 presidential election, a poster (see Figure 7.1) containing a stylized image of Barack Obama, designed by Los Angeles street artist Shepard Fairey, appeared with the words “hope” and “progress.” It became synonymous with the campaign. In January 2009 as Obama became president, the Smithsonian Institution acquired Fairey’s mixed media collage for the National Portrait Gallery.

    Let’s mathematically transform a digital image into a stylized portrait similar to Fairey’s. We must first understand a storage scheme for color images. A picture is a collection of dots, called pixels. Each pixel is a combination of red, green and blue...

  11. 8 Painting with M&Ms
    (pp. 61-72)

    Suppose you bake a sheet cake, slather the top with icing, and decide to ornament the top with M&Ms. In this chapter, we will learn how the sugary surface of your cake can be a workspace to perform some calculus or calculate an estimate to the value ofπ. Want a less mathcentric cake decoration? By the end, we will learn how to create a portrait of friends, family or, as we will see, presidents with M&Ms.

    Let’s start with a simple chocolatey problem that will open a door to ideas of calculus. A Hershey’s chocolate bar, as seen in...

  12. 9 Distorting Reality
    (pp. 73-85)

    Computers allow for easy methods of image manipulation. From distorting an image to enhancing an image, one can change a picture with the click of a mouse. Let’s start with a repeated image that becomes a bit of magic wrapped in a puzzle.

    Counting to 15 is easy enough, unless you’re counting leprechauns drawn by Pat Lyons. Below verify there are 15 of the Irish fairies.

    If you cut out the puzzle along the straight black lines and interchange the left and right pieces on the top row, then you get the following configuration. How many leprechauns do you count...

  13. 10 A Pretty Mathematical Face
    (pp. 86-97)

    How would you describe your facial appearance? Granted, it may depend how long ago you awoke and if you have had your morning shower or coffee. Still, would you consider yourself a cross between Tom Cruise, Audrey Hepburn, and Clark Gable, or Robert Downey Jr. and Ellen DeGeneres? Let’s see how mathematics can help answer such questions.

    We’ll work with a library of grayscale images of the 16 famous people seen in Figure 10.1. Our goal will be to find the combination of these pictures that best approximates a target image.

    Recall from earlier in the book that the grayscale...

  14. 11 March MATHness
    (pp. 98-104)

    The madness sweeps the country, infecting millions. Be careful. It happens every March and breaks out in many offices. We are talking March Madness, which is the Division I NCAA Men’s basketball tournament. Whether you participate or not, many people do. In 2013, the ESPN online pool alone pitted over 8 million brackets against each other.

    If you’re immune, here’s a quick tutorial. A March Madness bracket starts with match-ups of 64 teams in 32 games. For instance, Figure 11.1 is one quarter of a bracket in 2013. As you see, UNLV played California and Syracuse played Montana in the...

  15. 12 Ranking a Googol of Bits
    (pp. 105-123)

    Today, a natural place to turn to answer one’s questions is the Internet. We might turn to Google, for instance, to search on the meaning of the word googol, which is the number 10100after which Google was named, reflecting the company’s goal of organizing all information on the World Wide Web (WWW). How do search engines rank web pages? What are some of the math and computing issues involved in returning web pages in an order deemed relevant to our query? You could search the internet for an answer, or keep reading.

    Suppose that we submit the query computing...

  16. 13 A Byte to Go
    (pp. 124-124)

    We’ve bitten off a lot of math. Are you creating a candy mosaic to top a cake? Maybe you’re collecting photos to illuminate who you look like. Or, possibly you’re admiring the results of the mathematics involved in the results to your queries in search engines. Whatever the case may be, we have digested a lot of math through the preceding pages of this book. As your mind is mathematically nourished, it may get energized with new ideas! Applying and adapting concepts is an important part of mathematics and computing.

    It is now your turn. What ideas come to mind?...

  17. 14 Up to the Challenge
    (pp. 125-130)

    Challenge questions have appeared throughout the book. Below are answers to these questions although it is entirely possible that you devised another approach.

    2.1. Note that if 3 divides the sum of the digits that comprise a number, then 3 divides that number. Now, 3+9+8+7 = 27, which is divisible by 3. So, 3987 is divisible by 3. Since 4+3+6+5 = 18, which is divisible by 3,4365 is also divisible by 3. This relationship works the other way, too. That is, a number is divisible by 3 if 3 divides the sum of the digits that comprise that number. So,...

    (pp. 131-132)
  19. INDEX
    (pp. 133-134)
    (pp. 135-136)