Beautiful, Simple, Exact, Crazy: Mathematics in the Real World - Softcover

Khare, Dr. Apoorva; Lachowska, Prof. Anna

Softcover

ISBN 10: 0300190891 ISBN 13: 9780300190892

Verlag: Yale University Press, 2015

Inhaltsangabe

In this vibrant work, which is ideal for both teaching and learning, Apoorva Khare and Anna Lachowska explain the mathematics essential for understanding and appreciating our quantitative world. They show with examples that mathematics is a key tool in the creation and appreciation of art, music, and literature, not just science and technology. The book covers basic mathematical topics from logarithms to statistics, but the authors eschew mundane finance and probability problems. Instead, they explain how modular arithmetic helps keep our online transactions safe, how logarithms justify the twelve-tone scale commonly used in music, and how transmissions by deep space probes are similar to knights serving as messengers for their traveling prince. Ideal for coursework in introductory mathematics and requiring no knowledge of calculus, Khare and Lachowska’s enlightening mathematics tour will appeal to a wide audience.

Über diesen Titel

Über die Autorin bzw. den Autor

Apoorva Khare is currently a research associate in mathematics and statistics at Stanford University. Anna Lachowska is a lecturer in mathematics at Yale University.

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Beautiful, Simple, Exact, Crazy: Mathematics in the Real World

By Apoorva Khare, Anna Lachowska

Yale UNIVERSITY PRESS

Copyright © 2015 Yale University
All rights reserved.

ISBN: 978-0-300-19089-2

Contents

Preface, vii, 
Acknowledgments, xv, 
1 Algebra: The art and craft of computation, 1, 
2 Velocity: On the road, 19, 
3 Acceleration: After the apple falls, 39, 
4 Irrational: The golden mean and other roots, 59, 
5 Exponents: How much would you pay for the island of Manhattan?, 79, 
6 Logarithms I: Money grows on trees, but it takes time, 97, 
7 Logarithms II: Rescaling the world, 107, 
8 e: The queen of growth and decay, 129, 
9 Finite series: Summing up your mortgage, geometrically, 149, 
10 Infinite series: Fractals and the myth of forever, 167, 
11 Estimation: What is your first guess?, 193, 
12 Modular arithmetic I: Around the clock and the calendar, 213, 
13 Modular arithmetic II: How to keep (and break) secrets, 237, 
14 Probability: Dice, coins, cards, and winning streaks, 261, 
15 Permutations and combinations: Counting your choices, 277, 
16 Bayes' law: How to win a car ... or a goat, 297, 
17 Statistics: Babe Ruth and Barry Bonds, 325, 
18 Regression: Chasing connections in big data, 341, 
The bigger story, 353, 
Solutions to odd-numbered exercises, 357, 
Practice exams, 423, 
Index, 457, 

CHAPTER 1

Algebra: The art and craft of computation

In this book, we hope to show you that a wide variety of real-world problems and applications can be tackled systematically, comprehensively, and relatively simply by using just a few mathematical formulas and techniques. In order to introduce the mathematics and then to apply it to the real world, it is essential to be able to work with mathematical expressions and quantities in a systematic manner. Thus, we first need to be comfortable with basic operations like adding or multiplying polynomials; solving equations and systems of (linear or other) equations; and choosing an optimal way to simplify an algebraic expression. Developing these techniques is the goal of this chapter.

Sometimes these techniques produce unexpected results which some of you may have seen as "magic tricks." For instance, you can check that multiplying two consecutive odd numbers (or consecutive even numbers) yields one less than a perfect square (e.g., 5 · 7 + 1 = 36 = 62, 10 · 12 + 1 = 121 = 112). Is this always the case, or can we find two consecutive odd or even numbers for which this phenomenon does not occur? Note that it is impossible – even for the biggest computer – to verify this for all integers in finite time, because there are infinitely many numbers. But, as we will see, there is a simple way to perform just one calculation – and it will do the job for every single case.

Similarly, multiplying three successive integers and adding the middle integer to this product always yields a perfect cube! Why? Once again, we will see in the exercises in this chapter how one calculation reveals the answer for all possible cases.

Thus, the purpose of this chapter is to discuss mathematical techniques which will then be used throughout the remainder of this book. We will see real-world applications of linear and quadratic equations in Chapters 2, 3, and 4.

The distributive law

We start by discussing one of the most important principles used in simplifying and manipulating mathematical expressions. As a first example, consider the following elementary computation:

2 · 39 + 2 · 61 = ?

One way to solve this equation is to explicitly calculate that 2 · 39 = 78, 2 · 61 = 122, and then to add them and obtain 200. This solution involves carrying out two multiplications (neither of which seems trivial enough to do mentally) followed by one addition. However, here is a simpler approach, which involves only one addition and then one trivial multiplication: we first add: 39 + 61 = 100, and then multiply the result by 2:

2 · 39 + 2 · 61 = 2 · (39 + 61) = 2 · 100 = 200.

Why does the first equality hold in the preceding equation? This is the principle of distributivity. It says that 2 · 39 + 2 · 61 = 2 · (39 + 61). More generally, the distributive law says that given any three real numbers a, b, c,

a · b + a · c = a · (b + c), a · b - a · c = a · (b - c).

The distributive law is at the heart of many algebraic manipulations and simplifications which will be used throughout the book. Thus it is important that we understand it – as well as its implications – thoroughly. Here are further applications of the distributive law.

Example 1.1. Compute (without using a calculator):

1. 17 · 3 + 3 · 283.

2. 3 · 3 + 3 · 17 + 20 · 17.

Solution: While the problem itself is elementary, our goal here is to see how to solve it using the distributive law, and make it easy enough to perform the computation mentally. For the first part, we compute using the distributive law:

17 · 3 + 3 · 283 = 3 · 17 + 3 · 283 = 3 · (17 + 283) = 3 · 300 = 900.

For the second part, we group terms two at a time, using the distributive law:

(3 · 3 + 3 · 17) + 20 · 17 = 3 · (3 + 17) + 20 · 17 = 3 · 20 + 20 · 17 = 20 · (3 + 17) = 20 · 20 = 400.

A more advanced version of this rule is when both factors are sums or differences. For instance, we can multiply out 101 · 201 without using a calculator, as follows:

101 · 201 = (100 + 1) · (200 + 1) = (100 + 1) · 200 + (100 + 1) · 1

= 200 · (100 + 1) + 101 = 200 · 100 + 200 · 1 + 101

= 20, 000 + 200 + 101 = 20, 301.

You may have seen a variation of this method:

101 · 201 = (100 + 1) · (200 + 1) = 100 · 200 + 100 · 1 + 1 · 200 + 1 · 1 = 20, 301.

This is sometimes called the "FOIL" method – essentially, it is simply the distributive law applied twice – once to the terms in the first factor, and once to the terms in the second.

Variables

A useful feature of the distributive law is that it applies equally well to variables. Thus if x, y are any unknown real numbers, then we once again have (for instance): 2 · (x + y) = 2x + 2y. The distributive law can also be used to "contract" expressions into more compact forms. Here is an example: the expression

(x2 - 2)(x + 2) - 3x2 + 6

looks somewhat unwieldy. However, look closely: the last two terms can be written using the distributive law as -3(x2 - 2). Now applying the distributive law again, we get

(x2 - 2)(x + 2) - 3x2 + 6 = (x2 - 2)(x + 2) + (x2 - 2) · (-3)

= (x2 - 2)(x + 2 - 3) = (x2 - 2)(x - 1),

and so just like long expressions involving numbers (as in Example 1.1), variable expressions can also be simplified.

An advantage of using variables is that once verified, a statement involving variables automatically holds for every real value assumed by these variables. For instance, consider the following "magic trick."

Example 1.2.Get your number back! Here's a magic trick: start with any number, add 2 to it, multiply the sum by 5, subtract 10 from the product, and divide the difference by 5 – and lo and behold!...

Details zur Ausgabe

• Verlag: Yale University Press
• Erscheinungsdatum: 2015
• Sprache: Englisch
• ISBN 10: 0300190891
• ISBN 13: 9780300190892
• Einband: Taschenbuch
• Auflage: 1
• Anzahl der Seiten: 480
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: preigu; mail@preigu.de; ; Lengericher Landstr. 19; Osnabrück; 49078; Deutschland