The Joy of Mathematics: Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class

241 Pages·2017·5.35 MB·English

by  Alfred S. Posamentier et al.

ALSO BY ALFRED S. POSAMENTIER AND ROBERT GERETSCHLÄGER
The Circle
ALSO BY ALFRED S. POSAMENTIER AND INGMAR LEHMANN
The Fabulous Fibonacci Numbers Pi: A Biography of the World's Most
Mysterious Number Mathematical Curiosities
Magnificent Mistakes in Mathematics The Secrets of Triangles
Mathematical Amazements and Surprises The Glorious Golden Ratio
ALSO BY ALFRED S. POSAMENTIER AND BERND THALLER
Numbers
ALSO BY ALFRED S. POSAMENTIER
The Pythagorean Theorem
Math Charmers

Published 2017 by Prometheus Books
The Joy of Mathematics: Marvels, Novelties, and Neglected Gems That Are Rarely Taught in Math Class.
Copyright © 2017 by Alfred S. Posamentier, Robert Geretschläger, Charles Li, and Christian Spreitzer. All
rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in
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conveyed via the Internet or a website without prior written permission of the publisher, except in the case
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Cover design by Nicole Sommer-Lecht
Cover design © Prometheus Books
Unless otherwise indicated, all interior images are by the authors or contributors.
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Library of Congress Cataloging-in-Publication Data Names: Posamentier, Alfred S. | Geretschläger, Robert
| Li, Charles, 1981- | Spreitzer, Christian, 1979-.
Title: The joy of mathematics : marvels, novelties, and neglected gems that are rarely taught in math class /
by Alfred S. Posamentier, Robert Geretschläger, Charles Li, and Christian Spreitzer.
Description: Amherst, New York : Prometheus Books, 2017. | Includes bibliographical references and
index.
Identifiers: LCCN 2017008896 (print) | LCCN 2017019495 (ebook) | ISBN 9781633882980 (ebook) |
ISBN 9781633882973 (pbk.) Subjects: LCSH: Mathematics—Popular works.
Classification: LCC QA93 (ebook) | LCC QA93 .P6674 2017 (print) | DDC 510—dc23
LC record available at https://lccn.loc.gov/2017008896
Printed in the United States of America
INTRODUCTION
CHAPTER 1: ARITHMETIC NOVELTIES
When Is a Number Divisible by 3 or 9?
When Is a Number Divisible by 11?
Divisibility by Prime Numbers
Squaring Numbers Quickly
Squares and Sums
Using Squares to Multiply Arbitrary Numbers
An Alternative Method for Extracting a Square Root
Sensible Number Comparisons
Euclidean Algorithm to Find the GCD
Sums of Positive Integers
Sums of Odd Positive Integers
The Realm of Nonterminating Decimals
Atoms in the Universe of Numbers
Fun with Number Relationships
Friendly Numbers
Palindromic Numbers
Prime Numbers
Infinite Primes
The Neglected Triangular Numbers
Perfect Numbers
Making Mistaken Generalizations
The Fibonacci Numbers
CHAPTER 2: ALGEBRAIC EXPLANATIONS OF ACCEPTED CONCEPTS
Simple Algebra Helps Logical Reasoning
Division by Zero
Irrationality of the Square Root of 2
Bisection Method to Approximate Square Roots
Continued Fractions of Square Roots
Fermat's Method of Factoring
Comparing Means
Diophantine Equations
Falling Squares
Descartes's Rule of Signs
Horner's Method to Evaluate Polynomials
Generating Pythagorean Triples
The Frobenius Problem
CHAPTER 3: GEOMETRIC CURIOSITIES
Parallelograms and Triangles
Using a Grid to Calculate Areas
The Center of a Quadrilateral
Beyond the Formula for the Area of a Triangle
Heronian Triangles
A New Formula for the Area of Isosceles Triangles
Pick's Theorem
When Intersecting Lines Meet a Circle
Origins of Trigonometry
Sines of Small Angles
An Unconventional View of the Sine
Surprising Proofs of the Pythagorean Theorem
Beyond the Pythagorean Theorem—Part I
Beyond the Pythagorean Theorem—Part II
Beyond the Pythagorean Theorem—Part III
The Pythagorean Theorem Extended to Three Dimensions
Polyhedra: Sides, Faces, and Vertices
Lunes and the Right Triangle
Concurrency
Similarity and the Golden Ratio
A Relation between Points and Circles
Constructions with Compasses Alone
The Sphere and the Cylinder
Regular Polygons and Stars
Platonic Solids and Star Polyhedra
CHAPTER 4: PROBABILITY APPLIED TO EVERYDAY EXPERIENCES
How the Theory of Probability Began
Benford's Law
The Birthday Phenomenon
The Monty Hall Problem
Bertrand's Box
The False Positive Paradox
Pascal's Triangle
Random Walks
The Poker Wild-Card Paradox
CHAPTER 5: COMMON SENSE FROM A MATHEMATICAL
PERSPECTIVE
The Origins of Some Mathematics Symbols
The Counterintuitive
A Surprising Solution
Don't “Wine” over This Problem—A Problem-Solving Approach
Organized Thinking
Successive Percentages
Rule of 72
A Mathematical Conjecture
Unexpected Patterns
An Infinity Conundrum
The Concept of Infinity
Counting the Uncountable
Mathematics on a Bicycle
The Parabola: A Remarkable Curve
ACKNOWLEDGMENTS
APPENDIX
NOTES
INDEX
For many decades the mathematics curriculum has been stocked with lots of
essential building blocks to enable the student to navigate properly such
disciplines as science, finance, engineering, architecture, and everyday life—just
to name a few! With much to cover and the stress of moving ahead at a steady
clip, there are many interesting and important mathematics concepts, topics, and
applications that rarely ever get mentioned in the classroom.
When students are presented with finance applications in the school
curriculum, such as computing the effect of interest on a given principal in the
bank, there is a missed chance to enrich them with a mathematical peculiarity
such as the “Rule of 72,” which enables one to determine how much time is
required for a bank account to double the deposited money at a specified interest
rate.
There are amazingly simple geometric phenomena that are rarely ever shown
in classrooms, simply for lack of time. These might include some of the special
yet straightforward characteristics of quadrilaterals inscribed in a circle, such as
the incredible relationship between the diagonals of a quadrilateral and its sides,
where the product of the diagonals is equal to the sum of the products of the
opposite sides. Remember, this is only true when the quadrilateral's four vertices
lie on the same circle.
Another opportunity missed is to show how the random placement of a point
inside an equilateral triangle shares a common property with any other point in
that triangle, namely that the sum of the perpendiculars to each of the three sides
is always the same.
There are everyday applications that regularly seem to escape classroom
presentations. For example, mechanical algorithms to multiply numbers
mentally. Even though there are electronic calculators readily available, the
facility of manipulating numbers mentally is clearly an asset that seems to be
seen as relatively unimportant in today's technological world. Here we choose to