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When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible

4.6 out of 5 stars 12 customer reviews
ISBN-13: 978-0691130521
ISBN-10: 0691130523
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Editorial Reviews

From Booklist

How can a factory manager minimize breakdowns? How can a disoriented hiker reach her car in the least possible time? In answering questions such as these, engineer Nahin delivers maximal mathematical enjoyment with minimal perplexity and boredom. Classical minimization problems allow Nahin to showcase the ingenuity of ancient mathematicians--and to let general readers in on the thrill of riding high-school geometry and algebra to breakthrough insights. Knowledgeable readers will probably anticipate the eventual transition from subtle geometry to complex calculus. But even specialists may learn from Nahin's chronicle of how the often-forgotten tangents of Pierre de Fermat paved the way to the calculus of Newton and Leibniz. In addition, Nahin deftly interweaves episodes from the lives of its discoverers: a rash Belgian theorist loses his sight staring at the sun; a jealous Swiss mathematician denies his own son credit for groundbreaking work. A refreshingly lucid and humanizing approach to mathematics. Bryce Christensen
Copyright © American Library Association. All rights reserved --This text refers to an out of print or unavailable edition of this title.

Review

"This book was terrific fun to read! I thought I would skim the chapters to write my review, but I was hooked by the preface, and read through the first 100 pages in one sitting. . . . [Nahin shows] obvious delight and enjoyment--he is having fun and it is contagious."--Bonnie Shulman, MAA Online

"When Least is Best is clearly the result of immense effort. . . . [Nahin] just seems to get better and better. . . . The book is really a popular book of mathematics that touches on a broad range of problems associated with optimization."--Dennis S. Bernstein, IEEE Control Systems Magazine

"[When Least is Best is] a wonderful sourcebook from projects and is just plain fun to read."--Choice

"This book is highly recommended."--Clark Kimberling, Mathematical Intelligener

"A valuable and stimulating introduction to problems that have fascinated mathematicians and physicists for millennia."--D.R. Wilkins, Contemporary Physics

"Nahin delivers maximal mathematical enjoyment with minimal perplexity and boredom. . . . [He lets] general readers in on the thrill of riding high-school geometry and algebra to breakthrough insights. . . . A refreshingly lucid and humanizing approach to mathematics."--Booklist

"Anyone with a modest command of calculus, a curiosity about how mathematics developed, and a pad of paper for calculations will enjoy Nahin's lively book. His enthusiasm is infectious, his writing style is active and fluid, and his examples always have a point. . . . [H]e loves to tell stories, so even the familiar is enjoyably refreshed."--Donald R. Sherbert, SIAM Review
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Product Details

  • Paperback: 392 pages
  • Publisher: Princeton University Press (July 22, 2007)
  • Language: English
  • ISBN-10: 0691130523
  • ISBN-13: 978-0691130521
  • Product Dimensions: 6.1 x 0.9 x 9.2 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.6 out of 5 stars  See all reviews (12 customer reviews)
  • Amazon Best Sellers Rank: #595,487 in Books (See Top 100 in Books)

More About the Author

Paul Nahin was born in California, and did all his schooling there (Brea-Olinda High 1958, Stanford BS 1962, Caltech MS 1963, and - as a Howard Hughes Staff Doctoral Fellow - UC/Irvine PhD 1972, with all degrees in electrical engineering). He worked as a digital logic designer and radar systems engineer in the Southern California aerospace industry until 1971, when he started his academic career. He has taught at Harvey Mudd College, the Naval Postgraduate School, and the Universities of New Hampshire (where he is now emeritus professor of electrical engineering) and Virginia. In between and here-and-there he spent a post-doctoral year at the Naval Research Laboratory, and a summer and a year at the Center for Naval Analyses and the Institute for Defense Analyses as a weapon systems analyst, all in Washington, DC. He has published a couple dozen short science fiction stories in ANALOG, OMNI, and TWILIGHT ZONE magazines, and has written 16 books on mathematics and physics, published by IEEE Press, Springer, and the university presses of Johns Hopkins and Princeton. His most recent book, INSIDE INTERESTING INTEGRALS, discussing numerous techniques for doing definite integrals (up through and including contour integration in the complex plane) that commonly occur in physics, engineering, and mathematics, was published by Springer in September 2014. His next book, IN PRAISE OF SIMPLE PHYSICS, on the application in everyday life situations of elementary mathematics (up to and including freshman calculus) and the fundamental physical laws, is under contract with Princeton University Press, is currently at the copyeditor, and will appear May 2016. Another book, TIME MACHINE TALES, an up-dating of the 2nd edition of TIME MACHINES (1999), is under contract at Springer (in the Fiction & Science series) and will appear in 2017. He has given invited talks on mathematics at Bowdoin College, the Claremont Graduate School, the University of Tennessee, and Caltech, has appeared on National Public Radio's "Science Friday" show (discussing time travel) as well as on New Hampshire Public Radio's "The Front Porch" show (discussing imaginary numbers), and advised Boston's WGBH Public Television's "Nova" program on the script for their time travel episode. He gave the invited Sampson Lectures for 2011 in Mathematics at Bates College (Lewiston, Maine). When he isn't writing he is battling evil-doers on his PS4 and, now and then, he even wins ("Just Cause 3" is my current time-waster -- and I am having a LOT of trouble with that *%#*! wing suit!).

FINALLY - numerous readers have written over the years asking about the solutions manual to my Springer book, THE SCIENCE OF RADIO. Springer has kindly made it available in pdf format (3 MB), and if you write to me I'll send you a copy. paul.nahin@unh.edu

Customer Reviews

Top Customer Reviews

Format: Hardcover
Nahin's book is a tour de force about the deep intellectual threads that surround the notion of optimality. In physics, engineering, and mathematics, while touching on a wide range of applications, he asks over and over again: What is the optimal solution and why does it matter? Since I've spent most of my professional career thinking about optimality in one form or another, I was skeptical about how much new I would find in this book. But I was astounded to find something new and interesting on virtually every page. Some examples:
--Preface: Torricelli's funnel, which has finite volume and can be filled, but has infinite surface area and cannot be painted; and a slick proof that an irrational number raised to an irrational power can be rational.
--Chapter 1: An optimization problem that is not amenable to calculus, but whose solution can be discerned by some clever insight; an optimization problem that is amenable to calculus, but whose solution can be arrived at by algebra; and the use of the arithmetic mean-geometric mean inequality in optimization.
--Chapter 2: The ancient isoperimetric problem of Dido on maximal area, how it remained unsolved until modern times; the fact that there exists a figure in the plane whose area is equal to the area of the period at the end of this sentence and which contains a line segment one million light years in length that can be rotated 360 degrees within the figure (the shape of the figure is a little hard to picture); and the fact that there are two consecutive prime numbers the gap between which is greater than a googolplex (don't ask what they are).
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Format: Hardcover
Mr. Nahin states in his preface that 1st year undergraduate math and physics is enough to manage "a lot of mathematics in this book." He is fairly on the mark, discounting my comments about chapter six below. As usual, the reader must keep pencils and scrap paper ready to fully appreciate this book. I hoped to find a book based on applications of math and physics, an engineer's approach. This is one such fascinating book.

I was familiar with the AM-GM inequality technique to find extremas. However, Mr. Nahin dispenses of this method early and shows the reader so much more. And in this book, there is a constant exercise of looking at problems a different way.

If you like geometric solutions along with the typical lines of algebraic manipulations, you'll love this book. The first five chapters are packed with problems and solutions with excellent graphic representations. Integration requirements increase throughout.

In finding extremas in chapter six, the author goes beyond ordinary calculus with the calculus of variations including the Euler-Lagrange differential equation and Beltrami's identity. The focus problem is the minimal decent time curve. It is in section 6.4 that the author truly breaks from his stated reader requirements of "high school algebra, trigonometry, and geometry, as well as the elementary integration techniques." I think most authors of this book's scope typically underestimate reader requirements. As for my part, I did not understand the calculus of variations technique on the first reading. After reading sections 6.4 through 6.8 again, I gained an appreciation of how the method works. After one more reading of these sections, I might know just enough to be dangerous.
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Format: Hardcover
Science writing needs to avoid two obvious traps: the pedantic discourse for the supposed layman, or the oversimplified analogy bordering on the erroneous. Paul Nahin deserves full credit for circumventing these effectively in his book about optimization. He is clear about the background he assumes on his readers part, though he doesn't always provide adequate references for those who don't. He does, however, offer pertinent citations for those readers who wish to wade deeper. I only wish he was more careful with his equations, and even more so with his diagrams, which often confuse rather than clarify... but the good things first.

The choice of topics, their sequence and the examples signify not only their historical importance, but places several in a modern context, with an emphasis on numerical solutions. I especially liked the approach he takes in section 1.7 with the numerical-graphical technique. The muddy wheel in section 3.6 demonstrates how an interesting (and real!) problem can yield to analytical techniques. However, numerical methods can be stretched at times, which is evident in the justification of eliminating one of the two values for a minimal surface on page 268. One would have appreciated a physical explanation based on analytical techniques. From a historical perspective, the use, discovery, exploration, development and the final definition of the derivative, (in that sequence!) and how Fermat played a seminal role in it clears several misconceptions even before the Newton-Leibniz imbroglio.

There are two particular examples that I would like to underline, both for their simplicity and their beauty of exposition. The first is the projectile problems in sections 5.4 and 5.5.
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When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible
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