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When Least Is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible Hardcover – December 14, 2003

by Paul J. Nahin (Author)

11 customer reviews

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What is the best way to photograph a speeding bullet? Why does light move through glass in the least amount of time possible? How can lost hikers find their way out of a forest? What will rainbows look like in the future? Why do soap bubbles have a shape that gives them the least area?

By combining the mathematical history of extrema with contemporary examples, Paul J. Nahin answers these intriguing questions and more in this engaging and witty volume. He shows how life often works at the extremes--with values becoming as small (or as large) as possible--and how mathematicians over the centuries have struggled to calculate these problems of minima and maxima. From medieval writings to the development of modern calculus to the current field of optimization, Nahin tells the story of Dido's problem, Fermat and Descartes, Torricelli, Bishop Berkeley, Goldschmidt, and more. Along the way, he explores how to build the shortest bridge possible between two towns, how to shop for garbage bags, how to vary speed during a race, and how to make the perfect basketball shot.

Written in a conversational tone and requiring only an early undergraduate level of mathematical knowledge, When Least Is Best is full of fascinating examples and ready-to-try-at-home experiments. This is the first book on optimization written for a wide audience, and math enthusiasts of all backgrounds will delight in its lively topics.

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

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Product Details

Hardcover: 370 pages
Publisher: Princeton University Press (December 14, 2003)
Language: English
ISBN-10: 0691070784
ISBN-13: 978-0691070780
Product Dimensions: 9.5 x 6.3 x 1.1 inches
Shipping Weight: 1.5 pounds
Average Customer Review: 4.5 out of 5 stars See all reviews (11 customer reviews)
Amazon Best Sellers Rank: #933,087 in Books (See Top 100 in Books)
- #747 in Books > Science & Math > Mathematics > History

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4.5 out of 5 stars

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125 of 128 people found the following review helpful

Off the Charts

By Dennis S. Bernstein on March 10, 2004

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).Read more ›

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22 of 22 people found the following review helpful

More would be better !

By Michael LaDeau on August 2, 2006

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.Read more ›

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12 of 15 people found the following review helpful

Delightful read, just avoid the the diagrams

By Kaushik Basu on November 14, 2006

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.Read more ›

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Most Recent Customer Reviews

Five Stars

Excellent book. Paul Nahin's style is very clear, making the concepts interesting and accessible. Highly recommended.

Published 15 days ago by Wolf Woods

Five Stars

Great Book...as are all of Paul Nahin's Books

Published 8 months ago by Bruce C. Taylor

Four Stars

Well written, some interesting topics, some everyday topics.

Published 16 months ago by Science fan

When Least is Best -- but not easy

'When Least is Best' is an interesting discussion of alternates to calculus for minimum/maximum problems. While the math is not formidable, I found it uninviting. Read more

Published on July 30, 2013 by Paul F. Bruce

Hard stuff

The book is well written but you have to be a professional mathematician or a math major at the very least to understand it. Read more

Published on January 1, 2012 by simon

Pretty darn amazing

This book does so many things well, that I would get bored trying to explain them all. What really impressed me was the explanation of the Euler-Lagrange equation. Read more

Published on March 12, 2008 by Jonathan Fischoff

A fabulous book!

I've recently finished the above book and can't tell the reader just how much I enjoyed it, particularly the connections made that often get lost in the usual silo approach to math... Read more

Published on April 11, 2007 by Nicholas Warren

excellent - I want all his books

Finally, a solid book that challenges the lay reader just like the best math teachers do - by showing the elegance and power of mathematical reasoning. Read more

Published on April 23, 2004 by pete