Mathematics Book Review

The following reviews were published on either Sacramento Book Review monthly journal or posted on SacramentoBookReview.com or SanFranciscoBookReview.com

Mathematics in Ten Lessons

by Jerry P. King. 394 Pages. $18.95. Prometheus Book

Amazon.com Title: The Grand Tour of Pure, Blissful Logic

Perry P. King surprises the reader by enlightenment during a ten-lesson marathon beginning with fundamental of logical operators, through number theory, on to probability and calculus. There’s something for the taste of any mathematical palate. He illustrates pi and e (the natural logarithm), which enjoy special status. Not only are they true, irrational numbers, aside from the square roots of primes, Ferdinand Lindemann and Charles Hermite proved pi and e, respectfully, are transcendental.

It’s remarkable when a mathematician illustrates pi by drawing on sequences that calculate an approximation. It’s even more prodigious when he shows how to calculate e. Unlike pi, which can be explained by comparing the circumference to the diameter of a circle (pi=c/d), e is the limit of the sequence (1+1/n) to the nth power as n approaches infinity. A mouthful!

Mathematics is full of paradoxical arguments, too. He lists and explains six of the more famous ones. The author also tells the story of 10 year-old Carl Friedrich Gauss, who single-handedly solved the sum of a sequence problem by realizing multiple pairs within the scope of elementary school, shocking his teacher.

Philosophically, King succeeds to take you on “The Grand Tour.” Quod erat demonstrandum (Q.E.D.)

The Pythagorean Theorem

by Alfred S. Posamentier

Amazon.com Title: The Grand Tour of Pure, Blissful Logic 294 Pages.

I’m a little surprised to see Dr. Posamentier driven down a well-paved road. Perhaps he has something new to show us. As compared to his previous publications, Posamentier continues to deliver a depth of insight. His work is nothing short of comprehensive and remains the most standard work to represent Pythagorian thought in a modern era.

He provides treatment of Pythagorus with numeric proofs, algebraic and geometric. He also shows how equivalent relations form Pathagorus’ theorem. Perhaps, the most poignant area involves the famous trigonometric identity, Sin2 θ + Cos2 θ = 1, derived from the distance formula, which is also Pythagorian in nature. One of these trigonometric concerns leads to the invaluable triangle-area calculations of ½ xySin θ in the Cartesian plane. The Pythagorian Theorem is also a special case of the law of Cosines as illustrated by a2+b2- 2abcos θ= c2.

Although Posamentier has nothing new to uncover, he sheds light in the gray areas of familiar topics in Pythagorian geometry. As one might recall from the history of mathematics, Pythagorus understands counting through his marvelous use of geometric numbers. Posamentier seeks a similar path, tempered with a Pythagorian view. A dynamic read for stimulating your brain.

Here’s Looking at Euclid

by Alex Bellos; Category: Science & Nature; 298 Pages;

Amazon.com Title: The Elements Provides no Shortcut

Euclid is often referred to as the father of geometry. His life’s work, called the Elements, remains the foundation of modern geometry. Indeed, it was Pythagorus, who predates Euclid by some 200 years, who set the stage. We can see this in the way that Euclid proceeds, beginning with arguments of simple definitions, such as what is a point, a line and a plane. He elaborates by arguing that such statements in nature exist that cannot be proved by reasoning, such as a straight line is the shortest distance between two points. He describes the essence of these self-evident concepts as postulates. From these and a wealth of definitions of geometric form, flows the basis of the Euclidian postulation system.

This is what we are taught in school. So, what is Alex Bellos really trying to say? Here’s Looking at Euclid is an attempt to examine ancient writings to see how Euclid fits in. We are impressed by the similarities certain ancient concepts share with regard to number formation and geometric form. Consequently, mathematics may not originate from divine thinking.

Nature is the backbone of modern thinking. Just as Leonardo de Vinci looked at birds to learn how to fly and creatures to explore movement, Euclid must have been inspired by what nature bestows. Bellos goes on a quest to examine the origin of mathematical thinking by examining the relics of the past. And he does a rather divine job.