Numbers

David Berlinksi is among the leaders of writing so-called popular books about different aspects of math -- some that is highly advanced, as in The Advent of the Algorithm, and some that is very very basic, as in 2011's One, Two Three.

Berlinksi assigns the basic mathematical functions the group name "AEM" or Absolutely Elementary Mathematics. The four major functions of addition, subtraction, multiplication and division are outlined as the building blocks of far more complicated functions and equations. Berlinksi also digs even deeper, offering ways to think about even the idea of "number."

Berlinski holds doctorates in philosophy and mathematics, so he is a good choice to explain math concepts in terms that don't lean too heavily on equations. His purpose in One, Two, Three is to suggest answers for these simplest and most basic questions about AEM and to show how such answers can be deduced via logic from some very simple assumptions.

One, Two, Three is both aided by and labors under Berlinksi's habit of breezy and almost flippant writing. On the one hand he largely succeeds in getting complex ideas boiled down to terms that most people can understand, and presents his arguments in ways that can be followed without specialized knowledge. But on the other hand, his tone sometimes crosses over into flippancy in ways that can slow readers down while they finish rolling their eyes.

He too often sacrifices some clarity and direction in order to make a witty observation and in more than one place sticks in some jokes for their own sake rather than explanatory value (yes, O Tolerant Reader, this blog does the same thing quite often. However, it's the product of some moke running his mouth and not someone explaining a potentially difficult subject. Judge for yourself what kind of damage that can do to explaining an idea). Whether or not Berlinksi is actually all that impressed with his own wit, he gives a good enough imitation of being so to make several parts of One, Two, Three way more annoying and way less useful than they could have been.

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Of all the people who probably hide their heads at the goofs they have made, the person who named "imaginary" numbers is probably among them as far as the field of mathematics is concerned.

So-called "imaginary" numbers describe the square roots of negative numbers, which are impossible to calculate using plain integers. The square root of 1, for example, is 1 because 1x1=1. But the square root of -1 seems impossible to figure, because the only way to get to -1 is to multiply two different numbers together. A negative number multiplied by another negative number leaves a positive number, not a negative one. At some point, mathematicians decided that there would be a square root of -1, and it would just be a 1 that was on another "axis" than the regular positive-negative line. But since the number didn't seem to have any real-world analogue like positive and negative numbers did, it somehow got hung with the tag, "imaginary." So today we say that the square root of negative 1 is i. The square root of -4 is 2i, and so on.

Retired electrical engineering professor Paul Nahin outlines some of the development of i through the history of mathematics in An Imaginary Tale. Some early cultures refused to acknowledge the existence of a quantity that could be squared to form a negative number, and even into the Renaissance and enlightenment years the so-called "imaginary" numbers were considered at best unimportant. They were not useful except in specialized cases and it seemed even mathematicians had reservations about dealing with numbers that didn't represent any real quantity.

Today, i and its counterparts find widespread use in many areas of math, and the only reservations that seem to continue deal mostly with the use of the word "imaginary." Nahin explores how important i is in many fields of engineering, especially his own. This part of the book -- about the latter two-thirds -- is heavily laden with equations and formulas and is going to be beyond most non-mathematician or non-engineer readers. He probably would have had to lengthen the book considerably to bring that subject matter within the grasp of the lay reader, but that doesn't make the string of equations and engineering language any easier to navigate.

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Most of the time we use math we do just that: use math. We rarely think about things like why numbers come together the way they do. Or why certain mathematical functions and relationships seem to matter in the world of actual things as well as within the realm of pure equation and solution. But almost every mathematical advance throughout human history has often stemmed from and also sparked some serious thought.

Luke Heaton's A Brief History of Mathematical Thought skims through history and sketches some of the thinking that accompanied the ciphering. He begins as close to the beginning as possible, offering some ideas on how our stone-age ancestors may have begun to progress beyond the simple counting of objects into understanding the numbers behind the counting had relationships that could be regularized and predicted. At what point, for example, did some forgotten genius figure out that two of anything added to three of that same thing would always make five of that thing? If you had two rocks and were handed three, you did not need to count all five of them over again -- you could add the three to your two and know you had five whether you counted them or not. And once people had developed this understanding, how did it change their civilization and culture?

History is better in the earlier sections, such as the one mentioned above and others that deal with numerical development among the ancient Greeks, ancient Indians and other civilizations. It's also a good overview of how the switch to Arabic numerals and the use of the zero as a place-keeper propelled scientific thought far beyond what had been possible with cumbersome systems like Roman numerals. Later sections, though, deal with more esoteric subjects within math and their impact seems less obvious. Heaton offers reasons to spend some time pondering non-Euclidian geometry, for example, but has fewer explanations about how this particular wrinkle affects the way we live and work. Still, History is a good primer on what kind of thought can come from dwelling on even the most mundane of numerical tasks, as well as how that thought has shaped who we are today.