# Where numbers roam free and irrational

The Kingdom of Infinite Number: A Field Guide By Bryan Bunch W.H. Freeman & Co. 160 pp., $23.95

This delightful volume is dedicated to the proposition that there are no uninteresting numbers. Sighting a "transcendental" number in the wild can be as exciting as spotting a pileated woodpecker in the North woods.

The author, a journalist and part-time math instructor at Pace University, shows numbers in their natural habitats. Each "genus" of number is described, along with "commonly seen species," recognition marks, personalities, and associations. There is no number, no matter how drab or common it may seem to the unobservant, that does not have a story to tell.

The natural numbers - genus natural - are portrayed in the first section of the guide. These are the familiar counting numbers 1, 2, 3, and so forth.

The author suggests that humans probably have some built-in number line that allows us to understand intuitively that three objects are more than two. Even babies can apparently count before they can use language.

From the counting numbers, it seems a simple step to the integers, which include negative as well as positive numbers. The idea of negative numbers also seems "natural" enough, but in fact it was not until about AD 628 that Indian mathematicians set down the rules for dealing with numbers less than one.

Dividing one integer by another gives the genus rational. Ratio numbers are those that can be described as fractions of the form 1/2 or 7/8. One of the "field marks" of the rationals is that when expressed as a decimal fraction, they either terminate (1/2 = .5) or else repeat the same sequence of digits over and over (1/7 = 0.142857142857...).

In contrast, the set of real numbers can be distinguished because they seem to go on forever without repeating. The most famous of these is probably pi, which as of the end of the 20th century had been computed to more than 50 billion decimal places. Pi is also one of the so-called "transcendental" numbers, because it cannot be computed as the solution to any algebraic equation.

Complex numbers, as the name implies, have two parts - one real and one imaginary. Imaginary numbers are multiples of the square root of -1, an idea that was so hard for mathematicians to swallow that they called these numbers by this fanciful name.

The story really gets interesting when the author turns to infinity. Contrary to what seems like common sense, it turns out that there is more than one kind of infinite number, and - even more difficult to fathom - some kinds of infinities are bigger than others!

Mathematics began with counting sheep and goats, and measuring fields. From the earliest recorded history to the supercomputers of the 21st century, mathematics has evolved in steps that were both logical and profound. With this field guide in hand, the reader is equipped to rediscover the familiar and to become familiar with the exotic.

*Frederick Pratter is a numbers tracker and freelance writer in Missoula, Mont.

(c) Copyright 2000. The Christian Science Publishing Society