One of the most fundamental concepts influencing the development of human civilization is numbers. While societies today rely on their understanding of numbers for everything from mapping the universe to running word processing programs on computers to buying lunch, numbers are a human invention. Babylonian, Roman, and Arabic societies devised influential systems for representing numbers, yet the story of how numbers developed is far more complicated. Concepts such as zero, negative numbers, fractions, irrational numbers, and roots of numbers were often controversial in the past. Numbers deals with the development of numbers from fractions to algebraic numbers to transcendental numbers to complex numbers and their uses. The book also examines in detail the number pi, the evolution of the idea of infinity, and the representation of numbers in computers. The metric and American systems of measurement as well as the applications of some historical concepts of numbers in such modern forms as cryptography and hand calculators are also covered. Illustrations, thought-provoking text, and other supplemental material cover the key ideas, figures, and events in the historical development of numbers.

us have any direct experience—is probably more characteristic of our time than the decimal system of notation pioneered by Napier, Stevin, and Viète. PART TWO EXTENDING THE IDEA OF A NUMBER 5 an evolving concept of a number Knowledge of positive fractions and positive whole numbers, what we now call the positive rational numbers, is at least as old as civilization itself. The ancient Egyptians, Mesopotamians, and Chinese all found ways to compute with positive rational numbers. This is not

using the complex plane, however, extend far past computing absolute values. To appreciate the difficulty of using complex numbers before the invention of the Argand diagram, consider the problem of graphing functions. In the study of functions of a real variable, every student learns to associate a twodimensional graph with the algebraic description of the function of interest. The graph is important because it establishes a relationship between a geometric and an algebraic interpretation of the

valuable for describing most transcendental numbers, either. If we write out the number as a decimal, we find that the sequence of digits never ends and never forms a repeating pattern in the way that decimal expansions of rational numbers always do. That is why when we refer to the transcendental number formed by the ratio of the circumference of a circle to its diameter—the number that is approximately 3.14—we use the Greek letter π. Similarly the number that occupies a special place in

generates good, logically consistent mathematics. Hilbert’s goal was to look inside mathematics to discover the way it worked. Keep in mind that in Hilbert’s conception of formal mathematics the only question of interest is whether the theorems are direct, logical consequences of the axioms. This is a question best answered by examining the formal, abstract axioms that defined the subject. In this very rarefied atmosphere, the questions of the completeness and consistency of the axioms become

consistency of particular sets of axioms. Hilbert, to be sure, never expected an easy answer to the questions of completeness and consistency. The fact that one has 168 NUMBERS Text from Principia Mathematica showing the formal language developed by Whitehead and Russell to express their ideas (Courtesy of the Department of Special Collections, University of Vermont) Cantor’s Legacy 169 not yet found an unprovable statement does not mean that the axioms are complete. Nor does the fact that