Numbers are the first association in the minds of most people when they hear
the word mathematics. The word arithmetic comes into English from the Greek
word arithmos, meaning number. What is nowadays called arithmetic—that is,
calculation—had a different name among the Greek writers of ancient times: logis-
tike, the source of our modern word logistics. In the comedy The Acharnians by
Aristophanes, the hero Dicaeopolis reflects that, arriving early for meetings of the
Athenian assembly, "aporo, grapho, paratillomai, logizomai" ("I don't know what
to do with myself; I doodle, pull my hair, and calculate").
The different levels of sophistication in the use and study of numbers provide a
convenient division into chapters for the present part of the book. We distinguish
three different stages in the advancement of human knowledge about numbers:
(1) the elementary stage, in which a limited set of integers and fractions is used
for counting and measuring: (2) the stage of calculation, in which the common
operations of addition, subtraction, multiplication, and division are introduced;
and (3) the theoretical stage, in which numbers themselves become an object of
interest, different kinds of numbers are distinguished, and new number systems are
invented.
The first stage forms the subject matter of Chapter 5. Even when dealing
with immediate problems of trade, mere counting is probably not quite sufficient
numeracy for practical life; some way of comparing numbers in terms of size is
needed. And when administering a more populous society, planning large public
works projects, military campaigns, and the like, sophisticated ways of calculating
are essential. In Chapter G we discuss the second stage, the methods of calculating
used in different cultures.
In Chapter 7 we examine the third stage, number theory. This theory be-
gins with the mathematicians of ancient Greece, India, and China. We look at
the unique achievements of each civilization: prime, composite, triangular, square,
and pentagonal numbers among the Pythagoreans, combinatorics and congruences
among the Chinese and Hindus.
In Chapter 8 we discuss number systems and number theory in the modern
world. Here we see how algebra led to the concept of irrational (algebraic) num-
bers, and the geometric representation of such numbers brought along still more
(transcendental) numbers. When combined with geometry and calculus, this new
algebraic view of numbers led to the theory of complex variables, which in turn
made it possible to answer some very delicate questions on the relative density of
prime numbers (the famous "prime number theorem"). The continuing develop-
ment of these connections, as well as connections with the theory of trigonometric
series, has made it possible to settle some famous conjectures: the Wiles-Taylor
proof of Fermat's last theorem and a partial proof by I. M. Vinogradov of the famous
Goldbach conjecture, for example.
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