- NUMBER SYSTEMS 209
The reaction of the mathematical community to this simple but profound idea
was less than overwhelming. Wessel's work was forgotten for a full century. In
the meantime another mathematician by avocation, the French accountant Jean
Argand (1768 1822), published the small book Essai sur une maniere de representer
les quantites imaginaires dans les constructions geometriques at his own expense
in 1806, modestly omitting to name himself as its author, in which he advocated
essentially the same idea, thinking, as Wallis had done, of an imaginary number as
the mean proportional between a positive number and a negative number. Through
a complicated series of events this book and its author gradually became known
in the mathematical community. There was, however, resistance to the idea of
interpreting complex numbers geometrically, since they had arisen in algebra. But
geometry was essential to the algebra of complex numbers, as shown by the fact
that a proof of the fundamental theorem of algebra by Gauss in 1799 is based on
the idea of intersecting curves in a plane. The lemmas that Gauss used for the
proof had been proved earlier by Euler using the algebra of imaginary numbers,
but Gauss gave a new proof using only real numbers, precisely to avoid invoking
any properties of imaginary numbers.
Even though he avoided the algebra of imaginary numbers, Gauss still needed
the continuity properties of real numbers, which, as we just saw, were not fully
arithmetized until many years later.^15 Continuity was a geometric property not
explicitly found in Euclid, but Gauss expressed the opinion that continuity could
be arithmetized. In giving a fifth proof of this theorem half a century later, he
made full use of complex numbers. In fact, the complex plane is sometimes called
the Gaussian plane.
2.4. Infinite numbers. The problem of infinity has occupied mathematicians for
a very long time. Neither arithmetic nor geometry can place any prcassigned limit
on the sizes of objects. An integer can be as large as we like, and a line can
be bisected as many times as we like. These are potential infinities and potential
infinitesimals. Geometry can lead to the concept of an actual infinity and an actual
infinitesimal. A line, plane, or solid is an infinite set of points; and in a sense a point
is an infinitesimal (infinitely short) line, a line is an infinitesimal (infinitely narrow)
plane, and a plane is an infinitesimal (infinitely thin) solid. These notions of the
infinite and the infinitesimal present a logical problem for beings whose experience
extends over only a finite amount of space, whose senses cannot resolve impressions
below a certain threshold, and whose reasoning is presented using a finite set of
words. The difficulties of dividing by zero and the problem of incommensurables,
mentioned above, are two manifestations of this difficulty. We shall see others in
later chapters.
The infinite in Hindu mathematics. Early Hindu mathematics had a prominent
metaphysical component that manifested itself in the handling of the infinite.
The Hindus accepted an actual infinity and classified different kinds of infinities.
This part of Hindu mathematics is particularly noticeable with the Jainas. They
classified numbers as enumerable, unenumerable, and infinite, and space as one-
dimensional, two-dimensional, three-dimensional, and infinitely infinite. Further,
they seem to have given a classification of infinite numbers remarkably similar to
(^15) The Czech scholar Bernard Bolzano (1781-1848) showed how to approach the idea of con-
tinuity analytically in an 1817 paper. One could argue that his work anticipated Dedekind's
arithmetization of real numbers.