The History of Mathematics: A Brief Course

204 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICS

has given no proof. I still do not see what prevents the circumfer-

ence itself, or some particular part of it, from being measured [that

is, being commensurable with the radius], a part whose arc has a

ratio to its sine [half-chord] that can be expressed by an equation

of finite degree. But to express the ratio of the arc to the sine

in general by an equation of finite degree is impossible, as I shall

prove in this little work. [Gerhardt, 1971, Vol. 5, p. 97]

No representation of ð as the root of a polynomial with rational coefficients was

ever found. This ratio had a long history of numerical approximations from all over

the world, but no one ever found any nonidentical equation satisfied by C and D.

The fact that ð is transcendental was first proved in 1881 by Ferdinand Lindemann

(1852-1939). The complete set of real numbers thus consists of the positive and

negative rational numbers, all real roots of equations with integer coefficients (the

algebraic numbers), and the transcendental numbers. All transcendental numbers

and some algebraic numbers are irrational. Examples of transcendental numbers

are rather difficult to produce. The first number to be proved transcendental was

the base of natural logarithms e, and this proof was achieved only in 1873, by the

French mathematician Charles Hermite (1822-1901). It is still not known whether

the Euler constant 7 ~ 0.57712 is even irrational.

The arithmetization of the real numbers. Not until the nineteenth century, when

mathematicians took a retrospective look at the magnificent edifice of calculus that

they had created and tried to give it the same degree of logical rigor possessed

by algebra and Euclidean geometry, were attempts made to define real numbers

arithmetically, without mentioning ratios of lines. One such definition by Richard

Dedekind (1831-1916), a professor at the Zurich Polytechnikum, was inspired by a

desire for rigor when he began lecturing to students in 1858. He found the rigor he

sought without much difficulty, but did not bother to publish what he regarded as

mere common sense until 1872, when he wished to publish something in honor of

his father. In his book Stetigkeit und irrationale Zahlen (Continuity and Irrational

Numbers) he referred to Newton's definition of a real number:

... the way in which the irrational numbers are usually introduced

is based directly upon the conception of extensive magnitudes—

which itself is nowhere carefully defined—and explains number as

the result of measuring such a magnitude by another of the same

kind. Instead of this I demand that arithmetic shall be developed

out of itself.

As Dedekind saw the matter, it was really the totality of rational numbers
that defined a ratio of continuous magnitudes. Although one might not be able to
say that two continuous quantities a and b had a ratio equal to, or defined by, a
ratio m : ç of two integers, an inequality such as ma < nb could be interpreted
as saying that the real number a : b (whatever it was) was less than the rational
number n/m. Thus a positive real number could be defined as a way of dividing the
positive rational numbers into two classes, those that were larger than the number
and those that were equal to it or smaller, and every member of the first class was
larger than every member of the second class. But, so reasoned Dedekind, once
the positive rational numbers have been partitioned in this way, the two classes
themselves can be regarded as the number. They are a well-defined object, and one