Men of Mathematics (88 page)

“Instead of expressing the closely interconnected system of roots, considered as functions of the coefficients, by a formula involving many-valued radicals,
^I
we may seek to obtain the roots expressed
separately by as many distinct uniform [one-valued] functions of auxiliary variables, as in the case of the third degree. In this case, where the equation

x
³
—
3x + 2a
= 0

is under discussion, it suffices, as we know, to represent the coefficient
a
by the sine of an angle, say
A,
in order that the roots be isolated as the following well-determined functions

[Hermite is here recalling the familiar “trigonometric solution” of the cubic usually discussed in the second course of school algebra. The “auxiliary variable” is
A;
the “uniform functions” are here sines.]

“Now it is an entirely similar fact which we have to exhibit concerning the equation

x
⁵
—x—a = 0.

Only, instead of sines or cosines, it is the elliptic functions which it is necessary to introduce. . . .”

In short order Hermite then proceeds to solve
the general equation of the fifth degree,
using for the purpose elliptic functions (strictly, elliptic modular functions, but the distinction is of no importance here). It is almost impossible to convey to a nonmathematician the spectacular brilliance of such a feat; to give a very inadequate simile, Hermite found the famous “lost chord” when no mortal had the slightest suspicion that such an elusive thing existed anywhere in time and space. Needless to say his totally unforeseen success created a sensation in the mathematical world. Better, it inaugurated a new department of algebra and analysis in which the grand problem is to discover and investigate those functions in terms of which the general equation of the
nth
degree can be solved explicitly in finite form. The best result so far obtained is that of Hermite's pupil, Poincaré (in the 1880's), who created the functions giving the required solution. These turned out to be a “natural” generalization of the elliptic functions. The characteristic of those functions that was generalized was periodicity. Further details would take us too far afield here, but if there is space we shall recur to this point when we reach Poincaré.

Hermite's other sensational isolated result was that which established
the
transcendence
(explained in a moment) of the number denoted in mathematical analysis by the letter
e,
namely

where 1! means 1, 2! = 1 × 2, 3! = 1 × 2 × 3, 4! = 1 × 2 × 3 × 4, and so on; this number is the “base” of the so-called “natural” system of logarithms, and is approximately 2.718281828. . . . It has been said that it is impossible to conceive of a universe in which
e
and
π
(the ratio of the circumference of a circle to its diameter) are lacking. However that may be (as a matter of fact it is false), it is a fact that
e
turns up everywhere in current mathematics, pure and applied. Why this should be so, at least so far as applied mathematics is concerned, may be inferred from the following fact:
e
^x
,
considered as a function of
x,
is the
only
function of
x
whose rate of change with respect to
x
is equal to the function itself—that is,
e
^x
is the only function which is equal to its derivative.
^II

The concept of “transcendence” is extremely simple, also extremely important. Any root of an algebraic equation whose coefficients are rational integers
is called an
algebraic number.
Thus
2.78 are algebraic numbers, because they are roots of the respective algebraic equations
x
²
+ 1 = 0,
50x −139
= 0, in which the coefficients (1, 1 for the first, 50, −139 for the second) are rational integers. A “number” which is
not
algebraic is called transcendental. Otherwise expressed, a transcendental number is one which satisfies
no
algebraic equation with rational integer coefficients.

Now, given any “number” constructed according to some definite law, it is a meaningful question to ask whether it is algebraic or transcendental. Consider, for example, the following simply defined number,

in which the exponents 2, 6, 24, 120, . . . are the successive “factorials,” namely 2 = 1 × 2, 6 = 1 × 2 × 3, 24 = 1 × 2 × 3 × 4, 120 = 1 × 2 × 3 × 4 × 5, . . . , and the indicated series continues “to infinity” according to the same law as that for the terms given.
The next term is
the sum of the first three terms is .1 + .01 + .000001, or .110001, and it can be proved that the series does actually define some definite number which is less than .12. Is this number a root of
any
algebraic equation with rational integer coefficients? The answer is no, although to prove this without having been shown how to go about it is a severe test of high mathematical ability. On the other hand, the number defined by the infinite series

is
algebraic; it is the root of 99900
x—
1 = 0 (as may be verified by the reader who remembers how to sum an infinite convergent geometrical progression).

The first to prove that certain numbers are transcendental was Joseph Liouville (the same man who encouraged Hermite to write to Jacobi) who, in 1844, discovered a very extensive class of transcendental numbers, of which all those of the form

where
n
is a real number greater than 1 (the example given above corresponds to
n
= 10), are among the simplest. But it is probably a much more difficult problem to prove that
a particular
suspect, like
e
or π, is or is not transcendental than it is to invent a whole infinite class of transcendentals: the inventive mathematician dictates—to a certain extent—the working conditions, while the suspected number is entire master of the situation, and it is the mathematician in this case, not the suspect, who takes orders which he only dimly understands. So when Hermite proved in 1873 that
e
(defined a short way back) is transcendental, the mathematical world was not only delighted but astonished at the marvellous ingenuity of the proof.