Number; The Language of Science by Tobias Dantzig

Author:Tobias Dantzig [Dantzig, Tobias]
Language: eng
Format: epub
Published: 2011-02-17T05:00:00+00:00

Dantzig_Ch_10.qxd 2/17/05 2:11 PM Page 195

The Domain of Number

195

It was surmised by Girard early in the seventeenth century

that what was true for equations of the first four degrees was

generally true; and in the middle of the eighteenth century

d’Alembert formulated it in the statement that any algebraic

equation must possess at least one solution real or complex. He,

however, was unable to prove this assertion rigorously, and in

spite of the efforts of many who followed him it remained a pos-

tulate for another fifty years.

This assertion recalls the other statement: any equation can

be solved by means of radicals. This, too, we saw, was considered

obvious by many mathematicians even in the days of Lagrange.

Yet the comparison is unfair: here the generalization was of the

type called incomplete induction, and the falsity of the proposition

only brought out in relief the danger of this method. Entirely

different is the intuition which led to d’Alembert’s postulate.

This intuition is reflected in all the proofs of this fundamental

theorem of algebra that have been given since the days of

d’Alembert; namely, d’Alembert’s, Euler’s, and Lagrange’s insuffi-

cient demonstrations; the proofs which Argand gave for it in 1806

and 1816; the four proofs by which the great Gauss established the

proposition; and all subsequent improvements on these latter.

Different though these proofs are in principle, they all possess

one feature in common. Somewhere, somehow,—sometimes

openly, sometimes implicitly—the idea of continuity is intro-

duced, an idea which is foreign to algebra, an idea which belongs

to the realm of analysis.

Let me explain this by a simple example. If we set Z = z 2 + 1

and z = x + iy, we obtain upon substitution Z = ( x 2 – y 2 + 1) +

i(2 xy). Now when x and y vary in a continuous manner and assume all possible values between –∞ and +∞, the expressions

within the parentheses will also assume all possible values

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