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Re: Number line
Posted:
Feb 22, 2009 6:16 PM
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> --- Samuel.Kutler@sjca.edu wrote:
>
> From: "Kutler, Samuel" <Samuel.Kutler@sjca.edu>
> To: <MATH-HISTORY-LIST@ENTERPRISE.MAA.ORG>
> Subject: RE: Sir Andrew Wiles
> Date: Mon, 16 Feb 2009 06:23:13 -0500
>
> When did the term real number line appear, and when did someone say that
> there is a one-to-one correspondence between the real numbers and the
>points on the real number line?
Dedekind wrote about the number line in 1872. From the paper "Continuity and Irrational Numbers" in _Essays on the Theory of Numbers_ by Richard Dedekind
Translated from the German by Wooster Woodruff Beman
Chicago: Open Court Publishing Company, 1901
available on-line at http://books.google.com/books?id=tzYIAAAAIAAJ&printsec=frontcover&dq=inauthor:richard+inauthor:dedekind&lr=&as_brr=0&as_pt=ALLTYPES#PPP1,M1
<quote>
This analogy between rational numbers and the points of a straight line, as is well known, becomes a real correspondence when we select upon the straight line a definite origin or zero-point o and a definite unit of length for the measurement of segments. With the aid of the latter to every rational number a a corresponding length can be constructed and if we lay this off upon the straight line to the right or left of o according as a is positive or negative, we obtain a definite end-point p, which may be regarded as the point corresponding to the number a ; to the rational number zero corresponds the point o. In this way to every rational number a, i. e., to every individual in R, corresponds one and only one point/, i. e., an individual in L. To the two numbers a, b respectively correspond the two points /, q, and if a~>b, then / lies to the right of q.
</quote>
Dedekind in this paper did not go on to describe a one-to-one correspondence between the real numbers and the line, but such an idea is an obvious extension. However, Dedekind was interested not in discussing geometry but in introducing what we call the "Dedekind cut", which he describes in terms of cutting a line into two pieces.
Dedekind and Cantor seem to have influenced each other. Note the following quote from the preface of the same paper:
<quote>
While writing this preface (March 20, 1872), I am just in receipt of the interesting paper Ueber die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen, by G. Cantor (Math. Annalen, Vol. 5), for which I owe the ingenious author my hearty thanks.
</quote>
James A. Landau
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