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Topic: [HM] Proof of Theorem of Theaetetus.
Replies: 15   Last Post: Jul 25, 2006 6:12 PM

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Robert (Bob) Eldon Taylor

Posts: 101
Registered: 12/3/04
[HM] Proof of Theorem of Theaetetus.
Posted: Aug 26, 2003 2:24 PM
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Dear Friends,

In "Stetigkeit und irrationale Zahlen" section 4, Dedekind introduces
the Theorem of Theaetetus (Eu. X.9): if a whole number D is not the
square of a whole number, it is not the square of any rational
number. He gives an indirect proof summarized below. What is the
source of this proof?


Robert Eldon Taylor
philologos at mindspring dot com

First, if D is a whole number, but not the square of a whole number,
then it lies between two squares, i.e. there is a whole number, n,
such that nn < D < (n + 1)^2.

Assume there is a rational number whose square is = D, then there are
two positive whole numbers, t, u which satisfy the equation
tt - Duu = 0,
and one may assume that u is the smallest positive number which
possesses the property that its square may be transformed into the
square of a whole number t by multiplication with D. Now, since
nu < t < (n + 1) u

thus the number u' = t - nu is a positive whole number, and indeed
smaller than u. Further if one sets
t' = Du - nt

t' is likewise a positive whole number and there results

t't' - du'u' = (nn - D)(tt - Duu) = 0,
which stands in contradiction to the assumption about u.

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