Download PDF by Eisenhart L.P.: Coordinate geometry

Geometry Topology

By Eisenhart L.P.

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This sequence of intervals has no common point. 6 For example you can find y/2 in this way ; cf. 6. g. a\ — 1 and 61 = 2. Divide the interval [ai,6i] at the middle. Thereby we get two new intervals with rational numbers as end points, y/2 lies in exactly one of these intervals; call it [02, 62]. We actually know t h a t 02 = 1 and 62 = § in this case, but the m e t h o d is quite general. Now divide the interval [02,62] a t the middle. Again we get two new intervals with rational numbers as end points, and again y/2 lies in exactly one of these intervals; call it [03, 63].

We shall return to the application aspect of such problems at the end of the chapter. e. proofs that solutions to problems exist. 1, in which we keep fixed the length g of the base AC. We now ask how this triangle shall be designed so that it has the largest area possible, when its perimeter has fixed length L. For the 47 Isosceles triangles sum of the length of the sides AB and BC, it must hold that \AB\ + \BC\ = L-g. 2. Let the altitude of the triangle have the length h. Then the area of the triangle is given by |/i • g.

If we want to describe the real numbers completely from the rational numbers, it can, for our purpose, be done most expediently by the following procedure. We simply imagine t h a t we catch the real numbers in so-called nested intervals. , in which the length of the interval [ a n , 6 n ] approaches 0 for increasing n. We can now introduce the real numbers as such 'limit points' for nested interval sequences, in which we use only rational numbers as end points of the intervals. After this has been done, every such nested interval sequence catches exactly one real number, namely the only point in common for all the intervals.

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