EUCLID’S ALGORITHM FOR MAGNITUDES: THE GEOMETRIC VIEWPOINT

Author (s) : V.SARANYA

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Abstract :
The Geometric Viewpoint of Euclid’s Elements, Euclid develops the notion of commensurate and incommensurate magnitudes. A magnitude is an abstract entity which measures a geometric object, and, in practice, may refer to length, area of volume. Topics in Number Theory, Algebra, and Geometry For definition, let us focus on line segments in Euclidean geometry. There is the notion of congruence of two segments. A segment x is greater than a segment y if there are points on x which mark off a segment congruent to y. A segment x measures a segment y if a whole number of copies of x form a segment congruent to y. The segments x and y are commensurate if they have a common measure. If equal magnitudes are obtained at some stage then this is the greatest common measure of the original pair of magnitudes. If equal magnitudes are never obtained then the original magnitudes are incommensurable. Though structurally identical to the procedure for finding the greatest common divisor, the content is quite different, and, furthermore, it leads to the discovery of incommensurable magnitudes. This is the geometric basis for introducing irrational numbers; they are ratios of incommensurable magnitudes. It should be kept in mind that a segment is not a magnitude. Rather, an equivalence class of all segments congruent to each other is a magnitude.

Keywords : Geometric Viewpoint, Euclid’s Algorithm

Conference Date : 17/09/2021

Pages : 100

Paper ID : IESMDT98