Lectures On Fundamental Concepts of Algebra And Geometry

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— The real number system of algebra has now been character- ized by two methods. The first, given at the end of Lecture IX, described this number system at one stroke, as it were, by a set of assumptions concerniag the undefined terms, number, <, 4-, X . The other, the discussion of which we have just completed, followed a more historical line of de- velopment, and showed how it was possible to build up this system by successive extensions of the number concept, be- ginning with that of the positive integer. It could now be LECTURE XI 115 shown without serious difficulty that the two characteriza- tions of the real number system which have been given are abstractly equivalent. Bearing on Elementary Teaching. — It will be well now to ask ourselves what bearing this discussion has on the prob- lems of elementary teaching. Should either of these meth- ods be applied in making the pupil, at the first approach to the study of algebra, familiar with its conceptions and its processes ? Should we begin by placing before him a set of assumptions characterizing real algebra, or should we at- tempt to make clear to him the notions involved by succes- sively extending the number system in the way described above ?

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