The Journal of the Indian Mathematical Society

B. S. Madhavarao ¹

Affiliations
1 University of Mysore, IN

Abstract

The arithmetical theory of irrational numbers has been developed in three main forms, of which the first was given by Weierstrass in his Lectures on Analytical Functions, the second is that of Cantor and the third that due to Dedekind. Of these three theories, it has been shown that those of Cantor and Dedekind are fundamentally identical and it has been established that whereas the theory of Dedekind operates with the whole aggregate of rational numbers, the other operates with sequences selected out of that aggregate. We here propose to establish the fundamental identity of all the three theories of irrational number by establishing the same for the theories of Weierstrass and Cantor and to show that the entities used in the former definition are but particular types of the entities used in the latter.

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B. S. Madhavarao , M. Lakshmanamurthi

Abstract

Every non-singular cubic can be reduced to the canonical form

x³ + y³ + z³ + 6mxyz = 0;

and if three points (xi, yi, zi) (i = 1,2,3) on this cubic be collinear, we have the well-known condition due to Cayley, viz.

x₁x₂x₃ + y₁y₂y₃ + z₁z₂z₃ = 0.

We have shown here that this condition is necessary but not sufficient.

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