/library/oar/handle/123456789/23566 2025-11-08T20:12:10Z /library/oar/handle/123456789/24482 Title: Two simple proofs Abstract: My proofs of two simple results will be presented. These are: a) 0.999999 ... = 1 b) The area of a circle is Pi*R*squared where R is the radius of the circle. 2000-01-01T00:00:00Z /library/oar/handle/123456789/24481 Title: Playing with nines Abstract: In his letter which appeared in the issue of the Times of the 28th January, 1999, Mr. M. Pace pointed out an interesting property of the number nine in the set of integers modulo ten. In general, when we work on the scale of n (where n is 3 or more) the number n-1 exhibits the same properties. So, the number 15 shows these properties in the hexadecimal scale. So does seven in the integers modulo 8. 2000-01-01T00:00:00Z /library/oar/handle/123456789/24480 Title: Intersecting circles and ellipses Abstract: In the February 2000 issue of The Collection, Phaedra Cassar posed the following the question: Is it possible for four or more circles to be drawn such that each new circle added has a common region with all existing regions? We prove that this is not possible. Furthermore we show that for ellipses, the task is not possible for five or more ellipses. 2000-01-01T00:00:00Z /library/oar/handle/123456789/24475 Title: Infinite sets Abstract: A countable union of countable sets is countable. The set of rational numbers, Q, is countable. 2000-01-01T00:00:00Z