We systematically introduce the four arithmetical operations on integral points of the plane, state some of the main arithmetical laws satisfied, and then show how to obtain the extended rational numbers by suitably identitying integer points, in a similar way to the introduction of rational numbers. Notably this allows a uniform and completely unambiguous introduction of 1/0 into arithmetic.Everything is motivated by projective geometry--the idea that a line through the origin in two dimensional space can be specified by homogenous coordinates, and more or less gives us an extended rational number. However we must pay a price: the admission of the strange new object 0/0, which is separate from all the other extended rationals, and plays a curious but central role. I also suggest a contest: what is a good name for this object 0/0??Dear Viewers-- After pondering our notational challenge, I am leaning towards one of the first suggestions, made by teavea10: that 0/0 by called ``zoz'', short for ``zero over zero''. Can anyone think of any compelling reason to not adopt this somewhat novel but appealing nomenclature??
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.