Rational number arithmetic with infinity and more | Real numbers and limits Math Foundations 104

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??

Video Content:
00:00 Intro to arithmetic operations
3:26 Integral points
6:40 Laws involving addition and multiplication
9:40 Associative laws
11:35 More distributive laws
15:02 Special points
18:08 Homogeneity laws
20:44 Extended rational numbers
28:03 Definition of an extended rational number
32:00 Exercise 104.4

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