Explore the foundations of rational number arithmetic, including operations with infinity, in this 37-minute mathematics lecture. Delve into the systematic introduction of arithmetic operations on integral points of the plane, learn about key arithmetical laws, and discover how to obtain extended rational numbers through the identification of integer points. Gain insights into the uniform and unambiguous introduction of 1/0 into arithmetic, motivated by concepts from projective geometry. Examine the curious role of the 0/0 object and its significance in extended rational arithmetic. Follow along as the lecture covers topics such as integral points, associative and distributive laws, homogeneity laws, and the definition of extended rational numbers. Conclude with a practical exercise to reinforce your understanding of these advanced mathematical concepts.