Explore the rigorous extension of arithmetic with rational numbers to include infinity in this 32-minute mathematics lecture. Investigate how to carefully expand the domain to incorporate the symbol 1/0, represented by a "sideways 8". Begin with a simpler case of natural number arithmetic before delving into the more complex rational number scenario. Connect arithmetic to plane geometry and projective geometry of lines through the origin. Focus on integral points [a,b] in the plane to define suitable notions of rational addition and multiplication. Learn how the point [1,0] plays a role analogous to infinity and discover connections to the Stern-Brocot tree. Progress through topics including projective geometry, integral points, theorems, and distributive laws, with exercises interspersed to reinforce understanding.