The dual numbers, in my opinion, are a lot easier to understand with some background in differential geometry. In differential geometry, if I_x is the ideal of

The dual numbers, in my opinion, are a lot easier to understand with some background in differential geometry. In differential geometry, if I_x is the ideal of smooth functions vanishing at the point x in the ring of smooth functions, then I_x / I_x^2 is a real vector space called the cotangent space (these elements are given the suggestive dx moniker; the “infinitely short line” you mentioned above is just a modern interpretation of the infinitesimal from calculus) and dual of this vector space is the space of tangent vectors at this point. Another way to think about R[X,Y]/(X^2) dropping all the non-linear terms for X to create a flat (co)tangent space. Also the original motivation for sheaves was about creating a way to deal with multi-valued complex function. The complex log function is multi-valued so in intro complex analysis it’s studied locally by choosing a branch of the range where it’s singular valued. Thus it’s impossible to “do differential geometry” by talking about a global ring of analytic functions. But you can talk about the “local ring of analytic functions” at a point and specific branch and glue these locally ringed spaces together to get global insight.