3.6 Structure

I took time to show the operations and the properties of R and its subsets such as Z and Q because these are very common in other parts of mathematics when abstracted. This structure allows us to learn and prove things and then apply them to new mathematical collections as we encounter them. We start with three: groups, rings, and fields.

These will come into play when we consider modular arithmetic in section 3.7, complex numbers in section 3.9, and vector spaces, linear transformations, and matrices in Chapter 5, “Dimensions.”

3.6.1 Groups

Consider a collection of objects we call G. For example, G might be Z, Q, or R, as above. We also have some pairwise operation between elements of G that we denote by “★”. This is a placeholder for an action that operates on two objects. group

This “★” operation could be addition “+” or multiplication “...