3.10 Summary

There’s more to numbers than you might have thought as you’ve used them in your daily life. Starting with the simplest kind, the natural numbers N, we systematically added operations and properties to gain functionality. The idea of ‘‘closure’’ was important in driving us to understand the value of extending to larger collections of numbers that could handle the problems we wanted to solve.

We briefly delved into abstract algebra to look at groups, rings, and fields and to see the common structure they provide. The complex numbers are key to working with quantum computing and we began to look at their algebraic properties. Though they involve the imaginary i, they are very real in describing the way the universe evidently works.

The following table and diagram bring together all the forms of numbers we have seen and some of their properties. The diagram shows the inclusion relationships among the collections of numbers...