3.10 Summary
There’s more to numbers than you might have thought, despite using them daily. Starting with the simplest kind, the natural numbers N, we systematically added operations and properties to gain functionality. The idea of “closure” drove us to understand the value of extending to increasingly 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 see their structure. 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 brings together the number collections we have seen and some of their properties.
The following diagram shows the inclusion relationships among the collections of numbers. The expressions
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