ABSTRACT
The concept of a function is taken for granted and many readers will likely be surprised that the idea doesn’t go back to ancient Greece. In fact, the Greeks based their mathematics largely on ratios instead, and the functional view really is a modern perspective. We can think of nearly all of mathematics as functions acting on sets. What exactly those functions and sets are varies depending on the kind of mathematics being examined, but the approach holds. Something that oscillates may be modeled by a function like sinx or cosx. Examples include a child being pushed on a swing or a buoy in a pool moving under the action of a wave machine. The floor and ceiling functions often come in handy for expressing other mathematical functions and concepts. For example, the Dirichlet Pigeon-hole principle, can be stated more efficiently using the ceiling function, like so.
