I've added a PDF version of the picture above to the Graphics page and if you're like most people, it needs an explanation. The curriculum I'm covering, along with the supporting book, talks only about the greatest integer function which is denoted by the book as . I find that annoying for several reasons. First, it isn't that clear from the name what the function does. Wikipedia mention that a similar looking symbol was used by Gauss and was the standard until the floor and ceiling functions were introduced in 1962. Secondly, the notation doesn't give any indication what the function does. So when you follow the book all sorts of needless confusion follows because the notation and naming aren't really intuitive. Compare that with the floor and ceiling functions with notation and . The little "feet" at the bottom of the floor function are suggestive of rounding the values down and the "feet" at the top of the ceiling function suggest rounding the values up. The floor function is the one that occurs most naturally in the curriculum, such as in determining the number of integers less than 1,000,000 that are divisible by either 5 or 2 (using the inclusion/exclusion) but learning about just the greatest integer function causes confusion because the students aren't sure (poor naming) about how to plot it combined with where is the open circle and where is the closed circle. Students should learn about the floor and ceiling functions at the same time.

The diagram helps to make sense of the floor function and ceiling function and the relationship between them and the line . All three functions have been plotted together and from that diagram you can see:

- if and only if is an integer.
- if and only if is an integer and if and only if is an integer.
- when n is an integer then the intervals have the ceiling function above the floor function; in fact, .
- By visualizing the graph (or using as a guide) the student can construct floor and ceiling function more easily and will remember which sides gets the open and closed circles.

The drawback in changing the notation and introducing the more standard notation are students complaining that the book doesn't cover the material so they shouldn't be responsible for it. Teaching to US students has been quite an experience. It would be great if the makers of math textbooks would teach floor functions and ceiling functions rather than the greatest integer function. Half a century behind--time to change! It will make the material easier to learn and take away student excuses.