How do I remember floor and ceiling?

Imagine a vertical number line, and put x at a point in the middle. Then the next point above and below x are the “caps” for x, ceiling and floor (respectively).

^                ^
|                |
+ ceil(x)        + 6
|                |
+ x              . 5.7
|                |
+ floor(x)       + 5
|                |
v                v

This makes more sense with real numbers (decimals), when x is desired to be an integer. For example, merge sort partitions the array and we need integers for the indices. You can’t get the 5.5th index of a number, although you might get that dividing an 11-element array.