\[ \begin{array}{r|r} \lfloor 1.5 \rfloor = \answer {1} & \lceil 1.5 \rceil = \answer {2} \\ \lfloor -1.5 \rfloor = \answer {-2} & \lceil -1.5 \rceil = \answer {-1} \\ \lfloor \sqrt {5} \rfloor = \answer {2} & \lceil \sqrt {5} \rceil = \answer {3} \\ \lfloor -\sqrt {5} \rfloor = \answer {-3} & \lceil -\sqrt {5} \rceil = \answer {-2} \\ \lfloor e \rfloor = \answer {2} & \lceil e \rceil = \answer {3} \\ \lfloor 7 \rfloor = \answer {7} & \lceil 7 \rceil = \answer {7} \\ \lfloor \frac {\pi }{3} \rfloor = \answer {1} & \lceil \frac {\pi }{3} \rceil = \answer {2} \\ \lfloor -4 \rfloor = \answer {-4} & \lceil -4 \rceil = \answer {-4} \end{array} \]
The study of the real numbers often separates into knowledge about the whole part of a number and the fractional part. This generates the idea of the floor and ceiling functions.
Floor
The floor function maps the real numbers onto the integers.
\[ \lfloor \, \rfloor : \mathbb {R} \mapsto \mathbb {Z} \]
\[ \lfloor r \rfloor = N \text {, where } N \in \mathbb {Z} \text { and } N \leq r < N+1 \]
The floor function returns the greatest integer less than or equal to the argument.
Ceiling
The ceiling function maps the real numbers onto the integers.
\[ \lceil \, \rceil : \mathbb {R} \mapsto \mathbb {Z} \]
\[ \lceil r \rceil = N \text {, where } N \in \mathbb {Z} \text { and } N-1 < r \leq N \]
The ceiling function returns the least integer greater than or equal to the argument.
Evaluating
2025-05-17 18:25:19