Floor, Ceiling & Fractional Part Functions Cheat Sheet

Key Facts

• The floor function is defined by $\lfloor x \rfloor=n$ when $n \le x < n+1$ for an integer $n$.
• The ceiling function is defined by $\lceil x \rceil=n$ when $n-1 < x \le n$ for an integer $n$.
• For every real number $x$, $\lfloor x \rfloor \le x \le \lceil x \rceil$.
• If $x$ is an integer, then $\lfloor x \rfloor=\lceil x \rceil=x$.
• The fractional part of $x$ is $\{x\}=x-\lfloor x \rfloor$, so $0 \le \{x\}<1$.
• Every real number can be written as $x=\lfloor x \rfloor+\{x\}$.
• For any integer $k$, $\lfloor x+k \rfloor=\lfloor x \rfloor+k$ and $\lceil x+k \rceil=\lceil x \rceil+k$.
• Floor and ceiling are related by $\lceil x \rceil=-\lfloor -x \rfloor$ and $\lfloor x \rfloor=-\lceil -x \rceil$.

Vocabulary

Floor Function

The floor function $\lfloor x \rfloor$ gives the greatest integer less than or equal to $x$.

Ceiling Function

The ceiling function $\lceil x \rceil$ gives the least integer greater than or equal to $x$.

Fractional Part

The fractional part $\{x\}$ is the amount left after subtracting $\lfloor x \rfloor$ from $x$.

Step Function

A step function is a graph made of horizontal pieces that jump at certain input values.

Integer

An integer is a whole number, its opposite, or zero, such as $-3$, $0$, or $5$.

Endpoint

An endpoint is the boundary value of an interval, often shown with a closed dot if included and an open dot if not included.