Section 3.1 Elementary properties of the floor function
The floor function has many elementary properties listed in the textbook. Here are a few.
Proposition 3.1.1.
For any real numbers \(x\) and \(y\text{,}\) and integer \(n\text{:}\)
\(\displaystyle \lfloor n + x \rfloor = n + \lfloor x \rfloor \)
\(\displaystyle \lfloor x \rfloor + \lfloor y \rfloor
\leq \lfloor x + y \rfloor \leq \lfloor x \rfloor + \lfloor y \rfloor + 1 \)
\(\displaystyle \lfloor -x \rfloor = - \lceil x \rceil \)
Proof.
For the first statement, \(n + \lfloor x \rfloor\) is an integer which is less than or equal to \(n + x\text{.}\) Since \(\lfloor n + x \rfloor\) is defined as the greatest integer less than or equal to \(n + x\text{,}\) it must be that this integer \(n + \lfloor x \rfloor\) is less than or equal to the greatest integer, so \(n + \lfloor x \rfloor \leq \lfloor n + x \rfloor\text{.}\)
On the other hand, \(\lfloor n + x \rfloor - n\) is an integer which is less than or equal to \((n+x)-n = x\text{.}\) Therefore \(\lfloor n + x \rfloor - n \leq \lfloor x \rfloor\text{.}\) This shows that \(lfloor n + x \rfloor \leq n + \lfloor x \rfloor \text{.}\)
For the second statement, observe that \(\lfloor x \rfloor + \lfloor y \rfloor\) is an integer which is less than or equal to \(x + y\text{.}\) Therefore it is less than or equal to \(\lfloor x + y \rfloor\text{.}\)
As before, \(\lfloor x + y \rfloor - \lfloor x \rfloor\) is an integer which is less than or equal to \(x + y - \lfloor x \rfloor\text{.}\) We have \(x - \lfloor x \rfloor \leq 1\) (in fact, strictly less) (because if \(x - \lfloor x \rfloor \geq 1\) then \(\lfloor x \rfloor + 1 \leq x\text{,}\) contradicting the maximality of \(\lfloor x \rfloor\)). Putting it together we have
\begin{equation*}
\lfloor x + y \rfloor - \lfloor x \rfloor \leq x + y - \lfloor x \rfloor
\leq y + 1\text{.}
\end{equation*}
Therefore
\begin{equation*}
\lfloor x + y \rfloor - \lfloor x \rfloor \leq \lfloor y+1 \rfloor = \lfloor y + \rfloor + 1\text{.}
\end{equation*}
Finally, to see that \(\lfloor -x \rfloor = - \lceil x \rceil\text{,}\) observe that \(m \leq -x\) if and only if \(-m \geq x\text{.}\)
