Exercise 3.1.1 (Integral part of $x$)

Question

Find $\lfloor 3/2 \rfloor, \lfloor -3/2 \rfloor, \lfloor \pi \rfloor, \lfloor -7 \rfloor$, and $\lfloor x \rfloor$ for $0 \leq x < 1$.

Richard Ganaye · Accepted Answer

ewcommand{\Z}{\mathbb{Z}}

\begin{proof} By definition, for all real numbers $x, n$,
$$
n = \lfloor x floor \iff
\left\{
\begin{array}{l}
n \in \Z,\
n \leq x < n + 1.
\end{array}
ight.
$$
\begin{itemize}
\item $1 \leq \frac{3}{2} < 2$, thus $\left \lfloor\frac{3}{2}  ight floor = 1$.
\item $-2 \leq -\frac{3}{2} < -1$, thus $\left \lfloor -\frac{3}{2}  ight floor = -2$.
\item $ 3 \leq \pi< 4$, thus $\left \lfloor\pi ight floor = 3$.
\item $ -7 \leq -7< -6$, thus $\left \lfloor -7 ight floor = -7$.
\item If $ 0\leq x< 1$, then $\left \lfloor x ight floor = 0$.
\end{itemize}
\end{proof}

Exercise 3.1.1 (Integral part of $x$)

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Exercise 3.1.1 (Integral part of $x$)

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