4.3 Lower and upper integer part

1 Introduction 2 Mathematics is not only about computing 3 Basic concepts 4 Elementary functions 4.1 What is it a function? 4.2 The absolute value 4.3 Lower and upper integer part 4.4 Linear function 4.5 Quadratic function 4.6 Polynomial function 4.7 Roots 4.8 Rational function 4.9 Trigonometric functions 4.10 Exponantiation and logarithm 5 Analytical geometry 6 Warning 7 List of the used symbols Index Bibliography

Other frequently used and useful functions are the lower integer part and the upper integer part of a real number.

The lower integer part of a real number $x$ is defined as the gratest integer number which is smaller than or equal to $x$ and we denote it as $\lfloor x \rfloor$. Similarly, the upper integer part of a real number $x$ is defined as the smallest integer number that is greater than or equal to $x$ and we denote it as $\lceil x \rceil$. Thus, we could explicitly write (we just express symbolically what we have written in the previous sentences):

\begin{equation*} \begin{aligned} \lfloor x \rfloor & \href{The entry on the right repeats what was written in the previous paragraph by using only symbols. Here we have the set of all integers $m$, that are smaller than or equal to $x$ and we take the largest element (maximum) of such a set.}{\class{mathpopup bg-info-subtle}{=}} \max \{ m \in \Z \mid m \leq x \}, \\ \lceil x \rceil & \href{The entry on the right repeats what was written in the previous paragraph by using only symbols. Here we have the set of all integers $m$, which are grster than or equal to $x$ and we take the smallest set of such a set.}{\class{mathpopup bg-info-subtle}{=}} \min \{ m \in \Z \mid m \geq x \}. \end{aligned} \end{equation*}

The domain of the upper and lower integer part is the whole real line $\mathbb{R}$. Trivially, according to its definition, we are able to costruct $\lfloor x \rfloor$, resp. $\lceil x \rceil$, for every real number $x$. The set of values of these functions is then given by the set of all the integers, i.e. $\mathbb{Z}$. The graphs of the upper and lower integer parts are shown in the figure 4.2.

Figure 4.2: Graph of the lower (above) and upper (below) integer part.

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