Mathematical representation of Floor( ) and Ceil( ) for various decimal places. I know the basic rules for Floor( ) and Ceil( ):
$$
\text{Floor}(2.5) = 2\\
\text{Ceil}(2.5) = 3
$$
Is there a way to denote the Floor( ) and Ceil( ) function for various decimal places? For example, I know that one way is to write the following for two decimal places:
$$
\frac{\text{Floor}(2.4783 \times 100)}{100} = 2.47
$$
but I was wondering if there was a way to write it without the fractions. Thank you so much!
 A: Your function scales both input $x$ and output $y$ up by a factor of $100$:
$$
\lfloor 100 x \rfloor = 100y, 
$$
i.e. if we define these scaled coordinates $X = 100x$ and $Y = 100y$, then the equation relating inputs to outputs looks like
$$
\lfloor X \rfloor = Y, 
$$
which you can think of the prototype of the relationship between the variables. Putting all the transformations in one diagram looks like
$$
x \to X \to Y \leftarrow y, 
$$
so the only way to build the composition $x \to y$ is to invert that last arrow $Y \leftarrow y$ to produce $Y \to y$, namely to divide by the scaling factor, hence we introduce fractions.
As always, we can hide the fraction inside of a definition, which doesn't change the fact that we're dividing by the scaling factor, but cosmetically, it might look nicer. We could, for instance, for any $p > 0$, define a rounding function with precision $\frac1p$ by
$$
\operatorname{floor}_p(x) 
= \frac1p \lfloor p x \rfloor
$$
that has the property that $\operatorname{floor}_p(x) = y$ for all $y \leq x \leq y + \frac1p$. With this notation, your example would look like
$$
\operatorname{floor}_{100}(2.4783) = 2.47.
$$
A: You are asking about truncation.
In the linked Wikipedia article, the notation is as follows.
Given a number $ x\in \mathbb {R}_+ $  to be truncated
and $n\in \mathbb {N} _{0}$, the number of digits to be kept after the decimal point,
the truncated value of $x $ is
$\operatorname {trunc} (x,n)={\dfrac {\lfloor 10^{n}\cdot x\rfloor }{10^{n}}}$.
