How do you use floor/ceil in math, e.g. how does it work exactly? I read the Wiki, but I am still somewhat confused. Supposedly floor returns the number lowest,  and ceil the number highest. However, I am not sure I understand how this works in freehand form, or how I may implement this myself in a specific thing. Would someone please clarify how ceil and floor works, and how it is used/examples?  Thanks!!
 A: Simply put it's two ways of thinking of rounding off a number.
Ceiling rounds up to nearest integer.
Floor rounds down to nearest integer.
If the number is an integer, nothing happens.
A: It's easy to think about floor and ceil from the perspective of the number line.
Let's say you have some decimal number, $2.31$ (I'm going to be using this number as an example throughout my answer)
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So, as you can see, the functions just return the nearest integer values.
floor returns the nearest lowest integer and ceil returns the nearest highest integer.
All real numbers are made of a characteristic (an integer part) and mantissa (a fractional part)
$$\text{Number} = \text{Characteristic} + \text{Mantissa}$$
$$2.31 = 2 + 0.31$$
When floor a number, you can think of it as replacing the Mantissa with $0$
$$\lfloor 2.31 \rfloor = 2 + 0 = 2$$
and ceil can be thought of as replacing the mantissa with $1$.
$$\lceil 2.31 \rceil = 2 + 1 = 3$$
That's not a very popular way of thinking about it but it was 
the way I thought about it when I first started using it in programming.
Remember, the number remains the same when it is an integer.
ie, floor($3$) $=$ ceil($3$) $= 3$

Let's now look at the proper definitions along with the graphs for them.
Floor Function: Returns the greatest integer that is less than or equal to $x$
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Ceiling Function: Returns the least integer that is greater than or equal to $x$
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Don't let the infinite staircase scare you. It's much more simpler than it seems.
Those "line-segments" that you see are actually called piecewise-step functions.
Simply, the black dot represents 'including this number' and the white represents 'excluding this number'.
Meaning  that each segment actually is from x to all numbers less than x+1.

Let's look at 2.31 and how it would look on both the graphs at once.

You can see that the line $x=2.31$ hits the floor function at the "line-piece" for $2$ 
and hits the ceiling function at $3$
A: Here's an example of using the floor function:  I have a number $m$ and I am looking for the smallest number $\ell$ such that $2^\ell\gt m+1$.  To find such a number I would use the expression
$$\ell =[\log_2(m+1)]+1$$
Where the square brackets indicate the use of the floor function.
The floor and ceil functions in a programming language might use a combination of truncation (rounding towards zero) and comparisons to actually achieve their result.  For example, if int is a function that truncates a number to its integer part, then for $x\ge0$ we have floor(x)=int(x), but for $x\lt 0$ it would be
y=int(x)
if (y\ne x)
  floor(x)=y-1
else
  floor(x)=y

