How to represent the floor function using mathematical notation?

I'm curious as to how the floor function can be defined using mathematical notation.

What I mean by this, is, instead of a word-based explanation (i.e. "The closest integer that is not greater than x"), I'm curious to see the mathematical equivalent of the definition, if that is even possible.

For example, a word-based explanation of the factorial function would be "multiply the number by all integers below it", whereas a mathematical equivalent of the definition would be $n! = \prod^n_{i=1}i$.

So, in summary: how can I show what $\lfloor x \rfloor$ means without words?

Solution 1:

For a real number $x$, $$\lfloor x\rfloor=\max\{n\in\mathbb{Z}\mid n\leq x\}.$$ I'd like to add though, that "$\lfloor x\rfloor$" is mathematical notation, just as much as the right side of the above equation is; the right side might use more "basic" constructions, but you can then ask about $$\max,\qquad\in,\qquad \mathbb{Z},\qquad {}\mathbin{\mid}{},\qquad \leq$$ and so on. At some point you just have to start writing notation and explaining it in words and hope your readers understand. So I disagree with your phrasing of the question.

Solution 2:

$\lfloor x \rfloor = x - \arctan(\tan(\pi x))/\pi$ ?...

Solution 3:

I am an engineer, perhaps this helps if you do not expect too much.

$x-(x$ mod $1$)

Solution 4:

Recursive way: $\lfloor x\rfloor \equiv \begin{cases} 1+\lfloor x-1\rfloor& x\ge1 \\ -1+\lfloor x+1\rfloor& x<-1 \\ 0&0\le x<1\\ -1&-1\le x<0 \end{cases}$

Also directly(sort of) computes the answer.