What Does the Floor Function Identity Mean?

[eddybob123]

Hi all, I found this rather interesting formula online and I was wondering what it means. Could someone explain it to me? All help is appreciated:
http://functions.wolfram.com/IntegerFunctions/Floor/16/03/0001/

 

[rbj]

Hi all, I found this rather interesting formula online and I was wondering what it means. Could someone explain it to me? All help is appreciated:
http://functions.wolfram.com/IntegerFunctions/Floor/16/03/0001/

the $\theta$ function is the unit step function, so they are creating a staircase function out of the steps (the infinite summation), and that becomes the floor function.

 

[eddybob123]

But what does the z mod 1 mean? I've never seen people use 1 as a modular base before?

 

[micromass]

But what does the z mod 1 mean? I've never seen people use 1 as a modular base before?

The notation $z$ mod 1 means that you take the element $x\in [0,1)$ such that $z-x\in \mathbb{Z}$.

If you wish, you can put an equivalence relation $x\sim y~\Leftrightarrow ~x-y\in \mathbb{Z}$ on $\mathbb{R}$. We can then look at equivalence classes. It won't be the exact same thing as what I said in my first sentence though.

 

[eddybob123]

Are there any formulas that compare any positive real number r with floor[r]? I know that their difference is the fractional part of r, which is {r}, but I mean are there any formulas where you can obtain these values systematically?