# Floor function identity

1. Mar 22, 2013

### eddybob123

2. Mar 22, 2013

### rbj

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.

3. Mar 22, 2013

### eddybob123

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

4. Mar 23, 2013

### micromass

Staff Emeritus
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.

5. Mar 28, 2013

### 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?