# Floor function identity

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2. ### 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. ### eddybob123

115
But what does the z mod 1 mean? I've never seen people use 1 as a modular base before?

4. ### micromass

18,900
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. ### eddybob123

115
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?