# Floor function identity

by eddybob123
Tags: floor, function, identity
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 P: 115 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/Integer...or/16/03/0001/
P: 2,251
 Quote by 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/Integer...or/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.
 P: 115 But what does the z mod 1 mean? I've never seen people use 1 as a modular base before?
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Floor function identity

 Quote by eddybob123 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.
 P: 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?

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