What Does the Floor Function Identity Mean?

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The discussion centers on understanding the floor function identity and its components, particularly the notation z mod 1. The floor function is created from a staircase function derived from an infinite summation, with the unit step function being a key element. The term z mod 1 indicates the fractional part of z, representing the element in the interval [0,1) that satisfies z - x being an integer. Participants also inquire about systematic formulas relating any positive real number r to its floor value, noting that the difference between r and its floor is the fractional part. Overall, the conversation seeks clarity on the mathematical implications of these concepts.
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eddybob123 said:
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.
 
But what does the z mod 1 mean? I've never seen people use 1 as a modular base before?
 
eddybob123 said:
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.
 
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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