Floor function identity

In summary, the discussion is about the use of the unit step function and the floor function, with the equation involving the modular base of 1. The concept of equivalence classes is also mentioned. The question is posed about formulas that compare a positive real number with its floor value.
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  • #2
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 [itex]\theta[/itex] 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
But what does the z mod 1 mean? I've never seen people use 1 as a modular base before?
 
  • #4
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.
 
  • #5
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?
 

What is the floor function identity?

The floor function identity is a mathematical concept that represents the largest integer less than or equal to a given number. It is denoted by the symbol ⌊x⌋ and is also known as the greatest integer function.

What is the difference between the floor function identity and the ceiling function identity?

The floor function identity rounds a number down to the nearest integer, while the ceiling function identity rounds a number up to the nearest integer. For example, ⌊4.5⌋ = 4 and ⌈4.5⌉ = 5.

How is the floor function identity used in real life?

The floor function identity is often used in computer programming to round numbers down to the nearest integer. It is also used in various areas of mathematics, such as number theory and calculus.

What are the properties of the floor function identity?

The floor function identity has the following properties:

  • If x is an integer, then ⌊x⌋ = x
  • If x is a negative number, then ⌊x⌋ is the next smaller integer
  • If x is a positive number, then ⌊x⌋ is the greatest integer less than x
  • If x is not an integer, then ⌊x⌋ < x < ⌊x⌋ + 1

How is the floor function identity related to the modulo operation?

The floor function identity is closely related to the modulo operation, which calculates the remainder of a division. The expression x mod y is equivalent to x - ⌊x/y⌋ * y, where ⌊x/y⌋ is the floor function identity. This relationship is often used in computer programming to calculate the remainder of a division.

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