Floor function

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Research consensus and handling negative numbers

[ilmath]\newcommand{\Floor}[1]{ {\text{Floor}{\left({#1}\right)} } } [/ilmath]

Definition

For [ilmath]x\in\mathbb{R}_{\ge 0} [/ilmath] there is no variation on the meaning of the floor function, however for negative numbers there are varying conventions.

Non-negative

Defined as follows:

• [ilmath]\text{Floor}:\mathbb{R}_{\ge 0}\rightarrow\mathbb{N}_0[/ilmath] by [ilmath]\text{Floor}:x\mapsto[/ilmath][ilmath]\text{Max} [/ilmath][ilmath](T_x)[/ilmath] where [ilmath]T_x:\eq\big\{n\in\mathbb{N}_0\ \big\vert\ n\le x\big\}\subseteq\mathbb{N}_0\subseteq\mathbb{R}_{\ge 0} [/ilmath] - note that the maximum element is defined as [ilmath]T_x[/ilmath] is always finite.
• This has the property that [ilmath]x\le\Floor{x} [/ilmath].

Negative numbers

Researching this opened my eyes to a massive dispute.... consensus seems to be that [ilmath]x\le \Floor{x} [/ilmath] is maintained, rounding is a separate and massive issue!

Future work

Properties

1. [ilmath]\forall n\in\mathbb{N}_0\subseteq\mathbb{R}_{\ge 0}\big[\Floor{n}\eq n\big][/ilmath], or [ilmath]\text{Floor}\vert_{\mathbb{N}_0}\eq\text{Id}_{\mathbb{N}_0} [/ilmath] - its restriction to [ilmath]\mathbb{N}_0[/ilmath] is the identity map on [ilmath]\mathbb{N}_0[/ilmath]
2. [ilmath]\forall x,y\in\mathbb{R}_{\ge 0}\big[(x\le y)\implies\big(\Floor{x}\le\Floor{y}\big)\big][/ilmath] - monotonicity
3. [ilmath]\forall x\in\mathbb{R}_{\ge 0}\exists\epsilon\in[0,1)\subseteq\mathbb{R}\big[x\eq\Floor{x}+\epsilon\big][/ilmath] - the characteristic property of the floor function

I believe that [ilmath]3\implies 1[/ilmath] and [ilmath]3\implies 2[/ilmath] might be possible, so these are perhaps in the wrong order. I just wanted to write down some notes before they get put into the massive stack of unfiled paper