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!