Difference between revisions of "Floor function"
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| + | ===Properties=== | ||
| + | # {{M|\forall n\in\mathbb{N}_0\subseteq\mathbb{R}_{\ge 0}\big[\Floor{n}\eq n\big]}}, or {{M|\text{Floor}\vert_{\mathbb{N}_0}\eq\text{Id}_{\mathbb{N}_0} }} - its {{link|restriction|function}} to {{M|\mathbb{N}_0}} is the [[identity map]] on {{M|\mathbb{N}_0}} | ||
| + | # {{M|\forall x,y\in\mathbb{R}_{\ge 0}\big[(x\le y)\implies\big(\Floor{x}\le\Floor{y}\big)\big]}} - [[monotonic function|monotonicity]] | ||
| + | # {{M|\forall x\in\mathbb{R}_{\ge 0}\exists\epsilon\in[0,1)\subseteq\mathbb{R}\big[x\eq\Floor{x}+\epsilon\big]}} - the ''[[characteristic property]]'' of the floor function | ||
| + | I believe that {{M|3\implies 1}} and {{M|3\implies 2}} 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 | ||
{{Definition|Analysis|Real Analysis}} | {{Definition|Analysis|Real Analysis}} | ||
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Research consensus and handling negative numbers
[ilmath]\newcommand{\Floor}[1]{ {\text{Floor}{\left({#1}\right)} } } [/ilmath]
Contents
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!
References
Future work
Properties
- [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]
- [ilmath]\forall x,y\in\mathbb{R}_{\ge 0}\big[(x\le y)\implies\big(\Floor{x}\le\Floor{y}\big)\big][/ilmath] - monotonicity
- [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
