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
\newcommand{\Floor}[1]{ {\text{Floor}{\left({#1}\right)} } }
Contents
[hide]Definition
For x\in\mathbb{R}_{\ge 0} there is no variation on the meaning of the floor function, however for negative numbers there are varying conventions.
Non-negative
Defined as follows:
- \text{Floor}:\mathbb{R}_{\ge 0}\rightarrow\mathbb{N}_0 by \text{Floor}:x\mapsto\text{Max} (T_x) where T_x:\eq\big\{n\in\mathbb{N}_0\ \big\vert\ n\le x\big\}\subseteq\mathbb{N}_0\subseteq\mathbb{R}_{\ge 0} - note that the maximum element is defined as T_x is always finite.
- This has the property that x\le\Floor{x} .
Negative numbers
Researching this opened my eyes to a massive dispute.... consensus seems to be that x\le \Floor{x} is maintained, rounding is a separate and massive issue!
References
Future work
Properties
- \forall n\in\mathbb{N}_0\subseteq\mathbb{R}_{\ge 0}\big[\Floor{n}\eq n\big], or \text{Floor}\vert_{\mathbb{N}_0}\eq\text{Id}_{\mathbb{N}_0} - its restriction to \mathbb{N}_0 is the identity map on \mathbb{N}_0
- \forall x,y\in\mathbb{R}_{\ge 0}\big[(x\le y)\implies\big(\Floor{x}\le\Floor{y}\big)\big] - monotonicity
- \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 3\implies 1 and 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
