Undirected value: Difference between revisions
Cmloegcmluin (talk | contribs) →Alternative: typo, correction, clarification, and preference showcase |
Sintel bot (talk | contribs) Fix <math> environments for MathJax v4 (align->aligned, etc.) |
||
| (8 intermediate revisions by one other user not shown) | |||
| Line 41: | Line 41: | ||
This alternative is designed to evoke the infix ":" operator which as previously described is the one typically used for undirected ratios. | This alternative is designed to evoke the infix ":" operator which as previously described is the one typically used for undirected ratios. | ||
== | ==Definitions== | ||
=== As real === | === As real === | ||
| Line 52: | Line 52: | ||
<math> | <math> | ||
\overline{\underline{x}} = | \overline{\underline{x}} = | ||
\begin{cases} | \begin{cases} | ||
x & \text{if} \; |x| \ge 1 \\ | x & \text{if} \; |x| \ge 1 \\ | ||
\frac{1}{x} & \text{if} \; |x| < 1 \\ | \frac{1}{x} & \text{if} \; |x| < 1 \\ | ||
\end{cases} | \end{cases} | ||
</math> | </math> | ||
| Line 82: | Line 80: | ||
A '''subunison''' number, by extension, is a real number whose absolute value is less than 1. | A '''subunison''' number, by extension, is a real number whose absolute value is less than 1. | ||
{| class="wikitable" | {| class="wikitable" | ||
| Line 89: | Line 85: | ||
! | ! | ||
!superunison | !superunison | ||
! | !undirected | ||
!subunison | !subunison | ||
|- | |- | ||
| Line 97: | Line 93: | ||
|<math>\frac12</math> | |<math>\frac12</math> | ||
|- | |- | ||
! | !unsigned | ||
| | |<math>\infty</math> | ||
|n/a | |n/a | ||
|0 | |0 | ||
| Line 177: | Line 173: | ||
|} | |} | ||
The following identity shows the relationship between the undirected value and the absolute value, for positive real numbers. | |||
<math> | |||
\overline{\underline{x}} = b^{|log_{b}{x}|} \;\; \text{for any base} \; b>1 \; \text{and} \; x>0 \\ | |||
</math> | |||
== Graphs == | |||
[[File:Und x.png|300px|frame|left|plot of the undirected value of x]] | |||
[[File:Abs x.png|300px|frame|left|plot of the absolute value of x, for comparison]] | |||
<br clear=all> | |||
== History == | == History == | ||