Algebraic number: Difference between revisions

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<h2>~~IMPORTED REVISION FROM WIKISPACES~~</h2>

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:~~<br>

A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] {{nowrap|''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>''n'' − 1</sup>}} + … {{nowrap|+ ''a''<sub>''n''</sub>}} whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as '''algebraic numbers'''. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions|polynomial function]] {{nowrap|''f''(''x'') {{=}} ''a''<sub>0</sub>''x''<sup>''n''</sup>}} {{nowrap|+ ''a''<sub>1</sub>''x''<sup>''n'' − 1</sup>}} + … {{nowrap|+ ''a''<sub>''n''</sub>}} satisfies {{nowrap|''f''(''r'') {{=}} 0}}. If ''r'' is a {{w|real number}}, it is a ''real algebraic number''.

~~: This revision was by author [[User:xenwolf|xenwolf]] and made on~~ <tt>~~2010-11-15 04:23:45 UTC~~<~~/tt~~>.<br>

~~: The original revision id was~~ <tt>~~179525169~~</tt>.<br>

~~: The revision comment was:~~ <tt><~~/tt~~><br>

~~The revision contents~~ are ~~below~~, ~~presented both in the original Wikispaces Wikitext format, and in HTML exactly~~ as ~~Wikispaces rendered it~~.<br>

<h4>~~Original Wikitext content:~~</h4>

<~~div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"~~><~~pre style="margin:0px~~;~~border:none~~;~~background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html"~~>~~After the [[http:~~/~~/en.wikipedia.org/wiki/Fundamental_theorem_of_algebra~~|~~fundamental theorem of algebra]] each [[http://en~~.~~wikipedia.org/wiki/Polynomial~~|~~polynomial]] has~~ a number ~~of (complex) [[http://en.wikipedia.org/wiki/Root_of_a_function|roots]] equal to its degree~~.

Real algebraic numbers which are not also rational numbers turn up in various places in musical tuning theory. For instance, the intervals in the [[Target_tunings|target tunings]] minimax or least squares are real algebraic numbers. An example of this is 1/4-comma meantone, where 2 and 5 are not tempered but 3 is tempered to 2×5<sup>1/4</sup>, a root of {{nowrap|''x''<sup>4</sup> − 80}}. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the {{w|combination tone|combination tones}} they produce. Algebraic numbers are also relevant to JI-agnostic [[delta-rational]] harmony, as tunings of [[mos scale]]s with exact delta-rational values for a certain chord have generators that are algebraic numbers in the linear frequency domain.

~~One~~ method ~~to find these roots~~ is the [[~~http~~:~~//en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method|Durand–Kerner method~~]]~~.</pre></div>~~

Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as {{w|Newton's method}} can be used. A refinement of Newton's method is the {{w|Durand–Kerner method}}.

~~<h4>Original HTML content~~:~~</h4>~~

~~<div style="width~~:~~100%; max-height~~:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Algebraic number</title></head><body>After the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra" rel="nofollow">fundamental theorem of algebra</a> each <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Polynomial" rel="nofollow">polynomial</a> has a number of (complex) <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Root_of_a_function" rel="nofollow">roots</a> equal to its degree.<br />

[[Category:Math]]

~~<br />~~

[[Category:Number theory]]

~~<br />~~

[[Category:todo:increase applicability]]

One method to find these roots is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method" rel="nofollow">Durand–Kerner method</a>.</body></html></pre></div>

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-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Wikipedia}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] {{nowrap|''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>''n'' &minus; 1</sup>}} +&nbsp;… {{nowrap|+ ''a''<sub>''n''</sub>}} whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as '''algebraic numbers'''. A root is a value ''r'' for which the [[Wikipedia: Polynomial  #Polynomial functions|polynomial function]] {{nowrap|''f''(''x'') {{=}} ''a''<sub>0</sub>''x''<sup>''n''</sup>}} {{nowrap|+ ''a''<sub>1</sub>''x''<sup>''n'' &minus; 1</sup>}} +&nbsp;… {{nowrap|+ ''a''<sub>''n''</sub>}} satisfies {{nowrap|''f''(''r'') {{=}} 0}}. If ''r'' is a {{w|real number}}, it is a ''real algebraic number''.
-: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-11-15 04:23:45 UTC</tt>.<br>
-: The original revision id was <tt>179525169</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">After the [[http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra|fundamental theorem of algebra]] each [[http://en.wikipedia.org/wiki/Polynomial|polynomial]] has a number of (complex) [[http://en.wikipedia.org/wiki/Root_of_a_function|roots]] equal to its degree.
+Real algebraic numbers which are not also rational numbers turn up in various places in musical tuning theory. For instance, the intervals in the [[Target_tunings|target tunings]] minimax or least squares are real algebraic numbers. An example of this is 1/4-comma meantone, where 2 and 5 are not tempered but 3 is tempered to 2×5<sup>1/4</sup>, a root of {{nowrap|''x''<sup>4</sup> &minus; 80}}. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the {{w|combination tone|combination tones}} they produce. Algebraic numbers are also relevant to JI-agnostic [[delta-rational]] harmony, as tunings of [[mos scale]]s with exact delta-rational values for a certain chord have generators that are algebraic numbers in the linear frequency domain.
-One method to find these roots is the [[http://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method|Durand–Kerner method]].</pre></div>
+Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as {{w|Newton's method}} can be used. A refinement of Newton's method is the {{w|Durand–Kerner method}}.
-<h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Algebraic number&lt;/title&gt;&lt;/head&gt;&lt;body&gt;After the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra" rel="nofollow"&gt;fundamental theorem of algebra&lt;/a&gt; each &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Polynomial" rel="nofollow"&gt;polynomial&lt;/a&gt; has a number of (complex) &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Root_of_a_function" rel="nofollow"&gt;roots&lt;/a&gt; equal to its degree.&lt;br /&gt;
+[[Category:Math]]
-&lt;br /&gt;
+[[Category:Number theory]]
-&lt;br /&gt;
+[[Category:todo:increase applicability]]
-One method to find these roots is the &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method" rel="nofollow"&gt;Durand–Kerner method&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>