Algebraic number: Difference between revisions
Wikispaces>genewardsmith **Imported revision 181280597 - Original comment: ** |
Wikispaces>xenwolf **Imported revision 181486479 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2010-11-21 08:16:05 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>181486479</tt>.<br> | ||
: The revision comment was: <tt></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> | 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> | <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">A [[http://mathworld.wolfram.com/UnivariatePolynomial.html|univariate polynomial]] a0x^n + a1x^(n-1) + ... + an whose coefficients ai are integers (or equivalently, rational numbers) has | <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">A [[http://mathworld.wolfram.com/UnivariatePolynomial.html|univariate polynomial]] a0x^n + a1x^(n-1) + ... + an whose coefficients ai are integers (or equivalently, rational numbers) has **roots** which are known as **[[http://en.wikipedia.org/wiki/Algebraic_number|algebraic numbers]]**. A root is a value r for which the [[http://en.wikipedia.org/wiki/Polynomial_function#Polynomial_functions|polynomial function]] f(x) = a0x^n + a1x^(n-1) + ... + a0 satisfies f(r) = 0. If r is a [[http://Real%20number|real number]], it is a //real algebraic number//. | ||
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^(1/4), a root of x^4-80. [[Generators]] for [[linear temperaments]] which are real algebraic numbers can have interesting properties in terms of the [[http://en.wikipedia.org/wiki/Combination_tone|combination tones]] they produce. | 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^(1/4), a root of x^4-80. [[Generators]] for [[linear temperaments]] which are real algebraic numbers can have interesting properties in terms of the [[http://en.wikipedia.org/wiki/Combination_tone|combination tones]] they produce. | ||
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Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as [[http://en.wikipedia.org/wiki/Newton%27s_method|Newton's method]] can be used. A refinement of Newton's method 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 [[http://en.wikipedia.org/wiki/Newton%27s_method|Newton's method]] can be used. A refinement of Newton's method is the [[http://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method|Durand–Kerner method]].</pre></div> | ||
<h4>Original HTML content:</h4> | <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>A <a class="wiki_link_ext" href="http://mathworld.wolfram.com/UnivariatePolynomial.html" rel="nofollow">univariate polynomial</a> a0x^n + a1x^(n-1) + ... + an whose coefficients ai are integers (or equivalently, rational numbers) has < | <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>A <a class="wiki_link_ext" href="http://mathworld.wolfram.com/UnivariatePolynomial.html" rel="nofollow">univariate polynomial</a> a0x^n + a1x^(n-1) + ... + an whose coefficients ai are integers (or equivalently, rational numbers) has <strong>roots</strong> which are known as <strong><a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_number" rel="nofollow">algebraic numbers</a></strong>. A root is a value r for which the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Polynomial_function#Polynomial_functions" rel="nofollow">polynomial function</a> f(x) = a0x^n + a1x^(n-1) + ... + a0 satisfies f(r) = 0. If r is a <a class="wiki_link_ext" href="http://Real%20number" rel="nofollow">real number</a>, it is a <em>real algebraic number</em>. <br /> | ||
<br /> | <br /> | ||
Real algebraic numbers which are not also rational numbers turn up in various places in musical tuning theory. For instance, the intervals in the <a class="wiki_link" href="/Target%20tunings">target tunings</a> 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^(1/4), a root of x^4-80. <a class="wiki_link" href="/Generators">Generators</a> for <a class="wiki_link" href="/linear%20temperaments">linear temperaments</a> which are real algebraic numbers can have interesting properties in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Combination_tone" rel="nofollow">combination tones</a> they produce. <br /> | Real algebraic numbers which are not also rational numbers turn up in various places in musical tuning theory. For instance, the intervals in the <a class="wiki_link" href="/Target%20tunings">target tunings</a> 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^(1/4), a root of x^4-80. <a class="wiki_link" href="/Generators">Generators</a> for <a class="wiki_link" href="/linear%20temperaments">linear temperaments</a> which are real algebraic numbers can have interesting properties in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Combination_tone" rel="nofollow">combination tones</a> they produce. <br /> | ||
<br /> | <br /> | ||
Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Newton%27s_method" rel="nofollow">Newton's method</a> can be used. A refinement of Newton's method 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> | Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Newton%27s_method" rel="nofollow">Newton's method</a> can be used. A refinement of Newton's method 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> | ||