Algebraic number
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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. 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]].
Original HTML content:
<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 <em>roots</em> which are known as <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_number" rel="nofollow">algebraic numbers</a>. 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 /> 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. Generators 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 /> 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>