Algebraic number: Difference between revisions

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) + ... + an satisfies f(r) = 0. If r is a [http://en.wikipedia.org/wiki/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|Generators]] for [[linear_temperaments|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].

[[Category:math]]

[[Category:theory]]

[[Category:todo:increase_applicability]]