Algebraic number: Difference between revisions

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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) + ~~...~~ + an satisfies f(r) = 0. If r is a [~~http~~:~~//en.wikipedia.org/wiki/~~Real~~%20number~~ real number], it is a ''real algebraic number''.

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

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 2×51/4, a root of ''x''4 - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: 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].

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

[[Category:~~definition~~]]

[[Category:~~math~~]]

[[Category:Definition]]

[[Category:~~theory~~]]

[[Category:Math]]

[[Category:todo:~~increase_applicability~~]]

[[Category:Theory]]

[[Category:todo:increase applicability]]

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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) + ... + an satisfies f(r) = 0. If r is a [http://en.wikipedia.org/wiki/Real%20number real number], it is a ''real algebraic number''.
+A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial  #Polynomial functions|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number|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.
+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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: 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].
+Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as [[Wikipedia: Newton's method|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method|Durand–Kerner method]].
-[[Category:definition]]
-[[Category:math]]
+[[Category:Definition]]
-[[Category:theory]]
+[[Category:Math]]
-[[Category:todo:increase_applicability]]
+[[Category:Theory]]
+[[Category:todo:increase applicability]]