Algebraic number: Difference between revisions

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A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''0''x''''n'' ~~{{nowrap|~~+ ''a''1''x''''n'' − 1}} + … {{nowrap|+ ''a''''n''}} whose coefficients ''a''''i'' 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''0''x''''n''}} {{nowrap|+ ''a''1''x''''n'' − 1}} + … {{nowrap|+ ''a''''n''}} satisfies {{nowrap|''f''(''r'') {{=}} 0}}. If ''r'' is a {{w|real number}}, it is a ''real algebraic number''.

A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] {{nowrap|''a''0''x''''n'' + ''a''1''x''''n'' − 1}} + … {{nowrap|+ ''a''''n''}} whose coefficients ''a''''i'' 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''0''x''''n''}} {{nowrap|+ ''a''1''x''''n'' − 1}} + … {{nowrap|+ ''a''''n''}} satisfies {{nowrap|''f''(''r'') {{=}} 0}}. If ''r'' is a {{w|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 2×51/4, a root of {{nowrap|''x''4 − 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.

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 {{Wikipedia}}
-A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''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>}} 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''.
+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''.
 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.