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In abstract algebra, a tessarine or bicomplex number is a hypercomplex number in a commutative, associative algebra over real numbers with two imaginary units (designated i and k). == History == The subject of multiple imaginary units was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in Philosophical Magazine, William Rowan Hamilton communicated a system multiplying according to the quaternion group. In 1848 Thomas Kirkman reported〔Thomas Kirkman (1848) "On Pluquaternions and Homoid Products of ''n'' Squares", London and Edinburgh Philosophical Magazine 1848, p 447 ( Google books link )〕 on his correspondence with Arthur Cayley regarding equations on the units determining a system of hypercomplex numbers. In 1848 James Cockle introduced the tessarines in a series of articles in ''Philosophical Magazine''.〔James Cockle in London-Dublin-Edinburgh Philosophical Magazine, series 3 * 1848 (On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra ), 33:435–9. * 1849 (On a New Imaginary in Algebra ) 34:37–47. * 1849 (On the Symbols of Algebra and on the Theory of Tessarines ) 34:406–10. * 1850 (On Impossible Equations, on Impossible Quantities and on Tessarines ) 37:281–3. Links from Biodiversity Heritage Library. 〕 A tessarine is a hypercomplex number of the form : where Cockle used tessarines to isolate the hyperbolic cosine series and the hyperbolic sine series in the exponential series. He also showed how zero divisors arise in tessarines, inspiring him to use the term "impossibles." The tessarines are now best known for their subalgebra of real tessarines , also called split-complex numbers, which express the parametrization of the unit hyperbola. In 1892 Corrado Segre introduced〔. (see especially pages 455–67)〕 bicomplex numbers in ''Mathematische Annalen'', which form an algebra isomorphic to the tessarines (see section below). As commutative hypercomplex numbers, the tessarine algebra has been advocated by Clyde M. Davenport (1978, 1991, 2008) (exchange ''j'' and −''k'' in his multiplication table).〔C. M. Davenport: 1978, ‘An Extension of the Complex Calculus to Four Real Dimensions, with an Application to Special Relativity’, M. S. Thesis, University of Tennessee, Knoxville.〕〔Clyde Davenport (1991) ''A Hypercomplex Calculus with Applications to Special Relativity'' ISBN 0-9623837-0-8 〕〔Clyde Davenport (2008) (Commutative Hypercomplex Mathematics )〕 Davenport has noted the isomorphism with the direct sum of the complex number plane with itself. Tessarines have also been applied in digital signal processing.〔Soo-Chang Pei, Ja-Han Chang & Jian-Jiun Ding (2004) "Commutative reduced biquaternions and their Fourier transform for signal and image processing", ''IEEE Transactions on Signal Processing'' 52:2012–31〕〔Daniel Alfsmann (2006) (On families of 2^N dimensional hypercomplex algebras suitable for digital signal processing ), 14th European Signal Processing Conference, Florence, Italy〕〔Daniel Alfsmann & Heinz G Göckler (2007) (On Hyperbolic Complex LTI Digital Systems )〕 In 2009 mathematicians proved a fundamental theorem of tessarine algebra: a polynomial of degree ''n'' with tessarine coefficients has ''n''2 roots, counting multiplicity.〔Robert D. Poodiack & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes", The College Mathematics Journal 40(5):322–35〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Bicomplex number」の詳細全文を読む スポンサード リンク
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