2229:
2212:
3217:
1611:
830:
1354:. The unarion level in the Cayley-Dickson process must be a field, and starting with the real field, the usual complex numbers arises as division binarions, another field. Thus the process can begin again to form bibinarions.
396:
The product of two bicomplex numbers yields a quadratic form value that is the product of the individual quadratic forms of the numbers: a verification of this property of the quadratic form of a product refers to the
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487:
359:
986:
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1977:
539:. Multiplication being associative and commutative, the product of these imaginary units must have positive one for its square. Such an element as this product has been called a
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2015:
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Bicomplex numbers are employed in fluid mechanics. The use of bicomplex algebra reconciles two distinct applications of complex numbers: the representation of
2515:
Rochon, Dominic, and
Michael Shapiro (2004). "On algebraic properties of bicomplex and hyperbolic numbers." Anal. Univ. Oradea, fasc. math 11, no. 71: 110.
2328:
2232:
1338:
algebras shows agreement in the fourth dimension when the negative sign is used; consider the sample product given above under linear representation.
1075:
Cockle used tessarines to isolate the hyperbolic cosine series and the hyperbolic sine series in the exponential series. He also showed how
2190:
1606:{\displaystyle \sum _{k=1}^{n}(a_{k},b_{k})(u,v)^{k}\quad =\quad \left({\sum _{k=1}^{n}a_{i}u^{k}},\quad \sum _{k=1}^{n}b_{k}v^{k}\right).}
1687:
2082:
433:
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2215:
233:
905:
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was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in
2496:
2063:
1079:
arise in tessarines, inspiring him to use the term "impossibles". The tessarines are now best known for their subalgebra of
994:
2729:
1856:, there is a correspondence of polynomials and a correspondence of their roots. Hence the tessarine polynomials of degree
825:{\displaystyle k={\begin{pmatrix}0&i\\i&0\end{pmatrix}},\quad \ j={\begin{pmatrix}0&1\\1&0\end{pmatrix}}}
2724:
398:
2478:
2028:
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1207:
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2245:
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832:, which multiply according to the table given. When the identity matrix is identified with 1, then a tessarine
131:
1201:
is then the same as James Cockle's tessarines, represented using a different basis. Segre noted that elements
532:. Thus, the composing property of the quadratic form concurs with the composing property of the determinant.
3246:
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3063:
2377:
2143:
1318:, their equivalence with tessarines is apparent, particularly if the vectors in this basis are reordered as
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2362:
2689:
1923:
1883:
2244:
Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes",
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62:
17:
68:
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2021:
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405:. In fact, bicomplex numbers arise at the binarion level of the CayleyâDickson construction based on
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2136:
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2129:
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2108:
1350:
positions the algebra as a binarion construction based on another binarion construction, hence the
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408:
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2485:
The
Mathematics of Minkowski Space-Time with an Introduction to Commutative Hypercomplex Numbers
401:. This property of the quadratic form of a bicomplex number indicates that these numbers form a
2279:"Commutative reduced biquaternions and their Fourier transform for signal and image processing"
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861:
857:
705:
381:
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1085:
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1982:
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of dimension four. In fact the real algebra is older than the complex one; it was labelled
370:
31:
2488:
8:
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3039:
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2756:
2330:
On families of 2 dimensional hypercomplex algebras suitable for digital signal processing
1405:
1347:
402:
2423:
2053:
Bicomplex
Holomorphic Functions: the algebra, geometry and analysis of bicomplex numbers
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2746:
2510:
Bicomplex holomorphic functions:the algebra, geometry and analysis of bicomplex numbers
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2409:
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2336:. 14th European Signal Processing Conference, Florence, Italy: EURASIP. Archived from
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F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008)
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2163:[The real representation of complex elements and hyperalgebraic entities],
1920:. Since the linear space of CAPS can be viewed as the four dimensional space span {
1335:
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38:
2503:
Basics of functional analysis with bicomplex scalars, and bicomplex Schur analysis
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regarding equations on the units determining a system of hypercomplex numbers.
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362:
58:
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On
Certain Functions Resembling Quaternions and on a New Imaginary in Algebra
1178:
1143:
873:
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2161:"Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici"
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2056:
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377:
2398:"Stability of two-dimensional potential flows using bicomplex numbers"
1415:
are also isomorphic, however polynomials in the latter algebra split:
1787:{\displaystyle u_{1},u_{2},\dots ,u_{n},\ v_{1},v_{2},\dots ,v_{n}.}
3075:
3006:
2852:
2414:
2137:
On
Impossible Equations, on Impossible Quantities and on Tessarines
1669:
in this algebra is set, it reduces to two polynomial equations on
1165:. Segre used some of Hamilton's notation to develop his system of
2595:
2518:
2260:
The
Complex Algebra of Physical Space: A Framework for Relativity
701:
in 1848 while the complex algebra was not introduced until 1892.
1177:
be elements that square to â1 and that commute. Then, presuming
482:{\displaystyle {\begin{pmatrix}w&iz\\iz&w\end{pmatrix}}}
2549:
2396:
Kleine, Vitor G.; Hanifi, Ardeshir; Henningson, Dan S. (2022).
2277:
Pei, Soo-Chang; Chang, Ja-Han; Ding, Jian-Jiun (21 June 2004).
1358:
noted the simplification of nomenclature provided by the term
430:
The general bicomplex number can be represented by the matrix
2508:
Luna-ElizarrarĂĄs ME, Shapiro M, Struppa DC, Vajiac A. (2015)
1876:
Bicomplex number appears as the center of CAPS (complexified
1841:
from this set of roots will satisfy the original equation in
2501:
Alpay D, Luna-ElizarrarĂĄs ME, Shapiro M, Struppa DC. (2014)
1302:
When bicomplex numbers are expressed in terms of the basis
354:{\displaystyle (w,z)^{*}(w,z)=(w,-z)(w,z)=(w^{2}+z^{2},0),}
981:{\displaystyle t=w+xi+yj+zk,\quad w,x,y,z\in \mathbb {R} }
27:
Commutative, associative algebra of two complex dimensions
2505:, Cham, Switzerland: Springer Science & BusinessMedia
2123:
On the
Symbols of Algebra and on the Theory of Tessarines
1185:
must square to +1. The algebra constructed on the basis
2051:
M.E. Luna-ElizarrarĂĄs, M. Shapiro, D.C. Struppa (2013)
2395:
1396:) of complex numbers. Since the algebra of tessarines
1150:, which form an algebra isomorphic to the tessarines.
1068:{\displaystyle ij=ji=k,\quad i^{2}=-1,\quad j^{2}=+1.}
791:
742:
442:
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1985:
1926:
1886:
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1622:
1424:
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908:
888:
introduced the tessarines in a series of articles in
730:
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134:
71:
30:"Tessarine" redirects here. For real tessarines, see
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864:communicated a system multiplying according to the
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212:
117:
2077:(1848) "On Pluquaternions and Homoid Products of
1830:
3233:
1334:. Looking at the linear representation of these
2375:
2266:. Vol. 22. SpringerLink. pp. 537â561.
1388:and represent elements of it by ordered pairs (
2257:
125:, and the product of two bicomplex numbers as
2534:
2376:Alfsmann, Daniel; Göckler, Heinz G. (2007).
2276:
2233:Associative Composition Algebra/Bibinarions
1287:{\displaystyle g=(1-hi)/2,\quad g'=(1+hi)/2}
693:, the bicomplex numbers are an algebra over
2083:London and Edinburgh Philosophical Magazine
1616:In consequence, when a polynomial equation
1124:, which express the parametrization of the
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2379:On Hyperbolic Complex LTI Digital Systems
1903:
974:
413:
369:The bicomplex numbers form a commutative
213:{\displaystyle (u,v)(w,z)=(uw-vz,uz+vw).}
2326:
2327:Alfsmann, Daniel (4â8 September 2006).
681:Bicomplex numbers form an algebra over
535:Bicomplex numbers feature two distinct
14:
3234:
2286:IEEE Transactions on Signal Processing
2522:
2258:Baylis, W.E.; Kiselica, J.D. (2012).
2155:
899:is a hypercomplex number of the form
65:that defines the bicomplex conjugate
2130:On the True Amplitude of a Tessarine
1131:
872:reported on his correspondence with
546:
2730:Set-theoretically definable numbers
1972:{\displaystyle 1,e_{1},e_{2},e_{3}}
1369:
24:
2701:
2548:
2460:
1913:{\displaystyle Cl(3,\mathbb {C} )}
25:
3263:
2216:Abstract Algebra/Polynomial Rings
708:for the tessarine 4-algebra over
3215:
2227:
2210:
2020:Tessarines have been applied in
118:{\displaystyle (w,z)^{*}=(w,-z)}
2512:, Cham, Switzerland: BirkhÀuser
2389:
2369:
2320:
2246:The College Mathematics Journal
2029:two-dimensional potential flows
1871:
1834:{\displaystyle (u_{i},v_{j})\!}
1553:
1501:
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1181:of multiplication, the product
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2709:{\displaystyle {\mathcal {P}}}
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399:BrahmaguptaâFibonacci identity
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3064:Plane-based geometric algebra
2201:(see especially pages 455â67)
2144:Biodiversity Heritage Library
2116:On a New Imaginary in Algebra
2038:
2031:in the complex plane and the
1880:), which is Clifford algebra
879:
3024:{\displaystyle \mathbb {S} }
2947:{\displaystyle \mathbb {C} }
2908:{\displaystyle \mathbb {R} }
2870:{\displaystyle \mathbb {O} }
2842:{\displaystyle \mathbb {H} }
2814:{\displaystyle \mathbb {C} }
2786:{\displaystyle \mathbb {R} }
2674:{\displaystyle \mathbb {A} }
2641:{\displaystyle \mathbb {Q} }
2613:{\displaystyle \mathbb {Z} }
2585:{\displaystyle \mathbb {N} }
2264:Adv. Appl. Clifford Algebras
2033:complex exponential function
1852:Due to the isomorphism with
1662:{\displaystyle f(u,v)=(0,0)}
685:of dimension two, and since
420:{\displaystyle \mathbb {C} }
7:
2100:in London-Dublin-Edinburgh
525:{\displaystyle w^{2}+z^{2}}
10:
3268:
1364:A Taste of Jordan Algebras
847:
29:
3206:
3148:
3074:
3054:Algebra of physical space
2976:
2884:
2755:
2557:
2022:digital signal processing
1878:algebra of physical space
689:is of dimension two over
552:Tessarine multiplication
376:of dimension two that is
3110:Extended complex numbers
3093:Extended natural numbers
2361:: CS1 maint: location (
1161:(1853) and the works of
1113:{\displaystyle t=w+yj\ }
852:The subject of multiple
572:
567:
562:
559:
556:
2298:10.1109/TSP.2004.828901
2010:{\displaystyle 1,i,k,j}
1159:Lectures on Quaternions
724:, giving the matrices
365:in the first component.
3166:Transcendental numbers
3025:
3002:Hyperbolic quaternions
2948:
2909:
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2815:
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2432:10.1098/rspa.2022.0165
2292:(7). IEEE: 2012â2031.
2102:Philosophical Magazine
2011:
1973:
1914:
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1663:
1607:
1574:
1528:
1445:
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890:Philosophical Magazine
862:William Rowan Hamilton
858:Philosophical Magazine
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526:
483:
421:
382:direct sum of algebras
355:
214:
119:
63:CayleyâDickson process
3098:Extended real numbers
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2949:
2919:Split-complex numbers
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2872:
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2711:
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2652:Constructible numbers
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2615:
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2166:Mathematische Annalen
2012:
1974:
1915:
1866:multiplicity of roots
1836:
1789:
1664:
1608:
1554:
1508:
1425:
1346:The modern theory of
1289:
1139:Mathematische Annalen
1122:split-complex numbers
1115:
1070:
983:
827:
527:
484:
422:
356:
215:
120:
3247:Hypercomplex numbers
3242:Composition algebras
3130:Supernatural numbers
3040:Multicomplex numbers
3013:
2997:Dual-complex numbers
2936:
2897:
2859:
2831:
2803:
2775:
2757:Composition algebras
2725:Arithmetical numbers
2696:
2663:
2630:
2602:
2574:
1983:
1924:
1884:
1798:
1688:
1620:
1422:
1406:rings of polynomials
1348:composition algebras
1208:
1086:
995:
906:
728:
496:
434:
409:
234:
132:
69:
32:Split-complex number
3035:Split-biquaternions
2747:Eisenstein integers
2685:Closed-form numbers
2424:2022RSPSA.47820165K
1684:for each equation:
1673:. If the degree is
553:
403:composition algebra
61:constructed by the
3193:Profinite integers
3156:Irrational numbers
3021:
2944:
2905:
2867:
2839:
2811:
2783:
2740:Gaussian rationals
2720:Computable numbers
2706:
2671:
2638:
2610:
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2179:10.1007/bf01443559
2007:
1969:
1910:
1831:
1784:
1659:
1603:
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522:
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417:
351:
210:
115:
3229:
3228:
3140:Superreal numbers
3120:Levi-Civita field
3115:Hyperreal numbers
3059:Spacetime algebra
3045:Geometric algebra
2958:Bicomplex numbers
2924:Split-quaternions
2765:Division algebras
2735:Gaussian integers
2657:Algebraic numbers
2560:definable numbers
2497:978-3-7643-8613-9
2489:BirkhÀuser Verlag
2087:Google books link
2064:978-3-319-24868-4
1794:Any ordered pair
1738:
1677:, then there are
1400:is isomorphic to
1167:bicomplex numbers
1148:bicomplex numbers
1132:Bicomplex numbers
1109:
779:
679:
678:
547:As a real algebra
16:(Redirected from
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3219:
3218:
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3088:Cardinal numbers
3049:Clifford algebra
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866:quaternion group
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43:bicomplex number
39:abstract algebra
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3125:Ordinal numbers
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2797:Complex numbers
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2568:Natural numbers
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2461:Further reading
2458:
2457:
2402:Proc. R. Soc. A
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2390:
2382:
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2370:
2357:cite conference
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2343:on 16 July 2011
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1881:
1874:
1821:
1817:
1808:
1804:
1799:
1796:
1795:
1775:
1771:
1756:
1752:
1743:
1739:
1727:
1723:
1708:
1704:
1695:
1691:
1689:
1686:
1685:
1621:
1618:
1617:
1589:
1585:
1579:
1575:
1569:
1558:
1543:
1539:
1533:
1529:
1523:
1512:
1507:
1506:
1502:
1491:
1487:
1466:
1462:
1453:
1449:
1440:
1429:
1423:
1420:
1419:
1375:
1372:
1356:Kevin McCrimmon
1344:
1319:
1303:
1276:
1247:
1235:
1209:
1206:
1205:
1186:
1134:
1120:, also called
1087:
1084:
1083:
1081:real tessarines
1050:
1046:
1027:
1023:
996:
993:
992:
973:
907:
904:
903:
882:
854:imaginary units
850:
815:
814:
809:
803:
802:
797:
787:
786:
766:
765:
760:
754:
753:
748:
738:
737:
729:
726:
725:
549:
541:hyperbolic unit
537:imaginary units
516:
512:
503:
499:
497:
494:
493:
472:
471:
466:
457:
456:
448:
438:
437:
435:
432:
431:
412:
410:
407:
406:
384:
333:
329:
320:
316:
253:
249:
235:
232:
231:
133:
130:
129:
88:
84:
70:
67:
66:
59:complex numbers
46:
35:
28:
23:
22:
15:
12:
11:
5:
3265:
3255:
3254:
3249:
3244:
3227:
3226:
3224:
3223:
3213:
3211:Classification
3207:
3204:
3203:
3201:
3200:
3198:Normal numbers
3195:
3190:
3168:
3163:
3158:
3152:
3150:
3146:
3145:
3143:
3142:
3137:
3132:
3127:
3122:
3117:
3112:
3107:
3106:
3105:
3095:
3090:
3084:
3082:
3080:infinitesimals
3072:
3071:
3069:
3068:
3067:
3066:
3061:
3056:
3042:
3037:
3032:
3019:
3004:
2999:
2994:
2989:
2983:
2981:
2974:
2973:
2971:
2970:
2965:
2960:
2955:
2942:
2926:
2921:
2916:
2903:
2890:
2888:
2882:
2881:
2879:
2878:
2865:
2850:
2837:
2822:
2809:
2794:
2781:
2761:
2759:
2753:
2752:
2750:
2749:
2744:
2743:
2742:
2732:
2727:
2722:
2717:
2703:
2687:
2682:
2669:
2654:
2649:
2636:
2621:
2608:
2593:
2580:
2564:
2562:
2555:
2554:
2546:
2545:
2538:
2531:
2523:
2517:
2516:
2513:
2506:
2499:
2481:
2473:Marcel Dekker
2467:G. Baley Price
2462:
2459:
2456:
2455:
2388:
2368:
2319:
2269:
2250:
2237:
2220:
2203:
2173:(3): 413â467,
2157:Segre, Corrado
2148:
2141:
2140:
2133:
2126:
2119:
2112:
2090:
2075:Thomas Kirkman
2067:
2043:
2042:
2040:
2037:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1966:
1962:
1958:
1953:
1949:
1945:
1940:
1936:
1932:
1929:
1909:
1905:
1901:
1898:
1895:
1892:
1889:
1873:
1870:
1829:
1824:
1820:
1816:
1811:
1807:
1803:
1783:
1778:
1774:
1770:
1767:
1764:
1759:
1755:
1751:
1746:
1742:
1735:
1730:
1726:
1722:
1719:
1716:
1711:
1707:
1703:
1698:
1694:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1614:
1613:
1602:
1598:
1592:
1588:
1582:
1578:
1572:
1567:
1564:
1561:
1557:
1552:
1546:
1542:
1536:
1532:
1526:
1521:
1518:
1515:
1511:
1505:
1500:
1494:
1490:
1486:
1483:
1480:
1477:
1474:
1469:
1465:
1461:
1456:
1452:
1448:
1443:
1438:
1435:
1432:
1428:
1371:
1368:
1343:
1340:
1300:
1299:
1283:
1279:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1253:
1250:
1245:
1242:
1238:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1163:W. K. Clifford
1155:W. R. Hamilton
1133:
1130:
1126:unit hyperbola
1106:
1103:
1100:
1097:
1094:
1091:
1064:
1061:
1058:
1053:
1049:
1044:
1041:
1038:
1035:
1030:
1026:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
989:
988:
976:
972:
969:
966:
963:
960:
957:
954:
951:
947:
944:
941:
938:
935:
932:
929:
926:
923:
920:
917:
914:
911:
881:
878:
870:Thomas Kirkman
849:
846:
819:
813:
810:
808:
805:
804:
801:
798:
796:
793:
792:
790:
785:
782:
775:
770:
764:
761:
759:
756:
755:
752:
749:
747:
744:
743:
741:
736:
733:
677:
676:
673:
668:
662:
657:
651:
650:
645:
642:
637:
632:
626:
625:
619:
614:
611:
606:
600:
599:
594:
589:
584:
581:
577:
576:
571:
566:
561:
558:
548:
545:
519:
515:
511:
506:
502:
476:
470:
467:
465:
462:
459:
458:
455:
452:
449:
447:
444:
443:
441:
415:
367:
366:
363:quadratic form
350:
347:
344:
341:
336:
332:
328:
323:
319:
315:
312:
309:
306:
303:
300:
297:
294:
291:
288:
285:
282:
279:
276:
273:
270:
267:
264:
261:
256:
252:
248:
245:
242:
239:
225:bicomplex norm
221:
220:
209:
206:
203:
200:
197:
194:
191:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
143:
140:
137:
114:
111:
108:
105:
102:
99:
96:
91:
87:
83:
80:
77:
74:
26:
9:
6:
4:
3:
2:
3264:
3253:
3250:
3248:
3245:
3243:
3240:
3239:
3237:
3222:
3214:
3212:
3209:
3208:
3205:
3199:
3196:
3194:
3191:
3188:
3184:
3178:
3174:
3169:
3167:
3164:
3162:
3161:Fuzzy numbers
3159:
3157:
3154:
3153:
3151:
3147:
3141:
3138:
3136:
3133:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3111:
3108:
3104:
3101:
3100:
3099:
3096:
3094:
3091:
3089:
3086:
3085:
3083:
3081:
3077:
3073:
3065:
3062:
3060:
3057:
3055:
3052:
3051:
3050:
3046:
3043:
3041:
3038:
3036:
3033:
3008:
3005:
3003:
3000:
2998:
2995:
2993:
2990:
2988:
2985:
2984:
2982:
2980:
2975:
2969:
2966:
2964:
2963:Biquaternions
2961:
2959:
2956:
2930:
2927:
2925:
2922:
2920:
2917:
2892:
2891:
2889:
2883:
2854:
2851:
2826:
2823:
2798:
2795:
2770:
2766:
2763:
2762:
2760:
2758:
2754:
2748:
2745:
2741:
2738:
2737:
2736:
2733:
2731:
2728:
2726:
2723:
2721:
2718:
2691:
2688:
2686:
2683:
2658:
2655:
2653:
2650:
2625:
2622:
2597:
2594:
2569:
2566:
2565:
2563:
2561:
2556:
2551:
2544:
2539:
2537:
2532:
2530:
2525:
2524:
2521:
2514:
2511:
2507:
2504:
2500:
2498:
2494:
2490:
2486:
2482:
2480:
2479:0-8247-8345-X
2476:
2472:
2468:
2465:
2464:
2451:
2447:
2442:
2437:
2433:
2429:
2425:
2421:
2416:
2411:
2407:
2403:
2399:
2392:
2381:
2380:
2372:
2364:
2358:
2339:
2332:
2331:
2323:
2315:
2311:
2307:
2303:
2299:
2295:
2291:
2287:
2280:
2273:
2265:
2261:
2254:
2248:40(5):322â35.
2247:
2241:
2234:
2230:
2224:
2217:
2213:
2207:
2193:on 2013-09-12
2192:
2188:
2184:
2180:
2176:
2172:
2168:
2167:
2162:
2158:
2152:
2145:
2138:
2134:
2131:
2127:
2124:
2120:
2117:
2113:
2110:
2106:
2105:
2103:
2099:
2094:
2088:
2084:
2080:
2076:
2071:
2065:
2061:
2058:
2054:
2048:
2044:
2036:
2034:
2030:
2025:
2023:
2018:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1964:
1960:
1956:
1951:
1947:
1943:
1938:
1934:
1930:
1927:
1899:
1896:
1890:
1887:
1879:
1869:
1867:
1863:
1859:
1855:
1850:
1848:
1844:
1822:
1818:
1814:
1809:
1805:
1781:
1776:
1772:
1768:
1765:
1762:
1757:
1753:
1749:
1744:
1740:
1733:
1728:
1724:
1720:
1717:
1714:
1709:
1705:
1701:
1696:
1692:
1683:
1680:
1676:
1672:
1653:
1650:
1647:
1641:
1635:
1632:
1629:
1623:
1600:
1596:
1590:
1586:
1580:
1576:
1570:
1565:
1562:
1559:
1555:
1550:
1544:
1540:
1534:
1530:
1524:
1519:
1516:
1513:
1509:
1503:
1498:
1492:
1484:
1481:
1478:
1467:
1463:
1459:
1454:
1450:
1441:
1436:
1433:
1430:
1426:
1418:
1417:
1416:
1414:
1410:
1407:
1403:
1399:
1395:
1391:
1386:
1382:
1378:
1367:
1365:
1361:
1357:
1353:
1349:
1339:
1337:
1331:
1327:
1323:
1315:
1311:
1307:
1297:
1281:
1277:
1270:
1267:
1264:
1261:
1255:
1251:
1248:
1243:
1240:
1236:
1229:
1226:
1223:
1220:
1214:
1211:
1204:
1203:
1202:
1198:
1194:
1190:
1184:
1180:
1179:associativity
1176:
1172:
1168:
1164:
1160:
1156:
1151:
1149:
1145:
1144:Corrado Segre
1141:
1140:
1129:
1127:
1123:
1104:
1101:
1098:
1095:
1092:
1089:
1082:
1078:
1077:zero divisors
1062:
1059:
1056:
1051:
1047:
1042:
1039:
1036:
1033:
1028:
1024:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
970:
967:
964:
961:
958:
955:
952:
949:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
902:
901:
900:
898:
893:
891:
887:
877:
875:
874:Arthur Cayley
871:
867:
863:
859:
855:
845:
843:
839:
835:
817:
811:
806:
799:
794:
788:
783:
780:
773:
768:
762:
757:
750:
745:
739:
734:
731:
723:
719:
715:
711:
707:
702:
700:
696:
692:
688:
684:
674:
672:
669:
667:
663:
661:
658:
656:
653:
652:
649:
646:
643:
641:
638:
636:
633:
631:
628:
627:
624:
620:
618:
615:
612:
610:
607:
605:
602:
601:
598:
595:
593:
590:
588:
585:
582:
579:
578:
575:
570:
565:
555:
544:
542:
538:
533:
517:
513:
509:
504:
500:
492:
474:
468:
463:
460:
453:
450:
445:
439:
428:
427:with norm z.
404:
400:
394:
391:
387:
383:
379:
375:
374:
371:algebra over
364:
348:
342:
339:
334:
330:
326:
321:
317:
310:
304:
301:
298:
289:
286:
283:
280:
274:
268:
265:
262:
254:
246:
243:
240:
230:
229:
228:
226:
207:
201:
198:
195:
192:
189:
186:
183:
180:
177:
174:
171:
165:
159:
156:
153:
144:
141:
138:
128:
127:
126:
109:
106:
103:
100:
94:
89:
81:
78:
75:
64:
60:
54:
50:
44:
40:
33:
19:
3182:
3172:
2987:Dual numbers
2979:hypercomplex
2957:
2769:Real numbers
2509:
2502:
2484:
2470:
2408:(20220165).
2405:
2401:
2391:
2378:
2371:
2345:. Retrieved
2338:the original
2329:
2322:
2289:
2285:
2272:
2263:
2259:
2253:
2240:
2235:at Wikibooks
2223:
2218:at Wikibooks
2206:
2195:, retrieved
2191:the original
2170:
2164:
2151:
2098:James Cockle
2093:
2085:1848, p 447
2078:
2070:
2052:
2047:
2026:
2019:
1875:
1872:Applications
1861:
1857:
1853:
1851:
1846:
1845:, so it has
1842:
1678:
1674:
1670:
1615:
1412:
1408:
1401:
1397:
1393:
1389:
1384:
1380:
1376:
1373:
1363:
1362:in his text
1359:
1351:
1345:
1329:
1325:
1321:
1313:
1309:
1305:
1301:
1196:
1192:
1188:
1182:
1174:
1170:
1166:
1158:
1152:
1147:
1137:
1135:
1080:
990:
896:
894:
889:
886:James Cockle
883:
851:
841:
837:
833:
721:
717:
713:
709:
703:
698:
694:
690:
686:
682:
680:
670:
665:
659:
654:
647:
639:
634:
629:
622:
616:
608:
603:
596:
591:
586:
573:
568:
563:
534:
489:, which has
429:
395:
389:
385:
372:
368:
227:is given by
224:
222:
52:
48:
42:
36:
3149:Other types
2968:Bioctonions
2825:Quaternions
2347:18 February
2142:Links from
2111:, 33:435â9.
2104:, series 3
1352:bibinarions
1342:Bibinarions
1296:idempotents
1294: are
1153:Segre read
1146:introduced
868:. In 1848
491:determinant
3236:Categories
3103:Projective
3076:Infinities
2415:2203.05857
2385:. EURASIP.
2197:2013-09-12
2125:34:406â10.
2081:Squares",
2057:Birkhauser
2055:, page 6,
2039:References
1860:also have
1336:isomorphic
1136:In a 1892
880:Tessarines
712:specifies
699:tessarines
378:isomorphic
45:is a pair
18:Tessarines
3187:solenoids
3007:Sedenions
2853:Octonions
2306:1941-0476
2187:121807474
2139:37:281â3.
2132:36:290-2.
2118:34:37â47.
1766:…
1718:…
1556:∑
1510:∑
1427:∑
1224:−
1037:−
971:∈
897:tessarine
287:−
255:∗
223:Then the
178:−
107:−
90:∗
3252:Matrices
2596:Integers
2558:Sets of
2491:, Basel
2450:35702595
2314:13907861
2159:(1892),
1979:} over {
1366:(2004).
1360:binarion
1252:′
884:In 1848
716:= 1 and
3177:numbers
3009: (
2855: (
2827: (
2799: (
2771: (
2692: (
2690:Periods
2659: (
2626: (
2598: (
2570: (
2552:systems
2469:(1991)
2441:9185835
2420:Bibcode
1849:roots.
1142:paper,
848:History
380:to the
2977:Other
2550:Number
2495:
2477:
2448:
2438:
2312:
2304:
2185:
2062:
1737:
1404:, the
1374:Write
1169:: Let
1108:
991:where
778:
3185:-adic
3175:-adic
2932:Over
2893:Over
2887:types
2885:Split
2410:arXiv
2383:(PDF)
2341:(PDF)
2334:(PDF)
2310:S2CID
2282:(PDF)
2183:S2CID
2135:1850
2128:1850
2121:1849
2114:1849
2107:1848
1682:roots
1320:{ 1,
1304:{ 1,
1187:{ 1,
706:basis
3221:List
3078:and
2493:ISBN
2475:ISBN
2446:PMID
2363:link
2349:2010
2302:ISSN
2060:ISBN
1411:and
1173:and
41:, a
2436:PMC
2428:doi
2406:478
2294:doi
2175:doi
2017:}.
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