3019:
6287:
3892:
7551:
1898:
1527:
1221:
3488:
2891:
739:
1769:
3640:
3274:
2059:
1363:
5322:
4755:
3779:
5419:
1734:
5117:
2258:
1670:
4504:
2559:
5852:
4849:
4269:
4132:
2118:
1021:
3843:
4344:
1368:
1026:
652:
3371:
587:
5473:) is a point on the standard hyperbola. More generally, there is a hypersurface in M(2,R) of hyperbolic units, any one of which serves in a basis to represent the split-complex numbers as a
3879:
3705:
3104:
2751:
3364:
3009:
2402:
647:
2677:
5782:
312:
5022:
2342:
2169:
4678:
3987:
394:
4063:
2484:
631:
5925:
4385:
4024:
3063:
784:
524:
5687:
2950:
474:
7048:
4639:
5212:
4416:. While analogous to ordinary orthogonality, particularly as it is known with ordinary complex number arithmetic, this condition is more subtle. It forms the basis for the
4169:
7363:
7286:
7247:
7209:
7181:
7153:
7125:
7013:
6980:
6952:
6924:
5173:
4915:
1346:
5463:
2724:
812:
198:
5554:
3549:
3163:
987:
4960:
3539:
1290:
941:
234:
95:
1973:
145:
5513:
5260:
3156:
1935:
1568:
882:
358:
3849:
in the span of a corresponding hyperbolic sector. Such confusion may be perpetuated when the geometry of the split-complex plane is not distinguished from that of
2431:
6828:
Advances on Theory and
Practice of Robots and Manipulators: Proceedings of Romansy 2014 XX CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators
5575:
used split-complex numbers to represent sums of spins. Clifford introduced the use of split-complex numbers as coefficients in a quaternion algebra now called
4687:
5786:
expressing products on the unit hyperbola illustrates the additivity of rapidities for collinear velocities. Simultaneity of events depends on rapidity
6051:(see link in References). He developed two algebraic systems, each of which he called "approximate numbers", the second of which forms a real algebra.
5362:
1683:
5069:
6683:
2192:
1578:
4432:
2491:
5795:
4780:
1982:
4209:
4075:
2066:
1893:{\displaystyle \langle z,w\rangle =\operatorname {\mathrm {Re} } \left(zw^{*}\right)=\operatorname {\mathrm {Re} } \left(z^{*}w\right)=xu-yv~,}
6062:
In 1961 Warmus continued his exposition, referring to the components of an approximate number as midpoint and radius of the interval denoted.
5265:
5053:
Since addition and multiplication are continuous operations with respect to the usual topology of the plane, the split-complex numbers form a
6430:
5946:
are hyperbolic orthogonal and lie on the axes of a frame of reference in which the events simultaneous with the origin are proportional to
4298:
3712:
6290:
6874:
6689:
K. Carmody, (1997) "Circular and hyperbolic quaternions, octonions, and sedenions – further results", Appl. Math. Comput. 84:27–48.
6018:
In 1935 J.C. Vignaux and A. Durañona y Vedia developed the split-complex geometric algebra and function theory in four articles in
5326:
Addition and multiplication of split-complex numbers are then given by matrix addition and multiplication. The squared modulus of
4584:
The set of all transformations of the split-complex plane which preserve the modulus (or equivalently, the inner product) forms a
1522:{\displaystyle {\begin{aligned}(z+w)^{*}&=z^{*}+w^{*}\\(zw)^{*}&=z^{*}w^{*}\\\left(z^{*}\right)^{*}&=z.\end{aligned}}}
2957:
2592:
5713:
31:
6843:
6707:
6523:
6377:
6368:
Francesco Catoni; Dino
Boccaletti; Roberto Cannata; Vincenzo Catoni; Paolo Zampetti (2011). "Chapter 2: Hyperbolic Numbers".
6355:
6257:
4966:
3787:
7063:
5579:. He called its elements "motors", a term in parallel with the "rotor" action of an ordinary complex number taken from the
4862:
7058:
6669:
6059:
and observed that this second system was isomorphic to the "hyperbolic complex" numbers, the subject of this article.
533:
6798:
6769:
6744:
6630:
3854:
3680:
3079:
2898:
4615:
2291:
7018:
1303:
1216:{\displaystyle {\begin{aligned}(x+jy)+(u+jv)&=(x+u)+j(y+v)\\(x+jy)(u+jv)&=(xu+yv)+j(xv+yu).\end{aligned}}}
7436:
6805:
6725:
6023:
5973:
5035:
3281:
2347:
17:
7514:
6826:
J. Rooney (2014). "Generalised
Complex Numbers in Mechanics". In Marco Ceccarelli and Victor A. Glazunov (ed.).
6783:
C. Musès, "Hypernumbers II—Further concepts and computational applications", Appl. Math. Comput. 4 (1978) 45–66.
6790:, Chapter 1: Hyperbolic Complex Numbers in Two Dimensions, pages 1–16, North-Holland Mathematics Studies #190,
6464:
989:
The change of sign distinguishes the split-complex numbers from the ordinary complex ones. The hyperbolic unit
243:
1254:
910:
7585:
7575:
7397:
6507:
6318:
2297:
852:
6867:
6751:
6070:
Different authors have used a great variety of names for the split-complex numbers. Some of these include:
5980:) to construct other composition algebras including the split-octonions. His innovation was perpetuated by
4589:
2142:
3483:{\displaystyle (\cosh a,\sinh a){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=\left(e^{a},e^{-a}\right)}
7023:
4644:
3954:
365:
4033:
6804:
Poodiack, Robert D. & Kevin J. LeClair (2009) "Fundamental theorems of algebra for the perplexes",
6623:
3504:
2436:
604:
6644:
Bencivenga, Uldrico (1946) "Sulla rappresentazione geometrica delle algebre doppie dotate di modulo",
5878:
4361:
4000:
3039:
760:
500:
7431:
7387:
5652:
2886:{\displaystyle \left(a_{1},b_{1}\right)\left(a_{2},b_{2}\right)=\left(a_{1}a_{2},b_{1}b_{2}\right)~.}
417:
7029:
6030:(in Spanish). These expository and pedagogical essays presented the subject for broad appreciation.
734:{\displaystyle {\begin{aligned}D&\to \mathbb {R} ^{2}\\x+yj&\mapsto (x-y,x+y)\end{aligned}}}
7554:
7426:
6747:. Contains a description of normed algebras in indefinite signature, including the Lorentz numbers.
6692:
6269:
5572:
5192:
4607:
4149:
1571:
1351:
315:
7346:
7269:
7230:
7192:
7164:
7136:
7108:
6996:
6963:
6935:
6907:
6493:
Vignaux, J.(1935) "Sobre el numero complejo hiperbolico y su relacion con la geometria de Borel",
6306:
5140:
4889:
4417:
7580:
6860:
5426:
3653:
2684:
793:
158:
5518:
4581:
preserves the geometric structure, taking hyperbolas to themselves and the null cone to itself.
6780:
C. Musès, "Applied hypernumbers: Computational concepts", Appl. Math. Comput. 3 (1977) 211–226.
5594:
Since the late twentieth century, the split-complex multiplication has commonly been seen as a
5354:
5132:
From the definition it is apparent that the ring of split-complex numbers is isomorphic to the
2572:
2184:
1739:
956:
4929:
3652:. The isomorphism, as a planar mapping, consists of a counter-clockwise rotation by 45° and a
3513:
3111:
The diagonal basis for the split-complex number plane can be invoked by using an ordered pair
203:
67:
7499:
7335:
6443:
6367:
5872:
5564:
4401:
3648:, the split-complex plane and the direct sum of two real lines differ in their layout in the
3018:
1940:
1234:
112:
5990:
as base field, builds split-complex numbers as a composition algebra. Reviewing Albert for
5483:
5230:
3126:
1905:
1538:
328:
6985:
6775:
6729:
6652:
6584:
Miller, William; Boehning, Rochelle (1968). "Gaussian, parabolic, and hyperbolic numbers".
6538:
6481:
6402:
6056:
6034:
5991:
5856:
is the line of events simultaneous with the origin in the frame of reference with rapidity
5224:
4604:
399:
6568:
6551:
6347:
6250:
Geometry of Mobius
Transformations: Elliptic, Parabolic, and Hyperbolic actions of SL(2,R)
5337:
In fact there are many representations of the split-complex plane in the four-dimensional
3663:. The dilation in particular has sometimes caused confusion in connection with areas of a
2407:
8:
7463:
7373:
7330:
7312:
7090:
6230:
5994:, N. H. McCoy wrote that there was an "introduction of some new algebras of order 2 over
5969:
5061:
5043:
4920:
4585:
4201:
3493:
3023:
1675:
489:
6406:
7368:
7080:
6601:
6131:
5694:
5576:
5338:
4424:
1754:
6703:
7526:
7489:
7453:
7392:
7378:
7073:
7053:
6839:
6794:
6765:
6740:
6626:
6480:
N.H. McCoy (1942) Review of "Quadratic forms permitting composition" by A.A. Albert,
6373:
6351:
6253:
6118:
4772:
4513:
3664:
3645:
1355:
6342:
F. Catoni, D. Boccaletti, R. Cannata, V. Catoni, E. Nichelatti, P. Zampetti. (2008)
5357:
element with which to extend the real line to the split-complex plane. The matrices
7544:
7473:
7448:
7382:
7291:
7257:
7098:
7068:
6990:
6893:
6831:
6593:
6410:
6225:
5588:
5054:
4874:
4521:
3668:
3635:{\displaystyle \sigma :(u,v)\mapsto \left(ru,{\frac {v}{r}}\right),\quad r=e^{b}~.}
3269:{\displaystyle (u,v)=(x,y){\begin{pmatrix}1&1\\1&-1\end{pmatrix}}=(x,y)S~.}
3071:
1743:
640:
6668:
N. A. Borota, E. Flores, and T. J. Osler (2000) "Spacetime numbers the easy way",
6520:
7421:
7325:
6957:
6772:
6757:
6649:
6535:
6527:
6460:
6220:
5342:
5046:
since the null elements are not invertible. All of the nonzero null elements are
4882:
4574:
4543:
4517:
3946:
3649:
3542:
3508:
636:
6835:
6033:
In 1941 E.F. Allen used the split-complex geometric arithmetic to establish the
7468:
7458:
7443:
7262:
7130:
6901:
6819:
6711:
6468:
6426:
6393:
Fjelstad, Paul (1986), "Extending special relativity via the perplex numbers",
5965:
5584:
4538:
has norm 1 and lies on the right branch of the unit hyperbola. Numbers such as
4197:
3782:
2134:
1013:
950:
946:
831:
596:
5976:, used to generate division algebras, could be modified (with a factor gamma,
7569:
7531:
7504:
7413:
6286:
5981:
5595:
4878:
4570:
4353:
2123:
1761:
3891:
2054:{\displaystyle \operatorname {\mathrm {Re} } (z)={\tfrac {1}{2}}(z+z^{*})=x}
7494:
7296:
6811:
6302:
6127:
6052:
5628:
5580:
5178:
5047:
5034:. With this description, it is clear that the split-complex numbers form a
4509:
4028:
can be described with complex numbers, the geometry of the
Minkowski plane
3938:
6830:. Mechanisms and Machine Science. Vol. 22. Springer. pp. 55–62.
7320:
7102:
6675:
N. A. Borota and T. J. Osler (2002) "Functions of a spacetime variable",
6658:
6597:
6506:
Allen, E.F. (1941) "On a
Triangle Inscribed in a Rectangular Hyperbola",
5331:
3070:
On the basis {e, e*} it becomes clear that the split-complex numbers are
2263:
1230:
1226:
895:
98:
6605:
5317:{\displaystyle z\mapsto {\begin{pmatrix}x&y\\y&x\end{pmatrix}}.}
7301:
7158:
5133:
4924:
6331:
4750:{\displaystyle \exp \colon (\mathbb {R} ,+)\to \mathrm {SO} ^{+}(1,1)}
4289:. The hyperbola and conjugate hyperbola are separated by two diagonal
6027:
5599:
5568:
5346:
4349:
4290:
4137:
6414:
6047:
30:"Double number" redirects here. For the computer number format, see
7409:
7340:
7186:
6791:
5961:
5698:
4596:
3992:
1009:
748:
6646:
Atti della Reale
Accademia delle Scienze e Belle-Lettere di Napoli
4173:
The hyperbola consists of a right and left branch passing through
3774:{\displaystyle \{(a,b)\in \mathbb {R} \oplus \mathbb {R} :ab=1\}.}
6929:
6852:
6699:, A. W. Tucker editor, page 392, "Further Notes on Biquaternions"
5474:
5414:{\displaystyle m={\begin{pmatrix}a&c\\b&-a\end{pmatrix}}}
2126:; nevertheless the bilinear form is frequently referred to as an
1729:{\displaystyle \lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~.}
38:
6717:
De Boer, R. (1987) "An also known as list for perplex numbers",
5112:{\displaystyle \lVert zw\rVert =\lVert z\rVert \lVert w\rVert ~}
4595:. This group consists of the hyperbolic rotations, which form a
2122:
Since it is not positive-definite, this bilinear form is not an
6883:
6684:"Circular and hyperbolic quaternions, octonions, and sedenions"
2253:{\displaystyle z^{-1}={\frac {z^{*}}{{\lVert z\rVert }^{2}}}~.}
2130:. A similar abuse of language refers to the modulus as a norm.
1665:{\displaystyle \lVert z\rVert ^{2}=zz^{*}=z^{*}z=x^{2}-y^{2}~.}
4499:{\displaystyle \exp(j\theta )=\cosh(\theta )+j\sinh(\theta ).}
2895:
The split-complex conjugate in the diagonal basis is given by
2554:{\displaystyle \lVert e\rVert =\lVert e^{*}\rVert =e^{*}e=0~.}
528:
and component-wise operations of addition and multiplication,
6495:
Contribucion al
Estudio de las Ciencias Fisicas y Matematicas
5847:{\displaystyle \{z=\sigma je^{aj}:\sigma \in \mathbb {R} \}}
4844:{\displaystyle e^{j(\theta +\phi )}=e^{j\theta }e^{j\phi }.}
1245:
Just as for complex numbers, one can define the notion of a
6553:
The Number Theory of a System of
Hyperbolic Complex Numbers
4264:{\displaystyle \left\{z:\lVert z\rVert ^{2}=-a^{2}\right\}}
3672:
4127:{\displaystyle \left\{z:\lVert z\rVert ^{2}=a^{2}\right\}}
2262:
Split-complex numbers which are not invertible are called
6350:, Basel. Chapter 4: Trigonometry in the Minkowski plane.
5984:, Richard D. Schafer, and others. The gamma factor, with
5563:
The use of split-complex numbers dates back to 1848 when
4877:
terms, the split-complex numbers can be described as the
2113:{\displaystyle \lVert z\rVert ^{2}=\langle z,z\rangle ~.}
743:
relates proportional quadratic forms, but the mapping is
6818:, translated by E. Primrose from 1963 Russian original,
1240:
847:
is an ordered pair of real numbers, written in the form
6497:, Universidad Nacional de la Plata, Republica Argentina
6140:, W. Miller & R. Boehning (1968), G. Sobczyk (1995)
3541:
then corresponds under this linear transformation to a
6152:, P. Fjelstad (1986) and Poodiack & LeClair (2009)
5619:
represents an event in a spatio-temporal plane, where
5377:
5280:
3838:{\displaystyle \{\cosh a+j\sinh a:a\in \mathbb {R} \}}
3407:
3205:
2575:
for the split-complex plane. This basis is called the
2365:
2308:
2012:
7349:
7272:
7233:
7195:
7167:
7139:
7111:
7032:
6999:
6966:
6938:
6910:
5881:
5798:
5716:
5655:
5521:
5486:
5429:
5365:
5268:
5233:
5195:
5143:
5072:
4969:
4932:
4892:
4783:
4690:
4647:
4618:
4435:
4364:
4301:
4212:
4152:
4078:
4036:
4003:
3957:
3857:
3790:
3715:
3683:
3552:
3516:
3374:
3284:
3166:
3129:
3082:
3042:
2960:
2901:
2754:
2687:
2595:
2494:
2439:
2410:
2350:
2300:
2195:
2145:
2069:
2061:. Another expression for the squared modulus is then
1985:
1943:
1908:
1772:
1686:
1581:
1541:
1366:
1306:
1257:
1024:
959:
913:
855:
796:
763:
650:
607:
536:
503:
420:
368:
331:
246:
206:
161:
115:
70:
4553:
has modulus 1, multiplying any split-complex number
5631:. The future corresponds to the quadrant of events
4339:{\displaystyle \left\{z:\lVert z\rVert =0\right\}.}
3108:with addition and multiplication defined pairwise.
7357:
7280:
7241:
7203:
7175:
7147:
7119:
7042:
7007:
6974:
6946:
6918:
5919:
5846:
5776:
5681:
5649:, which has the split-complex polar decomposition
5548:
5507:
5457:
5413:
5316:
5254:
5223:One can easily represent split-complex numbers by
5206:
5167:
5111:
5016:
4954:
4909:
4843:
4749:
4672:
4633:
4498:
4379:
4338:
4263:
4163:
4126:
4057:
4018:
3981:
3873:
3837:
3773:
3699:
3644:Though lying in the same isomorphism class in the
3634:
3533:
3482:
3358:
3268:
3150:
3098:
3057:
3003:
2944:
2885:
2718:
2671:
2553:
2478:
2425:
2396:
2336:
2252:
2163:
2112:
2053:
1967:
1929:
1892:
1728:
1664:
1562:
1521:
1340:
1284:
1215:
981:
935:
876:
806:
778:
733:
625:
581:
518:
468:
388:
352:
306:
228:
192:
139:
89:
27:The reals with an extra square root of +1 adjoined
6750:Hazewinkle, M. (1994) "Double and dual numbers",
6020:Contribución a las Ciencias Físicas y Matemáticas
5678:
3530:
823:Split-complex numbers have many other names; see
7567:
6724:Anthony A. Harkin & Joseph B. Harkin (2004)
6429:(1985) "Transformations in Special Relativity",
5341:of 2x2 real matrices. The real multiples of the
3496:hyperbolas are brought into correspondence with
2746:, then split-complex multiplication is given by
582:{\displaystyle (\mathbb {R} ^{2},+,\times ,xy),}
6702:V.Cruceanu, P. Fortuny & P.M. Gadea (1996)
5349:in the matrix ring M(2,R). Any hyperbolic unit
4273:with an upper and lower branch passing through
3874:{\displaystyle \mathbb {R} \oplus \mathbb {R} }
3700:{\displaystyle \mathbb {R} \oplus \mathbb {R} }
3099:{\displaystyle \mathbb {R} \oplus \mathbb {R} }
3026:relates the action of the hyperbolic versor on
6583:
6332:Semi-complex analysis and mathematical physics
6311:London-Edinburgh-Dublin Philosophical Magazine
5998:generalizing Cayley–Dickson algebras." Taking
6868:
6532:Bulletin de l'Académie polonaise des sciences
6049:Bulletin de l’Académie polonaise des sciences
5693:can be reached from the origin by entering a
5060:The algebra of split-complex numbers forms a
4775:since the usual exponential formula applies:
4067:can be described with split-complex numbers.
6431:International Journal of Theoretical Physics
6015:corresponds to the algebra of this article.
5841:
5799:
5423:which square to the identity matrix satisfy
5103:
5097:
5094:
5088:
5082:
5073:
4319:
4313:
4231:
4224:
4097:
4090:
3832:
3791:
3765:
3716:
2980:
2961:
2520:
2507:
2501:
2495:
2232:
2226:
2152:
2146:
2101:
2089:
2077:
2070:
1785:
1773:
1717:
1711:
1708:
1702:
1696:
1687:
1589:
1582:
4857:does not lie on one of the diagonals, then
4520:has odd powers. For all real values of the
3941:with the Minkowski inner product is called
3359:{\displaystyle uv=(x+y)(x-y)=x^{2}-y^{2}~.}
3004:{\displaystyle \lVert (a,b)\rVert ^{2}=ab.}
2397:{\displaystyle e^{*}={\tfrac {1}{2}}(1+j).}
997:a real number but an independent quantity.
7550:
6875:
6861:
2672:{\displaystyle z=x+jy=(x-y)e+(x+y)e^{*}~.}
1354:which satisfies similar properties to the
7351:
7274:
7235:
7197:
7169:
7141:
7113:
7001:
6968:
6940:
6912:
6825:
6372:. Springer Science & Business Media.
5837:
5777:{\displaystyle e^{aj}\ e^{bj}=e^{(a+b)j}}
5218:
5197:
5145:
4971:
4894:
4701:
4367:
4154:
4039:
4006:
3960:
3867:
3859:
3828:
3746:
3738:
3693:
3685:
3092:
3084:
3045:
835:for functions of a split-complex number.
766:
667:
610:
542:
506:
382:
307:{\displaystyle N(z):=zz^{*}=x^{2}-y^{2},}
6570:Introduction to Hyperbolic Number Theory
6392:
6130:(1968), Kantor and Solodovnikov (1989),
5708:nanoseconds. The split-complex equation
5583:. Extending the analogy, functions of a
5030:in the quotient is the "imaginary" unit
3890:
3017:
1016:of split-complex numbers are defined by
6726:Geometry of Generalized Complex Numbers
6663:Vorlesungen uber Geometrie der Algebren
6344:The Mathematics of Minkowski Space-Time
6252:, pages 2, 161, Imperial College Press
5515:can be represented by the matrix
5017:{\displaystyle \mathbb {R} /(x^{2}-1).}
4868:
14:
7568:
6461:On Generalized Cayley-Dickson Algebras
6180:, Cruceanu, Fortuny & Gadea (1996)
5038:over the real numbers. The algebra is
4348:These two lines (sometimes called the
3709:plane with its "unit circle" given by
2337:{\displaystyle e={\tfrac {1}{2}}(1-j)}
400:algebra over the field of real numbers
32:double-precision floating-point format
6856:
6764:, pp 66, 157, Universitext, Springer
6708:Rocky Mountain Journal of Mathematics
6566:
6099:, J.C. Vignaux (1935), G. Cree (1949)
5587:contrast to functions of an ordinary
4293:which form the set of null elements:
2285:
2187:of an invertible element is given by
2164:{\displaystyle \lVert z\rVert \neq 0}
2133:A split-complex number is invertible
1241:Conjugate, modulus, and bilinear form
6549:
4516:has only even powers while that for
3845:of the split-complex plane has only
2587:can be written in the null basis as
1738:However, this quadratic form is not
7064:Set-theoretically definable numbers
6534:, Vol. 4, No. 5, pp. 253–257,
4673:{\displaystyle z\mapsto \pm z^{*}.}
4508:This formula can be derived from a
3982:{\displaystyle \mathbb {R} ^{1,1}.}
389:{\displaystyle x,y\in \mathbb {R} }
24:
7035:
6882:
6677:Mathematics and Computer Education
6670:Mathematics and Computer Education
6638:
6291:Abstract Algebra/2x2 real matrices
4722:
4719:
4058:{\displaystyle \mathbb {R} ^{1,1}}
2404:Recall that idempotent means that
1991:
1988:
1835:
1832:
1796:
1793:
751:since the multiplicative identity
25:
7597:
6739:Academic Press, San Diego. 1990.
6296:
5602:plane. In that model, the number
5262:can be represented by the matrix
4427:for the split-complex numbers is
2486:Both of these elements are null:
2479:{\displaystyle e^{*}e^{*}=e^{*}.}
626:{\displaystyle \mathbb {R} ^{2},}
7549:
6704:A Survey on Paracomplex Geometry
6370:Geometry of Minkowski Space-Time
6285:
6037:of a triangle inscribed in
5920:{\displaystyle z^{*}w+zw^{*}=0.}
4380:{\displaystyle \mathbb {R} ^{2}}
4019:{\displaystyle \mathbb {R} ^{2}}
3058:{\displaystyle \mathbb {R} ^{2}}
779:{\displaystyle \mathbb {R} ^{2}}
519:{\displaystyle \mathbb {R} ^{2}}
6806:The College Mathematics Journal
6788:Complex Numbers in N Dimensions
6686:, Appl. Math. Comput. 28:47–72.
6612:
6577:
6573:(MA thesis). McGill University.
6560:
6556:(MA thesis). McGill University.
6543:
6513:
6500:
6487:
6474:
6453:
6436:
6024:National University of La Plata
5972:property. He realized that the
5682:{\displaystyle z=\rho e^{aj}\!}
3609:
2945:{\displaystyle (a,b)^{*}=(b,a)}
816:from 0, which is normalized in
480:over the algebra product makes
469:{\displaystyle N(wz)=N(w)N(z).}
97:A split-complex number has two
7043:{\displaystyle {\mathcal {P}}}
6754:, Soviet/AMS/Kluwer, Dordrect.
6465:Pacific Journal of Mathematics
6420:
6386:
6361:
6336:
6324:
6279:
6262:
6242:
5766:
5754:
5272:
5162:
5149:
5008:
4989:
4981:
4975:
4904:
4898:
4804:
4792:
4744:
4732:
4714:
4711:
4697:
4651:
4634:{\displaystyle z\mapsto \pm z}
4622:
4512:expansion using the fact that
4490:
4484:
4469:
4463:
4451:
4442:
3731:
3719:
3574:
3571:
3559:
3399:
3375:
3321:
3309:
3306:
3294:
3254:
3242:
3197:
3185:
3179:
3167:
3013:
2976:
2964:
2939:
2927:
2915:
2902:
2650:
2638:
2629:
2617:
2563:It is often convenient to use
2388:
2376:
2331:
2319:
2042:
2023:
2005:
1999:
1437:
1427:
1384:
1371:
1203:
1185:
1176:
1158:
1148:
1133:
1130:
1115:
1108:
1096:
1087:
1075:
1065:
1050:
1044:
1029:
724:
700:
697:
662:
573:
537:
460:
454:
448:
442:
433:
424:
256:
250:
13:
1:
7398:Plane-based geometric algebra
6508:American Mathematical Monthly
6319:Biodiversity Heritage Library
6307:On a New Imaginary in Algebra
6270:On a New Imaginary in Algebra
6236:
5334:of the corresponding matrix.
5207:{\displaystyle \mathbb {R} .}
4164:{\displaystyle \mathbb {R} .}
838:
825:
402:. Two split-complex numbers
325:of all split-complex numbers
7358:{\displaystyle \mathbb {S} }
7281:{\displaystyle \mathbb {C} }
7242:{\displaystyle \mathbb {R} }
7204:{\displaystyle \mathbb {O} }
7176:{\displaystyle \mathbb {H} }
7148:{\displaystyle \mathbb {C} }
7120:{\displaystyle \mathbb {R} }
7008:{\displaystyle \mathbb {A} }
6975:{\displaystyle \mathbb {Q} }
6947:{\displaystyle \mathbb {Z} }
6919:{\displaystyle \mathbb {N} }
6752:Encyclopaedia of Mathematics
6521:"Calculus of Approximations"
6330:Francesco Antonuccio (1994)
6117:, Warmus (1956), for use in
5168:{\displaystyle \mathbb {R} }
4910:{\displaystyle \mathbb {R} }
4590:generalized orthogonal group
2266:. These are all of the form
1341:{\displaystyle z^{*}=x-jy~.}
7:
6836:10.1007/978-3-319-07058-2_7
6816:Complex Numbers in Geometry
6719:American Journal of Physics
6448:College Mathematics Journal
6395:American Journal of Physics
6214:
6065:
5974:Cayley–Dickson construction
5623:is measured in seconds and
5458:{\displaystyle a^{2}+bc=1.}
5227:. The split-complex number
3912: Conjugate hyperbola:
3886:
2952:and the squared modulus by
2719:{\displaystyle z=ae+be^{*}}
2583:. The split-complex number
1000:The collection of all such
807:{\displaystyle {\sqrt {2}}}
495:A similar algebra based on
193:{\displaystyle z^{*}=x-yj.}
10:
7602:
6762:A Taste of Jordan Algebras
6624:Kluwer Academic Publishers
6097:hyperbolic complex numbers
5558:
5549:{\displaystyle x\ I+y\ m.}
4853:If a split-complex number
3278:Now the quadratic form is
1535:of a split-complex number
29:
7540:
7482:
7408:
7388:Algebra of physical space
7310:
7218:
7089:
6891:
6737:Spinors and calibrations.
6710:26(1): 83–115, link from
6567:Smith, Norman E. (1949).
6248:Vladimir V. Kisil (2012)
4561:preserves the modulus of
4527:the split-complex number
2290:There are two nontrivial
2173:thus numbers of the form
982:{\displaystyle i^{2}=-1.}
7444:Extended complex numbers
7427:Extended natural numbers
6693:William Kingdon Clifford
6550:Cree, George C. (1949).
6467:20(3):415–22, link from
6055:reviewed the article in
5573:William Kingdon Clifford
4955:{\displaystyle x^{2}-1,}
3534:{\displaystyle e^{bj}\!}
2681:If we denote the number
2128:indefinite inner product
1572:isotropic quadratic form
1285:{\displaystyle z=x+jy~,}
936:{\displaystyle j^{2}=+1}
316:isotropic quadratic form
236:the product of a number
229:{\displaystyle j^{2}=1,}
90:{\displaystyle j^{2}=1.}
6586:The Mathematics Teacher
6444:Hyperbolic Number Plane
6144:anormal-complex numbers
6109:real hyperbolic numbers
4418:simultaneous hyperplane
3995:of the Euclidean plane
3937:A two-dimensional real
3158:and making the mapping
2137:its modulus is nonzero
1968:{\displaystyle w=u+jv.}
1247:split-complex conjugate
1225:This multiplication is
829:below. See the article
140:{\displaystyle z=x+yj.}
7500:Transcendental numbers
7359:
7336:Hyperbolic quaternions
7282:
7243:
7205:
7177:
7149:
7121:
7044:
7009:
6976:
6948:
6920:
6786:Olariu, Silviu (2002)
6620:Geometry of Lie Groups
6459:Robert B. Brown (1967)
6274:Philosophical Magazine
6174:, F. Antonuccio (1994)
6105:, U. Bencivenga (1946)
6093:, W.K. Clifford (1882)
6082:, James Cockle (1848)
5921:
5848:
5778:
5689:. The model says that
5683:
5550:
5509:
5508:{\displaystyle z=x+jy}
5459:
5415:
5318:
5256:
5255:{\displaystyle z=x+jy}
5219:Matrix representations
5208:
5187:over the real numbers
5169:
5113:
5018:
4956:
4911:
4845:
4751:
4674:
4635:
4500:
4420:concept in spacetime.
4392:Split-complex numbers
4381:
4340:
4265:
4165:
4128:
4059:
4020:
3983:
3934:
3899: Unit hyperbola:
3875:
3839:
3775:
3701:
3636:
3535:
3484:
3360:
3270:
3152:
3151:{\displaystyle z=x+jy}
3100:
3067:
3059:
3005:
2946:
2887:
2720:
2673:
2555:
2480:
2427:
2398:
2338:
2254:
2185:multiplicative inverse
2165:
2114:
2055:
1969:
1931:
1930:{\displaystyle z=x+jy}
1894:
1730:
1666:
1564:
1563:{\displaystyle z=x+jy}
1523:
1342:
1294:then the conjugate of
1286:
1217:
983:
937:
878:
877:{\displaystyle z=x+jy}
808:
780:
735:
627:
583:
520:
470:
390:
354:
353:{\displaystyle z=x+yj}
308:
240:with its conjugate is
230:
194:
141:
91:
7432:Extended real numbers
7360:
7283:
7253:Split-complex numbers
7244:
7206:
7178:
7150:
7122:
7045:
7010:
6986:Constructible numbers
6977:
6949:
6921:
6618:Rosenfeld, B. (1997)
6210:, K. McCrimmon (2004)
6186:, B. Rosenfeld (1997)
6184:split-complex numbers
6134:(1990), Rooney (2014)
5922:
5873:hyperbolic-orthogonal
5849:
5779:
5684:
5551:
5510:
5460:
5416:
5319:
5257:
5209:
5170:
5114:
5019:
4957:
4912:
4846:
4752:
4675:
4636:
4603:, combined with four
4501:
4402:hyperbolic-orthogonal
4382:
4341:
4266:
4166:
4129:
4060:
4021:
3984:
3894:
3876:
3840:
3776:
3702:
3637:
3536:
3485:
3361:
3271:
3153:
3101:
3060:
3021:
3006:
2947:
2888:
2721:
2674:
2556:
2481:
2428:
2399:
2339:
2278:for some real number
2255:
2183:have no inverse. The
2166:
2115:
2056:
1970:
1932:
1895:
1731:
1667:
1565:
1524:
1343:
1287:
1218:
984:
938:
879:
809:
781:
736:
628:
584:
521:
471:
391:
355:
309:
231:
195:
142:
92:
7586:Hypercomplex numbers
7576:Composition algebras
7464:Supernatural numbers
7374:Multicomplex numbers
7347:
7331:Dual-complex numbers
7270:
7231:
7193:
7165:
7137:
7109:
7091:Composition algebras
7059:Arithmetical numbers
7030:
6997:
6964:
6936:
6908:
6730:Mathematics Magazine
6598:10.5951/MT.61.4.0377
6482:Mathematical Reviews
6446:, also published in
6268:James Cockle (1848)
6198:, P. Lounesto (2001)
6172:semi-complex numbers
6168:, F.R. Harvey (1990)
6057:Mathematical Reviews
6035:nine-point hyperbola
5992:Mathematical Reviews
5879:
5796:
5714:
5653:
5519:
5484:
5427:
5363:
5266:
5231:
5193:
5141:
5070:
4967:
4930:
4890:
4869:Algebraic properties
4781:
4688:
4682:The exponential map
4645:
4616:
4433:
4389:and have slopes ±1.
4362:
4299:
4210:
4150:
4076:
4034:
4001:
3991:Just as much of the
3955:
3855:
3788:
3713:
3681:
3550:
3514:
3372:
3282:
3164:
3127:
3080:
3040:
2958:
2899:
2752:
2685:
2593:
2492:
2437:
2426:{\displaystyle ee=e}
2408:
2348:
2298:
2193:
2143:
2067:
1983:
1941:
1906:
1770:
1749:, so the modulus is
1684:
1579:
1539:
1364:
1350:The conjugate is an
1304:
1255:
1022:
957:
911:
853:
845:split-complex number
794:
761:
648:
605:
534:
501:
476:This composition of
418:
366:
329:
244:
204:
159:
113:
68:
43:split-complex number
7369:Split-biquaternions
7081:Eisenstein integers
7019:Closed-form numbers
6682:K. Carmody, (1988)
6648:, Ser (3) v.2 No7.
6407:1986AmJPh..54..416F
6231:Hypercomplex number
6178:paracomplex numbers
6115:approximate numbers
6028:República Argentina
5970:composition algebra
5577:split-biquaternions
5062:composition algebra
5036:commutative algebra
4863:polar decomposition
4567:hyperbolic rotation
4202:conjugate hyperbola
3675:of a sector in the
3030:to squeeze mapping
3024:commutative diagram
2292:idempotent elements
1676:composition algebra
1006:split-complex plane
490:composition algebra
7527:Profinite integers
7490:Irrational numbers
7355:
7278:
7239:
7201:
7173:
7145:
7117:
7074:Gaussian rationals
7054:Computable numbers
7040:
7005:
6972:
6944:
6916:
6697:Mathematical Works
6526:2012-03-09 at the
6442:Sobczyk, G.(1995)
6317::435–9, link from
6204:, S. Olariu (2002)
6202:twocomplex numbers
6192:, N. Borota (2000)
6138:hyperbolic numbers
5917:
5844:
5774:
5695:frame of reference
5679:
5546:
5505:
5465:For example, when
5455:
5411:
5405:
5314:
5305:
5252:
5204:
5165:
5109:
5014:
4952:
4907:
4841:
4747:
4670:
4631:
4577:). Multiplying by
4544:hyperbolic versors
4496:
4377:
4336:
4261:
4161:
4140:for every nonzero
4124:
4070:The set of points
4055:
4016:
3979:
3935:
3925: Asymptotes:
3871:
3835:
3771:
3697:
3632:
3531:
3480:
3435:
3356:
3266:
3233:
3148:
3096:
3074:to the direct sum
3068:
3055:
3001:
2942:
2883:
2716:
2669:
2551:
2476:
2423:
2394:
2374:
2334:
2317:
2286:The diagonal basis
2250:
2161:
2110:
2051:
2021:
1965:
1927:
1890:
1726:
1662:
1560:
1519:
1517:
1338:
1282:
1213:
1211:
979:
933:
874:
804:
776:
731:
729:
623:
579:
516:
466:
386:
350:
304:
226:
190:
137:
87:
7563:
7562:
7474:Superreal numbers
7454:Levi-Civita field
7449:Hyperreal numbers
7393:Spacetime algebra
7379:Geometric algebra
7292:Bicomplex numbers
7258:Split-quaternions
7099:Division algebras
7069:Gaussian integers
6991:Algebraic numbers
6894:definable numbers
6845:978-3-319-07058-2
6822:, pp. 18–20.
6735:F. Reese Harvey.
6519:M. Warmus (1956)
6379:978-3-642-17977-8
6356:978-3-7643-8613-9
6348:Birkhäuser Verlag
6258:978-1-84816-858-9
6190:spacetime numbers
6119:interval analysis
6111:, N. Smith (1949)
5927:Canonical events
5732:
5539:
5527:
5108:
4923:generated by the
4773:group isomorphism
4565:and represents a
4542:have been called
3665:hyperbolic sector
3646:category of rings
3628:
3599:
3509:hyperbolic versor
3352:
3262:
2879:
2726:for real numbers
2665:
2547:
2373:
2316:
2246:
2242:
2106:
2020:
1886:
1740:positive-definite
1722:
1658:
1356:complex conjugate
1334:
1278:
802:
788:is at a distance
109:, and is written
47:hyperbolic number
16:(Redirected from
7593:
7553:
7552:
7520:
7510:
7422:Cardinal numbers
7383:Clifford algebra
7364:
7362:
7361:
7356:
7354:
7326:Dual quaternions
7287:
7285:
7284:
7279:
7277:
7248:
7246:
7245:
7240:
7238:
7210:
7208:
7207:
7202:
7200:
7182:
7180:
7179:
7174:
7172:
7154:
7152:
7151:
7146:
7144:
7126:
7124:
7123:
7118:
7116:
7049:
7047:
7046:
7041:
7039:
7038:
7014:
7012:
7011:
7006:
7004:
6981:
6979:
6978:
6973:
6971:
6958:Rational numbers
6953:
6951:
6950:
6945:
6943:
6925:
6923:
6922:
6917:
6915:
6877:
6870:
6863:
6854:
6853:
6849:
6633:
6616:
6610:
6609:
6581:
6575:
6574:
6564:
6558:
6557:
6547:
6541:
6517:
6511:
6504:
6498:
6491:
6485:
6478:
6472:
6457:
6451:
6440:
6434:
6424:
6418:
6417:
6390:
6384:
6383:
6365:
6359:
6340:
6334:
6328:
6322:
6300:
6294:
6289:
6283:
6277:
6266:
6260:
6246:
6226:Split-quaternion
6162:, Carmody (1988)
6146:, W. Benz (1973)
6043:
6014:
6007:
5989:
5979:
5956:
5945:
5934:
5926:
5924:
5923:
5918:
5910:
5909:
5891:
5890:
5870:
5866:
5853:
5851:
5850:
5845:
5840:
5826:
5825:
5789:
5783:
5781:
5780:
5775:
5773:
5772:
5745:
5744:
5730:
5729:
5728:
5707:
5703:
5692:
5688:
5686:
5685:
5680:
5677:
5676:
5648:
5642:
5626:
5618:
5589:complex variable
5555:
5553:
5552:
5547:
5537:
5525:
5514:
5512:
5511:
5506:
5464:
5462:
5461:
5456:
5439:
5438:
5420:
5418:
5417:
5412:
5410:
5409:
5330:is given by the
5329:
5323:
5321:
5320:
5315:
5310:
5309:
5261:
5259:
5258:
5253:
5215:
5213:
5211:
5210:
5205:
5200:
5186:
5176:
5174:
5172:
5171:
5166:
5161:
5160:
5148:
5128:
5124:
5121:for any numbers
5118:
5116:
5115:
5110:
5106:
5055:topological ring
5033:
5029:
5023:
5021:
5020:
5015:
5001:
5000:
4988:
4974:
4961:
4959:
4958:
4953:
4942:
4941:
4918:
4916:
4914:
4913:
4908:
4897:
4875:abstract algebra
4860:
4856:
4850:
4848:
4847:
4842:
4837:
4836:
4824:
4823:
4808:
4807:
4770:
4762:
4756:
4754:
4753:
4748:
4731:
4730:
4725:
4704:
4679:
4677:
4676:
4671:
4666:
4665:
4640:
4638:
4637:
4632:
4602:
4594:
4580:
4564:
4560:
4556:
4552:
4541:
4537:
4526:
4522:hyperbolic angle
4505:
4503:
4502:
4497:
4423:The analogue of
4415:
4399:
4395:
4388:
4386:
4384:
4383:
4378:
4376:
4375:
4370:
4345:
4343:
4342:
4337:
4332:
4328:
4288:
4280:
4270:
4268:
4267:
4262:
4260:
4256:
4255:
4254:
4239:
4238:
4195:
4188:
4180:
4172:
4170:
4168:
4167:
4162:
4157:
4143:
4133:
4131:
4130:
4125:
4123:
4119:
4118:
4117:
4105:
4104:
4066:
4064:
4062:
4061:
4056:
4054:
4053:
4042:
4027:
4025:
4023:
4022:
4017:
4015:
4014:
4009:
3990:
3988:
3986:
3985:
3980:
3975:
3974:
3963:
3949:, often denoted
3944:
3932:
3924:
3919:
3911:
3906:
3898:
3882:
3880:
3878:
3877:
3872:
3870:
3862:
3844:
3842:
3841:
3836:
3831:
3780:
3778:
3777:
3772:
3749:
3741:
3708:
3706:
3704:
3703:
3698:
3696:
3688:
3669:hyperbolic angle
3662:
3661:
3641:
3639:
3638:
3633:
3626:
3625:
3624:
3605:
3601:
3600:
3592:
3540:
3538:
3537:
3532:
3529:
3528:
3499:
3489:
3487:
3486:
3481:
3479:
3475:
3474:
3473:
3458:
3457:
3440:
3439:
3365:
3363:
3362:
3357:
3350:
3349:
3348:
3336:
3335:
3275:
3273:
3272:
3267:
3260:
3238:
3237:
3157:
3155:
3154:
3149:
3122:
3107:
3105:
3103:
3102:
3097:
3095:
3087:
3066:
3064:
3062:
3061:
3056:
3054:
3053:
3048:
3033:
3029:
3010:
3008:
3007:
3002:
2988:
2987:
2951:
2949:
2948:
2943:
2923:
2922:
2892:
2890:
2889:
2884:
2877:
2876:
2872:
2871:
2870:
2861:
2860:
2848:
2847:
2838:
2837:
2820:
2816:
2815:
2814:
2802:
2801:
2787:
2783:
2782:
2781:
2769:
2768:
2745:
2733:
2729:
2725:
2723:
2722:
2717:
2715:
2714:
2678:
2676:
2675:
2670:
2663:
2662:
2661:
2586:
2571:as an alternate
2570:
2566:
2560:
2558:
2557:
2552:
2545:
2535:
2534:
2519:
2518:
2485:
2483:
2482:
2477:
2472:
2471:
2459:
2458:
2449:
2448:
2432:
2430:
2429:
2424:
2403:
2401:
2400:
2395:
2375:
2366:
2360:
2359:
2343:
2341:
2340:
2335:
2318:
2309:
2281:
2277:
2259:
2257:
2256:
2251:
2244:
2243:
2241:
2240:
2235:
2223:
2222:
2213:
2208:
2207:
2182:
2172:
2170:
2168:
2167:
2162:
2119:
2117:
2116:
2111:
2104:
2085:
2084:
2060:
2058:
2057:
2052:
2041:
2040:
2022:
2013:
1995:
1994:
1974:
1972:
1971:
1966:
1936:
1934:
1933:
1928:
1899:
1897:
1896:
1891:
1884:
1865:
1861:
1857:
1856:
1839:
1838:
1826:
1822:
1821:
1820:
1800:
1799:
1748:
1735:
1733:
1732:
1727:
1720:
1671:
1669:
1668:
1663:
1656:
1655:
1654:
1642:
1641:
1626:
1625:
1613:
1612:
1597:
1596:
1570:is given by the
1569:
1567:
1566:
1561:
1528:
1526:
1525:
1520:
1518:
1501:
1500:
1495:
1491:
1490:
1472:
1471:
1462:
1461:
1445:
1444:
1422:
1421:
1409:
1408:
1392:
1391:
1347:
1345:
1344:
1339:
1332:
1316:
1315:
1297:
1291:
1289:
1288:
1283:
1276:
1222:
1220:
1219:
1214:
1212:
1003:
992:
988:
986:
985:
980:
969:
968:
945:In the field of
942:
940:
939:
934:
923:
922:
904:
893:
889:
883:
881:
880:
875:
819:
815:
813:
811:
810:
805:
803:
798:
787:
785:
783:
782:
777:
775:
774:
769:
754:
740:
738:
737:
732:
730:
676:
675:
670:
641:ring isomorphism
634:
632:
630:
629:
624:
619:
618:
613:
594:
590:
588:
586:
585:
580:
551:
550:
545:
527:
525:
523:
522:
517:
515:
514:
509:
487:
479:
475:
473:
472:
467:
413:
409:
405:
397:
395:
393:
392:
387:
385:
359:
357:
356:
351:
324:
313:
311:
310:
305:
300:
299:
287:
286:
274:
273:
239:
235:
233:
232:
227:
216:
215:
199:
197:
196:
191:
171:
170:
154:
146:
144:
143:
138:
108:
104:
96:
94:
93:
88:
80:
79:
63:
57:) is based on a
21:
7601:
7600:
7596:
7595:
7594:
7592:
7591:
7590:
7566:
7565:
7564:
7559:
7536:
7515:
7505:
7478:
7469:Surreal numbers
7459:Ordinal numbers
7404:
7350:
7348:
7345:
7344:
7306:
7273:
7271:
7268:
7267:
7265:
7263:Split-octonions
7234:
7232:
7229:
7228:
7220:
7214:
7196:
7194:
7191:
7190:
7168:
7166:
7163:
7162:
7140:
7138:
7135:
7134:
7131:Complex numbers
7112:
7110:
7107:
7106:
7085:
7034:
7033:
7031:
7028:
7027:
7000:
6998:
6995:
6994:
6967:
6965:
6962:
6961:
6939:
6937:
6934:
6933:
6911:
6909:
6906:
6905:
6902:Natural numbers
6887:
6881:
6846:
6758:Kevin McCrimmon
6641:
6639:Further reading
6636:
6617:
6613:
6582:
6578:
6565:
6561:
6548:
6544:
6528:Wayback Machine
6518:
6514:
6510:48(10): 675–681
6505:
6501:
6492:
6488:
6479:
6475:
6458:
6454:
6441:
6437:
6425:
6421:
6415:10.1119/1.14605
6391:
6387:
6380:
6366:
6362:
6341:
6337:
6329:
6325:
6301:
6297:
6284:
6280:
6267:
6263:
6247:
6243:
6239:
6221:Minkowski space
6217:
6208:split binarions
6166:Lorentz numbers
6150:perplex numbers
6068:
6038:
6009:
5999:
5985:
5977:
5966:split-octonions
5947:
5936:
5928:
5905:
5901:
5886:
5882:
5880:
5877:
5876:
5868:
5864:
5836:
5818:
5814:
5797:
5794:
5793:
5787:
5753:
5749:
5737:
5733:
5721:
5717:
5715:
5712:
5711:
5705:
5701:
5690:
5669:
5665:
5654:
5651:
5650:
5638:
5632:
5624:
5603:
5561:
5520:
5517:
5516:
5485:
5482:
5481:
5434:
5430:
5428:
5425:
5424:
5404:
5403:
5395:
5389:
5388:
5383:
5373:
5372:
5364:
5361:
5360:
5343:identity matrix
5327:
5304:
5303:
5298:
5292:
5291:
5286:
5276:
5275:
5267:
5264:
5263:
5232:
5229:
5228:
5221:
5196:
5194:
5191:
5190:
5188:
5185:
5181:
5156:
5152:
5144:
5142:
5139:
5138:
5136:
5126:
5122:
5071:
5068:
5067:
5031:
5027:
4996:
4992:
4984:
4970:
4968:
4965:
4964:
4937:
4933:
4931:
4928:
4927:
4893:
4891:
4888:
4887:
4885:
4883:polynomial ring
4871:
4858:
4854:
4829:
4825:
4816:
4812:
4788:
4784:
4782:
4779:
4778:
4764:
4763:to rotation by
4760:
4726:
4718:
4717:
4700:
4689:
4686:
4685:
4661:
4657:
4646:
4643:
4642:
4617:
4614:
4613:
4600:
4592:
4578:
4575:squeeze mapping
4569:(also called a
4562:
4558:
4554:
4550:
4539:
4528:
4524:
4434:
4431:
4430:
4425:Euler's formula
4405:
4400:are said to be
4397:
4393:
4371:
4366:
4365:
4363:
4360:
4359:
4357:
4306:
4302:
4300:
4297:
4296:
4282:
4274:
4250:
4246:
4234:
4230:
4217:
4213:
4211:
4208:
4207:
4190:
4182:
4174:
4153:
4151:
4148:
4147:
4145:
4141:
4113:
4109:
4100:
4096:
4083:
4079:
4077:
4074:
4073:
4043:
4038:
4037:
4035:
4032:
4031:
4029:
4010:
4005:
4004:
4002:
3999:
3998:
3996:
3964:
3959:
3958:
3956:
3953:
3952:
3950:
3947:Minkowski space
3942:
3933:
3926:
3922:
3920:
3913:
3909:
3907:
3900:
3896:
3889:
3866:
3858:
3856:
3853:
3852:
3850:
3827:
3789:
3786:
3785:
3781:The contracted
3745:
3737:
3714:
3711:
3710:
3692:
3684:
3682:
3679:
3678:
3676:
3671:corresponds to
3659:
3657:
3650:Cartesian plane
3620:
3616:
3591:
3581:
3577:
3551:
3548:
3547:
3543:squeeze mapping
3521:
3517:
3515:
3512:
3511:
3497:
3466:
3462:
3453:
3449:
3448:
3444:
3434:
3433:
3425:
3419:
3418:
3413:
3403:
3402:
3373:
3370:
3369:
3344:
3340:
3331:
3327:
3283:
3280:
3279:
3232:
3231:
3223:
3217:
3216:
3211:
3201:
3200:
3165:
3162:
3161:
3128:
3125:
3124:
3112:
3091:
3083:
3081:
3078:
3077:
3075:
3072:ring-isomorphic
3049:
3044:
3043:
3041:
3038:
3037:
3035:
3031:
3027:
3016:
2983:
2979:
2959:
2956:
2955:
2918:
2914:
2900:
2897:
2896:
2866:
2862:
2856:
2852:
2843:
2839:
2833:
2829:
2828:
2824:
2810:
2806:
2797:
2793:
2792:
2788:
2777:
2773:
2764:
2760:
2759:
2755:
2753:
2750:
2749:
2735:
2731:
2727:
2710:
2706:
2686:
2683:
2682:
2657:
2653:
2594:
2591:
2590:
2584:
2568:
2564:
2530:
2526:
2514:
2510:
2493:
2490:
2489:
2467:
2463:
2454:
2450:
2444:
2440:
2438:
2435:
2434:
2409:
2406:
2405:
2364:
2355:
2351:
2349:
2346:
2345:
2307:
2299:
2296:
2295:
2288:
2279:
2267:
2236:
2225:
2224:
2218:
2214:
2212:
2200:
2196:
2194:
2191:
2190:
2174:
2144:
2141:
2140:
2138:
2080:
2076:
2068:
2065:
2064:
2036:
2032:
2011:
1987:
1986:
1984:
1981:
1980:
1942:
1939:
1938:
1907:
1904:
1903:
1852:
1848:
1847:
1843:
1831:
1830:
1816:
1812:
1808:
1804:
1792:
1791:
1771:
1768:
1767:
1760:The associated
1746:
1742:but rather has
1685:
1682:
1681:
1650:
1646:
1637:
1633:
1621:
1617:
1608:
1604:
1592:
1588:
1580:
1577:
1576:
1540:
1537:
1536:
1516:
1515:
1502:
1496:
1486:
1482:
1478:
1477:
1474:
1473:
1467:
1463:
1457:
1453:
1446:
1440:
1436:
1424:
1423:
1417:
1413:
1404:
1400:
1393:
1387:
1383:
1367:
1365:
1362:
1361:
1311:
1307:
1305:
1302:
1301:
1295:
1256:
1253:
1252:
1243:
1237:over addition.
1210:
1209:
1151:
1112:
1111:
1068:
1025:
1023:
1020:
1019:
1001:
990:
964:
960:
958:
955:
954:
947:complex numbers
918:
914:
912:
909:
908:
902:
900:hyperbolic unit
891:
887:
854:
851:
850:
841:
826:§ Synonyms
817:
797:
795:
792:
791:
789:
770:
765:
764:
762:
759:
758:
756:
752:
728:
727:
693:
678:
677:
671:
666:
665:
658:
651:
649:
646:
645:
637:quadratic space
614:
609:
608:
606:
603:
602:
600:
592:
546:
541:
540:
535:
532:
531:
529:
510:
505:
504:
502:
499:
498:
496:
481:
477:
419:
416:
415:
414:that satisfies
411:
410:have a product
407:
403:
381:
367:
364:
363:
361:
330:
327:
326:
322:
321:The collection
295:
291:
282:
278:
269:
265:
245:
242:
241:
237:
211:
207:
205:
202:
201:
166:
162:
160:
157:
156:
152:
114:
111:
110:
106:
102:
75:
71:
69:
66:
65:
61:
59:hyperbolic unit
35:
28:
23:
22:
18:Hyperbolic unit
15:
12:
11:
5:
7599:
7589:
7588:
7583:
7581:Linear algebra
7578:
7561:
7560:
7558:
7557:
7547:
7545:Classification
7541:
7538:
7537:
7535:
7534:
7532:Normal numbers
7529:
7524:
7502:
7497:
7492:
7486:
7484:
7480:
7479:
7477:
7476:
7471:
7466:
7461:
7456:
7451:
7446:
7441:
7440:
7439:
7429:
7424:
7418:
7416:
7414:infinitesimals
7406:
7405:
7403:
7402:
7401:
7400:
7395:
7390:
7376:
7371:
7366:
7353:
7338:
7333:
7328:
7323:
7317:
7315:
7308:
7307:
7305:
7304:
7299:
7294:
7289:
7276:
7260:
7255:
7250:
7237:
7224:
7222:
7216:
7215:
7213:
7212:
7199:
7184:
7171:
7156:
7143:
7128:
7115:
7095:
7093:
7087:
7086:
7084:
7083:
7078:
7077:
7076:
7066:
7061:
7056:
7051:
7037:
7021:
7016:
7003:
6988:
6983:
6970:
6955:
6942:
6927:
6914:
6898:
6896:
6889:
6888:
6880:
6879:
6872:
6865:
6857:
6851:
6850:
6844:
6823:
6820:Academic Press
6809:
6802:
6784:
6781:
6778:
6755:
6748:
6733:
6722:
6715:
6712:Project Euclid
6700:
6690:
6687:
6680:
6673:
6666:
6656:
6640:
6637:
6635:
6634:
6611:
6592:(4): 377–382.
6576:
6559:
6542:
6512:
6499:
6486:
6473:
6469:Project Euclid
6452:
6435:
6427:Louis Kauffman
6419:
6401:(5): 416–422,
6385:
6378:
6360:
6335:
6323:
6295:
6278:
6261:
6240:
6238:
6235:
6234:
6233:
6228:
6223:
6216:
6213:
6212:
6211:
6205:
6199:
6193:
6187:
6181:
6175:
6169:
6163:
6156:countercomplex
6153:
6147:
6141:
6135:
6124:double numbers
6121:
6112:
6106:
6103:bireal numbers
6100:
6094:
6083:
6067:
6064:
5968:and noted the
5964:was using the
5916:
5913:
5908:
5904:
5900:
5897:
5894:
5889:
5885:
5843:
5839:
5835:
5832:
5829:
5824:
5821:
5817:
5813:
5810:
5807:
5804:
5801:
5771:
5768:
5765:
5762:
5759:
5756:
5752:
5748:
5743:
5740:
5736:
5727:
5724:
5720:
5675:
5672:
5668:
5664:
5661:
5658:
5637: : |
5585:motor variable
5560:
5557:
5545:
5542:
5536:
5533:
5530:
5524:
5504:
5501:
5498:
5495:
5492:
5489:
5454:
5451:
5448:
5445:
5442:
5437:
5433:
5408:
5402:
5399:
5396:
5394:
5391:
5390:
5387:
5384:
5382:
5379:
5378:
5376:
5371:
5368:
5313:
5308:
5302:
5299:
5297:
5294:
5293:
5290:
5287:
5285:
5282:
5281:
5279:
5274:
5271:
5251:
5248:
5245:
5242:
5239:
5236:
5220:
5217:
5203:
5199:
5183:
5164:
5159:
5155:
5151:
5147:
5105:
5102:
5099:
5096:
5093:
5090:
5087:
5084:
5081:
5078:
5075:
5013:
5010:
5007:
5004:
4999:
4995:
4991:
4987:
4983:
4980:
4977:
4973:
4951:
4948:
4945:
4940:
4936:
4906:
4903:
4900:
4896:
4870:
4867:
4840:
4835:
4832:
4828:
4822:
4819:
4815:
4811:
4806:
4803:
4800:
4797:
4794:
4791:
4787:
4746:
4743:
4740:
4737:
4734:
4729:
4724:
4721:
4716:
4713:
4710:
4707:
4703:
4699:
4696:
4693:
4669:
4664:
4660:
4656:
4653:
4650:
4630:
4627:
4624:
4621:
4495:
4492:
4489:
4486:
4483:
4480:
4477:
4474:
4471:
4468:
4465:
4462:
4459:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4374:
4369:
4335:
4331:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4305:
4259:
4253:
4249:
4245:
4242:
4237:
4233:
4229:
4226:
4223:
4220:
4216:
4198:unit hyperbola
4196:is called the
4160:
4156:
4122:
4116:
4112:
4108:
4103:
4099:
4095:
4092:
4089:
4086:
4082:
4052:
4049:
4046:
4041:
4013:
4008:
3978:
3973:
3970:
3967:
3962:
3921:
3908:
3895:
3888:
3885:
3869:
3865:
3861:
3834:
3830:
3826:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3783:unit hyperbola
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3748:
3744:
3740:
3736:
3733:
3730:
3727:
3724:
3721:
3718:
3695:
3691:
3687:
3631:
3623:
3619:
3615:
3612:
3608:
3604:
3598:
3595:
3590:
3587:
3584:
3580:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3527:
3524:
3520:
3478:
3472:
3469:
3465:
3461:
3456:
3452:
3447:
3443:
3438:
3432:
3429:
3426:
3424:
3421:
3420:
3417:
3414:
3412:
3409:
3408:
3406:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3355:
3347:
3343:
3339:
3334:
3330:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3265:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3236:
3230:
3227:
3224:
3222:
3219:
3218:
3215:
3212:
3210:
3207:
3206:
3204:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3147:
3144:
3141:
3138:
3135:
3132:
3094:
3090:
3086:
3052:
3047:
3015:
3012:
3000:
2997:
2994:
2991:
2986:
2982:
2978:
2975:
2972:
2969:
2966:
2963:
2941:
2938:
2935:
2932:
2929:
2926:
2921:
2917:
2913:
2910:
2907:
2904:
2882:
2875:
2869:
2865:
2859:
2855:
2851:
2846:
2842:
2836:
2832:
2827:
2823:
2819:
2813:
2809:
2805:
2800:
2796:
2791:
2786:
2780:
2776:
2772:
2767:
2763:
2758:
2713:
2709:
2705:
2702:
2699:
2696:
2693:
2690:
2668:
2660:
2656:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2577:diagonal basis
2550:
2544:
2541:
2538:
2533:
2529:
2525:
2522:
2517:
2513:
2509:
2506:
2503:
2500:
2497:
2475:
2470:
2466:
2462:
2457:
2453:
2447:
2443:
2422:
2419:
2416:
2413:
2393:
2390:
2387:
2384:
2381:
2378:
2372:
2369:
2363:
2358:
2354:
2333:
2330:
2327:
2324:
2321:
2315:
2312:
2306:
2303:
2287:
2284:
2249:
2239:
2234:
2231:
2228:
2221:
2217:
2211:
2206:
2203:
2199:
2160:
2157:
2154:
2151:
2148:
2135:if and only if
2109:
2103:
2100:
2097:
2094:
2091:
2088:
2083:
2079:
2075:
2072:
2050:
2047:
2044:
2039:
2035:
2031:
2028:
2025:
2019:
2016:
2010:
2007:
2004:
2001:
1998:
1993:
1990:
1979:is defined by
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1926:
1923:
1920:
1917:
1914:
1911:
1889:
1883:
1880:
1877:
1874:
1871:
1868:
1864:
1860:
1855:
1851:
1846:
1842:
1837:
1834:
1829:
1825:
1819:
1815:
1811:
1807:
1803:
1798:
1795:
1790:
1787:
1784:
1781:
1778:
1775:
1725:
1719:
1716:
1713:
1710:
1707:
1704:
1701:
1698:
1695:
1692:
1689:
1661:
1653:
1649:
1645:
1640:
1636:
1632:
1629:
1624:
1620:
1616:
1611:
1607:
1603:
1600:
1595:
1591:
1587:
1584:
1559:
1556:
1553:
1550:
1547:
1544:
1514:
1511:
1508:
1505:
1503:
1499:
1494:
1489:
1485:
1481:
1476:
1475:
1470:
1466:
1460:
1456:
1452:
1449:
1447:
1443:
1439:
1435:
1432:
1429:
1426:
1425:
1420:
1416:
1412:
1407:
1403:
1399:
1396:
1394:
1390:
1386:
1382:
1379:
1376:
1373:
1370:
1369:
1337:
1331:
1328:
1325:
1322:
1319:
1314:
1310:
1298:is defined as
1281:
1275:
1272:
1269:
1266:
1263:
1260:
1242:
1239:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1152:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1113:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1069:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1027:
1014:multiplication
1004:is called the
978:
975:
972:
967:
963:
951:imaginary unit
932:
929:
926:
921:
917:
873:
870:
867:
864:
861:
858:
840:
837:
832:Motor variable
801:
773:
768:
746:
726:
723:
720:
717:
714:
711:
708:
705:
702:
699:
696:
694:
692:
689:
686:
683:
680:
679:
674:
669:
664:
661:
659:
657:
654:
653:
622:
617:
612:
597:quadratic form
578:
575:
572:
569:
566:
563:
560:
557:
554:
549:
544:
539:
513:
508:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
384:
380:
377:
374:
371:
349:
346:
343:
340:
337:
334:
303:
298:
294:
290:
285:
281:
277:
272:
268:
264:
261:
258:
255:
252:
249:
225:
222:
219:
214:
210:
189:
186:
183:
180:
177:
174:
169:
165:
136:
133:
130:
127:
124:
121:
118:
86:
83:
78:
74:
51:perplex number
26:
9:
6:
4:
3:
2:
7598:
7587:
7584:
7582:
7579:
7577:
7574:
7573:
7571:
7556:
7548:
7546:
7543:
7542:
7539:
7533:
7530:
7528:
7525:
7522:
7518:
7512:
7508:
7503:
7501:
7498:
7496:
7495:Fuzzy numbers
7493:
7491:
7488:
7487:
7485:
7481:
7475:
7472:
7470:
7467:
7465:
7462:
7460:
7457:
7455:
7452:
7450:
7447:
7445:
7442:
7438:
7435:
7434:
7433:
7430:
7428:
7425:
7423:
7420:
7419:
7417:
7415:
7411:
7407:
7399:
7396:
7394:
7391:
7389:
7386:
7385:
7384:
7380:
7377:
7375:
7372:
7370:
7367:
7342:
7339:
7337:
7334:
7332:
7329:
7327:
7324:
7322:
7319:
7318:
7316:
7314:
7309:
7303:
7300:
7298:
7297:Biquaternions
7295:
7293:
7290:
7264:
7261:
7259:
7256:
7254:
7251:
7226:
7225:
7223:
7217:
7188:
7185:
7160:
7157:
7132:
7129:
7104:
7100:
7097:
7096:
7094:
7092:
7088:
7082:
7079:
7075:
7072:
7071:
7070:
7067:
7065:
7062:
7060:
7057:
7055:
7052:
7025:
7022:
7020:
7017:
6992:
6989:
6987:
6984:
6959:
6956:
6931:
6928:
6903:
6900:
6899:
6897:
6895:
6890:
6885:
6878:
6873:
6871:
6866:
6864:
6859:
6858:
6855:
6847:
6841:
6837:
6833:
6829:
6824:
6821:
6817:
6813:
6810:
6808:40(5):322–35.
6807:
6803:
6800:
6799:0-444-51123-7
6796:
6793:
6789:
6785:
6782:
6779:
6777:
6774:
6771:
6770:0-387-95447-3
6767:
6763:
6759:
6756:
6753:
6749:
6746:
6745:0-12-329650-1
6742:
6738:
6734:
6732:77(2):118–29.
6731:
6727:
6723:
6720:
6716:
6713:
6709:
6705:
6701:
6698:
6694:
6691:
6688:
6685:
6681:
6678:
6674:
6671:
6667:
6664:
6660:
6657:
6654:
6651:
6647:
6643:
6642:
6632:
6631:0-7923-4390-5
6628:
6625:
6621:
6615:
6607:
6603:
6599:
6595:
6591:
6587:
6580:
6572:
6571:
6563:
6555:
6554:
6546:
6540:
6537:
6533:
6529:
6525:
6522:
6516:
6509:
6503:
6496:
6490:
6483:
6477:
6470:
6466:
6462:
6456:
6449:
6445:
6439:
6432:
6428:
6423:
6416:
6412:
6408:
6404:
6400:
6396:
6389:
6381:
6375:
6371:
6364:
6357:
6353:
6349:
6345:
6339:
6333:
6327:
6320:
6316:
6312:
6308:
6304:
6299:
6292:
6288:
6282:
6275:
6271:
6265:
6259:
6255:
6251:
6245:
6241:
6232:
6229:
6227:
6224:
6222:
6219:
6218:
6209:
6206:
6203:
6200:
6197:
6196:Study numbers
6194:
6191:
6188:
6185:
6182:
6179:
6176:
6173:
6170:
6167:
6164:
6161:
6157:
6154:
6151:
6148:
6145:
6142:
6139:
6136:
6133:
6129:
6125:
6122:
6120:
6116:
6113:
6110:
6107:
6104:
6101:
6098:
6095:
6092:
6088:
6084:
6081:
6077:
6073:
6072:
6071:
6063:
6060:
6058:
6054:
6050:
6045:
6041:
6036:
6031:
6029:
6025:
6021:
6016:
6012:
6006:
6002:
5997:
5993:
5988:
5983:
5982:Adrian Albert
5975:
5971:
5967:
5963:
5958:
5954:
5950:
5943:
5939:
5932:
5914:
5911:
5906:
5902:
5898:
5895:
5892:
5887:
5883:
5874:
5861:
5859:
5854:
5833:
5830:
5827:
5822:
5819:
5815:
5811:
5808:
5805:
5802:
5791:
5784:
5769:
5763:
5760:
5757:
5750:
5746:
5741:
5738:
5734:
5725:
5722:
5718:
5709:
5700:
5696:
5673:
5670:
5666:
5662:
5659:
5656:
5646:
5641:
5636:
5630:
5629:light-seconds
5622:
5617:
5614:
5610:
5606:
5601:
5597:
5596:Lorentz boost
5592:
5590:
5586:
5582:
5578:
5574:
5570:
5567:revealed his
5566:
5556:
5543:
5540:
5534:
5531:
5528:
5522:
5502:
5499:
5496:
5493:
5490:
5487:
5478:
5476:
5472:
5468:
5452:
5449:
5446:
5443:
5440:
5435:
5431:
5421:
5406:
5400:
5397:
5392:
5385:
5380:
5374:
5369:
5366:
5358:
5356:
5352:
5348:
5344:
5340:
5335:
5333:
5324:
5311:
5306:
5300:
5295:
5288:
5283:
5277:
5269:
5249:
5246:
5243:
5240:
5237:
5234:
5226:
5216:
5201:
5180:
5157:
5153:
5135:
5130:
5119:
5100:
5091:
5085:
5079:
5076:
5065:
5063:
5058:
5056:
5051:
5049:
5048:zero divisors
5045:
5041:
5037:
5026:The image of
5024:
5011:
5005:
5002:
4997:
4993:
4985:
4978:
4962:
4949:
4946:
4943:
4938:
4934:
4926:
4922:
4901:
4884:
4880:
4876:
4866:
4864:
4851:
4838:
4833:
4830:
4826:
4820:
4817:
4813:
4809:
4801:
4798:
4795:
4789:
4785:
4776:
4774:
4768:
4757:
4741:
4738:
4735:
4727:
4708:
4705:
4694:
4691:
4683:
4680:
4667:
4662:
4658:
4654:
4648:
4628:
4625:
4619:
4611:
4609:
4606:
4598:
4591:
4587:
4582:
4576:
4572:
4571:Lorentz boost
4568:
4547:
4545:
4535:
4531:
4523:
4519:
4515:
4511:
4506:
4493:
4487:
4481:
4478:
4475:
4472:
4466:
4460:
4457:
4454:
4448:
4445:
4439:
4436:
4428:
4426:
4421:
4419:
4413:
4409:
4403:
4390:
4372:
4355:
4354:perpendicular
4351:
4346:
4333:
4329:
4325:
4322:
4316:
4310:
4307:
4303:
4294:
4292:
4286:
4278:
4271:
4257:
4251:
4247:
4243:
4240:
4235:
4227:
4221:
4218:
4214:
4205:
4203:
4199:
4193:
4186:
4178:
4158:
4139:
4134:
4120:
4114:
4110:
4106:
4101:
4093:
4087:
4084:
4080:
4071:
4068:
4050:
4047:
4044:
4011:
3994:
3976:
3971:
3968:
3965:
3948:
3945:-dimensional
3940:
3930:
3917:
3904:
3893:
3884:
3863:
3848:
3847:half the area
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3784:
3768:
3762:
3759:
3756:
3753:
3750:
3742:
3734:
3728:
3725:
3722:
3689:
3674:
3670:
3666:
3655:
3651:
3647:
3642:
3629:
3621:
3617:
3613:
3610:
3606:
3602:
3596:
3593:
3588:
3585:
3582:
3578:
3568:
3565:
3562:
3556:
3553:
3545:
3544:
3525:
3522:
3518:
3510:
3506:
3501:
3495:
3490:
3476:
3470:
3467:
3463:
3459:
3454:
3450:
3445:
3441:
3436:
3430:
3427:
3422:
3415:
3410:
3404:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3367:
3366:Furthermore,
3353:
3345:
3341:
3337:
3332:
3328:
3324:
3318:
3315:
3312:
3303:
3300:
3297:
3291:
3288:
3285:
3276:
3263:
3257:
3251:
3248:
3245:
3239:
3234:
3228:
3225:
3220:
3213:
3208:
3202:
3194:
3191:
3188:
3182:
3176:
3173:
3170:
3159:
3145:
3142:
3139:
3136:
3133:
3130:
3120:
3116:
3109:
3088:
3073:
3050:
3025:
3020:
3011:
2998:
2995:
2992:
2989:
2984:
2973:
2970:
2967:
2953:
2936:
2933:
2930:
2924:
2919:
2911:
2908:
2905:
2893:
2880:
2873:
2867:
2863:
2857:
2853:
2849:
2844:
2840:
2834:
2830:
2825:
2821:
2817:
2811:
2807:
2803:
2798:
2794:
2789:
2784:
2778:
2774:
2770:
2765:
2761:
2756:
2747:
2743:
2739:
2711:
2707:
2703:
2700:
2697:
2694:
2691:
2688:
2679:
2666:
2658:
2654:
2647:
2644:
2641:
2635:
2632:
2626:
2623:
2620:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2588:
2582:
2578:
2574:
2561:
2548:
2542:
2539:
2536:
2531:
2527:
2523:
2515:
2511:
2504:
2498:
2487:
2473:
2468:
2464:
2460:
2455:
2451:
2445:
2441:
2420:
2417:
2414:
2411:
2391:
2385:
2382:
2379:
2370:
2367:
2361:
2356:
2352:
2328:
2325:
2322:
2313:
2310:
2304:
2301:
2293:
2283:
2275:
2271:
2265:
2260:
2247:
2237:
2229:
2219:
2215:
2209:
2204:
2201:
2197:
2188:
2186:
2181:
2177:
2158:
2155:
2149:
2136:
2131:
2129:
2125:
2124:inner product
2120:
2107:
2098:
2095:
2092:
2086:
2081:
2073:
2062:
2048:
2045:
2037:
2033:
2029:
2026:
2017:
2014:
2008:
2002:
1996:
1978:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1924:
1921:
1918:
1915:
1912:
1909:
1900:
1887:
1881:
1878:
1875:
1872:
1869:
1866:
1862:
1858:
1853:
1849:
1844:
1840:
1827:
1823:
1817:
1813:
1809:
1805:
1801:
1788:
1782:
1779:
1776:
1765:
1763:
1762:bilinear form
1758:
1756:
1752:
1745:
1741:
1736:
1723:
1714:
1705:
1699:
1693:
1690:
1679:
1677:
1672:
1659:
1651:
1647:
1643:
1638:
1634:
1630:
1627:
1622:
1618:
1614:
1609:
1605:
1601:
1598:
1593:
1585:
1574:
1573:
1557:
1554:
1551:
1548:
1545:
1542:
1534:
1529:
1512:
1509:
1506:
1504:
1497:
1492:
1487:
1483:
1479:
1468:
1464:
1458:
1454:
1450:
1448:
1441:
1433:
1430:
1418:
1414:
1410:
1405:
1401:
1397:
1395:
1388:
1380:
1377:
1374:
1359:
1357:
1353:
1348:
1335:
1329:
1326:
1323:
1320:
1317:
1312:
1308:
1299:
1292:
1279:
1273:
1270:
1267:
1264:
1261:
1258:
1250:
1248:
1238:
1236:
1232:
1228:
1223:
1206:
1200:
1197:
1194:
1191:
1188:
1182:
1179:
1173:
1170:
1167:
1164:
1161:
1155:
1153:
1145:
1142:
1139:
1136:
1127:
1124:
1121:
1118:
1105:
1102:
1099:
1093:
1090:
1084:
1081:
1078:
1072:
1070:
1062:
1059:
1056:
1053:
1047:
1041:
1038:
1035:
1032:
1017:
1015:
1011:
1007:
998:
996:
976:
973:
970:
965:
961:
952:
948:
943:
930:
927:
924:
919:
915:
906:
901:
897:
884:
871:
868:
865:
862:
859:
856:
848:
846:
836:
834:
833:
828:
827:
821:
799:
771:
750:
744:
741:
721:
718:
715:
712:
709:
706:
703:
695:
690:
687:
684:
681:
672:
660:
655:
643:
642:
638:
635:also forms a
620:
615:
598:
576:
570:
567:
564:
561:
558:
555:
552:
547:
511:
493:
491:
485:
463:
457:
451:
445:
439:
436:
430:
427:
421:
401:
378:
375:
372:
369:
347:
344:
341:
338:
335:
332:
319:
317:
301:
296:
292:
288:
283:
279:
275:
270:
266:
262:
259:
253:
247:
223:
220:
217:
212:
208:
187:
184:
181:
178:
175:
172:
167:
163:
150:
134:
131:
128:
125:
122:
119:
116:
100:
84:
81:
76:
72:
60:
56:
55:double number
52:
48:
44:
40:
33:
19:
7516:
7506:
7321:Dual numbers
7313:hypercomplex
7252:
7103:Real numbers
6827:
6815:
6812:Isaak Yaglom
6787:
6761:
6736:
6718:
6696:
6679:36: 231–239.
6676:
6672:34: 159–168.
6662:
6645:
6619:
6614:
6589:
6585:
6579:
6569:
6562:
6552:
6545:
6531:
6515:
6502:
6494:
6489:
6476:
6455:
6447:
6438:
6422:
6398:
6394:
6388:
6369:
6363:
6343:
6338:
6326:
6314:
6310:
6303:James Cockle
6298:
6293:at Wikibooks
6281:
6273:
6264:
6249:
6244:
6207:
6201:
6195:
6189:
6183:
6177:
6171:
6165:
6159:
6155:
6149:
6143:
6137:
6123:
6114:
6108:
6102:
6096:
6090:
6086:
6079:
6075:
6069:
6061:
6053:D. H. Lehmer
6048:
6046:
6039:
6032:
6019:
6017:
6010:
6004:
6000:
5995:
5986:
5959:
5952:
5948:
5941:
5937:
5930:
5862:
5857:
5855:
5792:
5785:
5710:
5704:and waiting
5644:
5643:| <
5639:
5634:
5620:
5615:
5612:
5608:
5604:
5593:
5581:circle group
5565:James Cockle
5562:
5479:
5470:
5466:
5422:
5359:
5350:
5336:
5325:
5222:
5179:cyclic group
5131:
5120:
5066:
5059:
5052:
5039:
5025:
4963:
4872:
4852:
4777:
4766:
4758:
4684:
4681:
4612:
4583:
4566:
4548:
4533:
4529:
4510:power series
4507:
4429:
4422:
4411:
4407:
4391:
4347:
4295:
4284:
4276:
4272:
4206:
4204:is given by
4191:
4184:
4176:
4135:
4072:
4069:
3939:vector space
3936:
3928:
3915:
3902:
3846:
3643:
3546:
3502:
3494:parametrized
3491:
3368:
3277:
3160:
3118:
3114:
3110:
3069:
2954:
2894:
2748:
2741:
2737:
2680:
2589:
2580:
2576:
2562:
2488:
2289:
2273:
2269:
2264:null vectors
2261:
2189:
2179:
2175:
2132:
2127:
2121:
2063:
1976:
1901:
1766:
1764:is given by
1759:
1750:
1737:
1680:
1673:
1575:
1532:
1531:The squared
1530:
1360:
1349:
1300:
1293:
1251:
1246:
1244:
1224:
1018:
1005:
999:
994:
953:i satisfies
944:
907:
899:
896:real numbers
885:
849:
844:
842:
830:
824:
822:
742:
644:
494:
483:
320:
148:
58:
54:
50:
46:
42:
36:
7483:Other types
7302:Bioctonions
7159:Quaternions
6659:Walter Benz
6622:, page 30,
6128:I.M. Yaglom
5863:Two events
5480:The number
5477:of M(2,R).
5469:= 0, then (
5353:provides a
5332:determinant
4608:reflections
4588:called the
4189:. The case
3492:so the two
3034:applied to
3014:Isomorphism
1674:It has the
1235:distributes
1231:associative
1227:commutative
101:components
99:real number
64:satisfying
7570:Categories
7437:Projective
7410:Infinities
6721:55(4):296.
6665:, Springer
6450:26:268–80.
6433:24:223–36.
6309:34:37–47,
6237:References
6160:hyperbolic
6132:Hazewinkel
6080:tessarines
5569:tessarines
5134:group ring
4925:polynomial
4291:asymptotes
3667:. Indeed,
2581:null basis
1975:Here, the
1678:property:
1358:. Namely,
1352:involution
905:satisfies
839:Definition
486:, +, ×, *)
7521:solenoids
7341:Sedenions
7187:Octonions
6087:algebraic
5907:∗
5888:∗
5834:∈
5831:σ
5809:σ
5663:ρ
5600:spacetime
5398:−
5347:real line
5273:↦
5104:‖
5098:‖
5095:‖
5089:‖
5083:‖
5074:‖
5003:−
4944:−
4834:ϕ
4821:θ
4802:ϕ
4796:θ
4715:→
4695::
4663:∗
4655:±
4652:↦
4626:±
4623:↦
4610:given by
4488:θ
4482:
4467:θ
4461:
4449:θ
4440:
4350:null cone
4320:‖
4314:‖
4244:−
4232:‖
4225:‖
4138:hyperbola
4098:‖
4091:‖
3864:⊕
3825:∈
3813:
3798:
3743:⊕
3735:∈
3690:⊕
3575:↦
3554:σ
3468:−
3428:−
3394:
3382:
3338:−
3316:−
3226:−
3089:⊕
2981:‖
2962:‖
2920:∗
2712:∗
2659:∗
2624:−
2532:∗
2521:‖
2516:∗
2508:‖
2502:‖
2496:‖
2469:∗
2456:∗
2446:∗
2357:∗
2326:−
2294:given by
2233:‖
2227:‖
2220:∗
2202:−
2156:≠
2153:‖
2147:‖
2102:⟩
2090:⟨
2078:‖
2071:‖
2038:∗
1997:
1977:real part
1876:−
1854:∗
1841:
1818:∗
1802:
1786:⟩
1774:⟨
1744:signature
1718:‖
1712:‖
1709:‖
1703:‖
1697:‖
1688:‖
1644:−
1623:∗
1610:∗
1590:‖
1583:‖
1498:∗
1488:∗
1469:∗
1459:∗
1442:∗
1419:∗
1406:∗
1389:∗
1324:−
1313:∗
974:−
707:−
698:↦
663:→
562:×
398:forms an
379:∈
289:−
271:∗
179:−
168:∗
149:conjugate
6930:Integers
6892:Sets of
6792:Elsevier
6606:27957849
6524:Archived
6484:#0006140
6215:See also
6066:Synonyms
5962:Max Zorn
5960:In 1933
5699:rapidity
5225:matrices
4879:quotient
4759:sending
4605:discrete
4601:SO(1, 1)
4599:denoted
4597:subgroup
3993:geometry
3887:Geometry
3654:dilation
1010:Addition
898:and the
749:isometry
7511:numbers
7343: (
7189: (
7161: (
7133: (
7105: (
7026: (
7024:Periods
6993: (
6960: (
6932: (
6904: (
6886:systems
6814:(1968)
6776:2014924
6760:(2004)
6695:(1882)
6661:(1973)
6653:0021123
6539:0081372
6403:Bibcode
6305:(1849)
5559:History
5475:subring
5345:form a
5214:
5189:
5177:of the
5175:
5137:
4919:by the
4917:
4886:
4881:of the
4593:O(1, 1)
4387:
4358:
4171:
4146:
4065:
4030:
4026:
3997:
3989:
3951:
3943:(1 + 1)
3881:
3851:
3707:
3677:
3658:√
3106:
3076:
3065:
3036:
1747:(1, −1)
1533:modulus
814:
790:
786:
757:
633:
601:
595:is the
589:
530:
526:
497:
396:
362:
49:, also
39:algebra
7311:Other
6884:Number
6842:
6797:
6768:
6743:
6629:
6604:
6376:
6354:
6276:33:438
6256:
6091:motors
5731:
5538:
5526:
5107:
5064:since
4861:has a
4549:Since
4532:= exp(
4352:) are
4200:. The
3923:
3918:‖ = −1
3910:
3897:
3627:
3505:action
3351:
3261:
2878:
2664:
2546:
2245:
2105:
1902:where
1885:
1721:
1657:
1333:
1277:
886:where
753:(1, 1)
639:. The
591:where
200:Since
7519:-adic
7509:-adic
7266:Over
7227:Over
7221:types
7219:Split
6602:JSTOR
5875:when
5598:of a
5355:basis
5044:field
4921:ideal
4771:is a
4586:group
4573:or a
4414:⟩ = 0
4283:(0, −
4136:is a
3931:‖ = 0
3905:‖ = 1
3022:This
2573:basis
1249:. If
7555:List
7412:and
6840:ISBN
6795:ISBN
6766:ISBN
6741:ISBN
6627:ISBN
6374:ISBN
6352:ISBN
6313:(3)
6254:ISBN
6076:real
6008:and
5951:exp(
5940:exp(
5935:and
5929:exp(
5871:are
5867:and
5339:ring
5125:and
4765:exp(
4641:and
4518:sinh
4514:cosh
4479:sinh
4458:cosh
4396:and
4281:and
4275:(0,
4187:, 0)
4181:and
4179:, 0)
3810:sinh
3795:cosh
3673:area
3503:The
3391:sinh
3379:cosh
3123:for
2730:and
2567:and
2433:and
2344:and
1937:and
1755:norm
1233:and
1012:and
949:the
894:are
890:and
406:and
360:for
147:The
105:and
45:(or
41:, a
6832:doi
6594:doi
6411:doi
6158:or
6042:= 1
6013:= 1
5697:of
5627:in
5471:b,c
5040:not
4873:In
4692:exp
4557:by
4437:exp
4404:if
4356:in
4194:= 1
4144:in
3656:by
3507:of
2734:by
2579:or
2274:j a
2180:j x
1751:not
995:not
993:is
755:of
747:an
745:not
599:on
314:an
155:is
151:of
37:In
7572::
7101::
6838:.
6773:MR
6728:,
6706:,
6650:MR
6600:.
6590:61
6588:.
6536:MR
6530:,
6463:,
6409:,
6399:54
6397:,
6346:,
6315:33
6272:,
6126:,
6089:)
6078:)
6044:.
6040:zz
6026:,
6022:,
6003:=
5957:.
5953:aj
5942:aj
5931:aj
5915:0.
5860:.
5790:;
5611:+
5607:=
5591:.
5571:.
5453:1.
5129:.
5057:.
5050:.
5042:a
4865:.
4767:jθ
4546:.
4534:jθ
4410:,
4183:(−
3883:.
3500:.
3117:,
2740:,
2282:.
2272:±
2178:±
2171:),
1757:.
1753:a
1229:,
1008:.
977:1.
843:A
820:.
593:xy
492:.
488:a
412:wz
318:.
260::=
85:1.
53:,
7523:)
7517:p
7513:(
7507:p
7381:/
7365:)
7352:S
7288::
7275:C
7249::
7236:R
7211:)
7198:O
7183:)
7170:H
7155:)
7142:C
7127:)
7114:R
7050:)
7036:P
7015:)
7002:A
6982:)
6969:Q
6954:)
6941:Z
6926:)
6913:N
6876:e
6869:t
6862:v
6848:.
6834::
6801:.
6714:.
6655:.
6608:.
6596::
6471:.
6413::
6405::
6382:.
6358:.
6321:.
6085:(
6074:(
6011:e
6005:R
6001:F
5996:F
5987:R
5978:γ
5955:)
5949:j
5944:)
5938:j
5933:)
5912:=
5903:w
5899:z
5896:+
5893:w
5884:z
5869:w
5865:z
5858:a
5842:}
5838:R
5828::
5823:j
5820:a
5816:e
5812:j
5806:=
5803:z
5800:{
5788:a
5770:j
5767:)
5764:b
5761:+
5758:a
5755:(
5751:e
5747:=
5742:j
5739:b
5735:e
5726:j
5723:a
5719:e
5706:ρ
5702:a
5691:z
5674:j
5671:a
5667:e
5660:=
5657:z
5647:}
5645:x
5640:y
5635:z
5633:{
5625:y
5621:x
5616:j
5613:y
5609:x
5605:z
5544:.
5541:m
5535:y
5532:+
5529:I
5523:x
5503:y
5500:j
5497:+
5494:x
5491:=
5488:z
5467:a
5450:=
5447:c
5444:b
5441:+
5436:2
5432:a
5407:)
5401:a
5393:b
5386:c
5381:a
5375:(
5370:=
5367:m
5351:m
5328:z
5312:.
5307:)
5301:x
5296:y
5289:y
5284:x
5278:(
5270:z
5250:y
5247:j
5244:+
5241:x
5238:=
5235:z
5202:.
5198:R
5184:2
5182:C
5163:]
5158:2
5154:C
5150:[
5146:R
5127:w
5123:z
5101:w
5092:z
5086:=
5080:w
5077:z
5032:j
5028:x
5012:.
5009:)
5006:1
4998:2
4994:x
4990:(
4986:/
4982:]
4979:x
4976:[
4972:R
4950:,
4947:1
4939:2
4935:x
4905:]
4902:x
4899:[
4895:R
4859:z
4855:z
4839:.
4831:j
4827:e
4818:j
4814:e
4810:=
4805:)
4799:+
4793:(
4790:j
4786:e
4769:)
4761:θ
4745:)
4742:1
4739:,
4736:1
4733:(
4728:+
4723:O
4720:S
4712:)
4709:+
4706:,
4702:R
4698:(
4668:.
4659:z
4649:z
4629:z
4620:z
4579:λ
4563:z
4559:λ
4555:z
4551:λ
4540:λ
4536:)
4530:λ
4525:θ
4494:.
4491:)
4485:(
4476:j
4473:+
4470:)
4464:(
4455:=
4452:)
4446:j
4443:(
4412:w
4408:z
4406:⟨
4398:w
4394:z
4373:2
4368:R
4334:.
4330:}
4326:0
4323:=
4317:z
4311::
4308:z
4304:{
4287:)
4285:a
4279:)
4277:a
4258:}
4252:2
4248:a
4241:=
4236:2
4228:z
4222::
4219:z
4215:{
4192:a
4185:a
4177:a
4175:(
4159:.
4155:R
4142:a
4121:}
4115:2
4111:a
4107:=
4102:2
4094:z
4088::
4085:z
4081:{
4051:1
4048:,
4045:1
4040:R
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3977:.
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2024:(
2018:2
2015:1
2009:=
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2003:z
2000:(
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1989:R
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