4500:
4734:
5115:
4238:
4855:
4172:
40:
4509:
4137:
1437:
4879:
2833:
4495:{\displaystyle {\begin{alignedat}{3}{\sqrt {i}}&={\exp }{\bigl (}{\tfrac {1}{2}}{\pi i}{\bigr )}^{1/2}&&{}={\exp }{\bigl (}{\tfrac {1}{4}}\pi i{\bigr )},\\-{\sqrt {i}}&={\exp }{\bigl (}{\tfrac {1}{4}}{\pi i}-\pi i{\bigr )}&&{}={\exp }{\bigl (}{-{\tfrac {3}{4}}\pi i}{\bigr )}.\end{alignedat}}}
1170:
Although the construction is called "imaginary", and although the concept of an imaginary number may be intuitively more difficult to grasp than that of a real number, the construction is valid from a mathematical standpoint. Real number operations can be extended to imaginary and complex numbers, by
3939:
1192:
5311:
2537:
part. (A scalar is a quantity with no orientation, a vector is a quantity oriented like a line, and a bivector is a quantity oriented like a plane.) The square of any vector is a positive scalar, representing its length squared, while the square of any bivector is a negative scalar.
4729:{\displaystyle {\begin{alignedat}{3}{\sqrt {i}}&=\ (1+i){\big /}{\sqrt {2}}&&{}={\phantom {-}}{\tfrac {\sqrt {2}}{2}}+{\tfrac {\sqrt {2}}{2}}i,\\-{\sqrt {i}}&=-(1+i){\big /}{\sqrt {2}}&&{}=-{\tfrac {\sqrt {2}}{2}}-{\tfrac {\sqrt {2}}{2}}i.\end{alignedat}}}
2684:
5110:{\displaystyle {\sqrt{i}}={\exp }{\bigl (}{\tfrac {1}{6}}\pi i{\bigr )}={\tfrac {\sqrt {3}}{2}}+{\tfrac {1}{2}}i,\quad {\exp }{\bigl (}{\tfrac {5}{6}}\pi i{\bigr )}=-{\tfrac {\sqrt {3}}{2}}+{\tfrac {1}{2}}i,\quad {\exp }{\bigl (}{-{\tfrac {1}{2}}\pi i}{\bigr )}=-i.}
6231:
2066:
7360:
3443:
2928:
3652:
2680:
or for the principal branch of the complex square root function. Attempting to apply the calculation rules of the principal (real) square root function to manipulate the principal branch of the complex square root function can produce false results:
4850:
5147:
3013:
5752:
3907:
6410:
2157:
4132:{\displaystyle i^{n}={\exp }{\bigl (}{\tfrac {1}{2}}\pi i{\bigr )}^{n}={\exp }{\bigl (}{\tfrac {1}{2}}n\pi i{\bigr )}={\cos }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}+{i\sin }{\bigl (}{\tfrac {1}{2}}n\pi {\bigr )}.}
6114:
1432:{\displaystyle {\begin{alignedat}{3}i^{3}&=i^{2}i&&=(-1)i&&=-i,\\i^{4}&=i^{3}i&&=\;\!(-i)i&&=\ \,1,\\i^{5}&=i^{4}i&&=\ \,(1)i&&=\ \ i,\end{alignedat}}}
6764:
5887:, with each branch in the domain corresponding to one infinite strip in the codomain. Functions depending on the complex logarithm therefore depend on careful choice of branch to define and evaluate clearly.
6710:
6123:
1960:
7550:
6339:
3245:
5597:
5531:
1616:
of each other. Although the two solutions are distinct numbers, their properties are indistinguishable; there is no property that one has that the other does not. One of these two solutions is labelled
3240:
959:
896:
719:
656:
485:
425:
2402:
6005:
5848:
3758:
3512:
5363:
function relates complex addition in the domain to complex multiplication in the codomain. Real values in the domain represent scaling in the codomain (multiplication by a real scalar) with
7227:
1073:
1016:
833:
776:
4741:
2584:, and can thus be taken as a representative of the imaginary unit. Any sum of a scalar and bivector can be multiplied by a vector to scale and rotate it, and the algebra of such sums is
6643:
6519:
5935:
6603:
593:
539:
2828:{\displaystyle -1=i\cdot i={\sqrt {-1}}\cdot {\sqrt {-1}}\mathrel {\stackrel {\mathrm {fallacy} }{=}} {\textstyle {\sqrt {(-1)\cdot (-1)}}}={\sqrt {1}}=1\qquad {\text{(incorrect).}}}
6458:
2840:
1735:
3488:
1693:
1538:
6559:
2512:
5658:
3180:
2257:
161:
1831:
136:
4225:
2639:
3086:
3059:
1144:
1099:
365:
2667:
2334:
2297:
5626:
3780:
6334:
2933:
2426:
1197:
2085:
6038:
2588:
to the algebra of complex numbers. In this interpretation points, vectors, and sums of scalars and bivectors are all distinct types of geometric objects.
5306:{\displaystyle \exp \left(2\pi i{\frac {k+{\frac {1}{4}}}{n}}\right)=\cos \left({\frac {4k+1}{2n}}\pi \right)+i\sin \left({\frac {4k+1}{2n}}\pi \right).}
4514:
7495:
7468:
5543:
5435:
3211:, which as part of the complex plane is typically drawn with a vertical orientation, perpendicular to the real axis which is drawn horizontally.
5860:, with each complex number in the domain corresponding to multiple values in the codomain, separated from each-other by any integer multiple of
7604:
7410:
7120:
7085:
7054:
7023:
6946:
6915:
4243:
5944:
5763:
3661:
6715:
6664:
7529:
7176:
6806:
281:
were treated with skepticism β so the square root of a negative number was previously considered undefined or nonsensical. The name
7554:
7609:
6571:
904:
841:
664:
601:
5756:
Other functions based on the complex exponential are well-defined with imaginary inputs. For example, a number raised to the
5648:
176:
433:
373:
2346:
7428:
1472:
6226:{\displaystyle |\Gamma (1+i)|={\sqrt {\frac {\pi }{\sinh \pi }}}\approx 0.5216,\quad \arg {\Gamma (1+i)}\approx -0.3016.}
5371:, while imaginary values in the domain represent rotation in the codomain (multiplication by a unit complex number) with
2061:{\displaystyle I={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\quad J={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}.}
7355:{\textstyle \pi \cot \pi z={\frac {1}{z}}+{\frac {1}{z-1}}+{\frac {1}{z+1}}+{\frac {1}{z-2}}+{\frac {1}{z+2}}+\cdots .}
2472:
in the plane, while multiplication by a unit-magnitude complex number corresponds to rotation about the origin. Every
2447:
1660:
1508:
232:
7599:
3438:{\displaystyle {\begin{aligned}(a+bi)+(c+di)&=(a+c)+(b+d)i,\\(a+bi)(c+di)&=(ac-bd)+(ad+bc)i.\end{aligned}}}
3129:
7569:
1024:
967:
784:
727:
6862:
6608:
5893:
7594:
6970:
6463:
1175:
as an unknown quantity while manipulating an expression (and using the definition to replace any occurrence of
1113:
547:
493:
2923:{\textstyle {\sqrt {x{\vphantom {ty}}}}\cdot \!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x\cdot y{\vphantom {ty}}}}}
2641:
and still is in some modern works. However, great care needs to be taken when manipulating formulas involving
6867:
6823:
5420:
3647:{\displaystyle i\,re^{\varphi i}=re^{(\varphi +\pi /2)i},\quad -i\,re^{\varphi i}=re^{(\varphi -\pi /2)i}.}
51:: Real numbers are conventionally drawn on the horizontal axis, and imaginary numbers on the vertical axis.
5405:
2469:
2166:
1705:
1613:
270:
228:
6415:
3461:
1666:
6328:
5857:
2473:
1930:
and all of the ordinary rules of complex arithmetic can be derived from the rules of matrix arithmetic.
1854:
1514:
5883:
consisting of multiple copies of the complex plane stitched together along the negative real axis as a
277:, obtainable by physical measurements or basic arithmetic, were considered to be numbers at all β even
5601:
This fact can be used to demonstrate, among other things, the apparently counterintuitive result that
6903:
5409:
2300:
2259:
an algebraic structure with addition and multiplication and sharing many properties with the ring of
1850:
1789:
1590:
6523:
5379:
radian. The complex exponential is thus a periodic function in the imaginary direction, with period
4845:{\displaystyle \left(\pm {\frac {1+i}{\sqrt {2}}}\right)^{2}={\frac {1+2i-1}{2}}={\frac {2i}{2}}=i.}
2479:
6961:
The interpretation of the imaginary unit as the ratio of two perpendicular vectors was proposed by
6889:
5652:
5360:
3185:
2337:
2228:
1834:
141:
7136:
1814:
119:
7445:
6828:
3034:
is real but negative, these problems can be avoided by writing and manipulating expressions like
2222:
164:
70:
5535:
3927:
7426:
Ivan, M.; Thornber, N.; Kouba, O.; Constales, D. (2013). "Arggh! Eye factorial . . . Arg(i!)".
4186:
2614:
1081:
347:
7525:
7406:
7182:
7172:
7116:
7081:
7077:
7050:
7019:
6962:
6942:
6938:
6911:
6907:
6897:
6802:
6788:
5853:
3232:
3064:
3037:
2553:
2522:
286:
172:
7200:
7112:
7046:
7015:
6979:
6875:
2648:
2306:
2269:
7437:
7376:
6990:
6856:
6837:
6787:
4232:
3200:
2530:
1609:
278:
248:
184:
113:
5604:
4660:
4551:
3008:{\textstyle {\sqrt {x{\vphantom {ty}}}}{\big /}\!{\sqrt {y{\vphantom {ty}}}}={\sqrt {x/y}}}
2960:
2217:(weighted sums of the powers of a variable) are a basic tool in algebra. Polynomials whose
6241:
5880:
5344:
5321:
4146:
2592:
2526:
2341:
1945:
1785:
1738:
1555:
7441:
7546:
6975:
6023:
3228:
3105:
2411:
2197:
1563:
1504:
298:
260:
96:
88:
5747:{\displaystyle \pi \coth \pi z=\lim _{n\to \infty }\sum _{k=-n}^{n}{\frac {1}{z+ki}}.}
3507:, any arbitrary complex number is rotated by a quarter turn clockwise. In polar form:
7588:
7224:
Euler expressed the partial fraction decomposition of the trigonometric cotangent as
5332:
3497:
2452:
2437:
1746:
1594:
1551:
213:
48:
7449:
7514:
6893:
6246:
3920:
3916:
3235:. The sum, difference, or product of Gaussian integers is also a Gaussian integer:
3189:
3093:
1838:
1808:
290:
274:
7381:
7364:
6331:, separating the real part and imaginary part, we have a system of two equations:
7102:
7071:
7040:
7009:
6994:
6932:
3456:, any arbitrary complex number in the complex plane is rotated by a quarter turn
17:
7481:
7454:
6308:
Because the real and imaginary parts are always separate, we regroup the terms,
5348:
4228:
3204:
3196:
3089:
2669:
is reserved either for the principal square root function, which is defined for
2443:
2405:
2218:
2170:
1796:
205:
180:
80:
2468:, with a right angle between them. Addition by a complex number corresponds to
5884:
4142:
3902:{\displaystyle i^{4n}=1,\quad i^{4n+1}=i,\quad i^{4n+2}=-1,\quad i^{4n+3}=-i.}
3116:
When the imaginary unit is repeatedly added or subtracted, the result is some
2585:
2570:, which when multiplied rotates the divisor a quarter turn into the dividend,
2214:
1483:
168:
6405:{\displaystyle {\begin{aligned}x^{2}-y^{2}&=0\\2xy&=1.\end{aligned}}}
3124:; any such numbers can be added and the result is also an imaginary integer:
7396:
7186:
6015:
5868:
One way of obtaining a single-valued function is to treat the codomain as a
4867:
4854:
4171:
2476:
transformation of the plane can be represented by a complex-linear function
1567:
39:
6841:
5538:
decomposes the exponential of an imaginary number representing a rotation:
2446:
as the horizontal axis and the imaginary numbers as the vertical axis of a
7400:
116:
are an important mathematical concept; they extend the real number system
7166:
5869:
5124:
2642:
2534:
92:
31:
7461:; and "Decimal expansion of the negated imaginary part of i!", Sequence
2556:). The quotient of any two perpendicular vectors of the same magnitude,
7577:
3117:
2442:
The complex numbers can be represented graphically by drawing the real
2260:
1767:
is inherently positive or negative in the sense that real numbers are.
1153:
2152:{\displaystyle aI+bJ={\begin{pmatrix}a&-b\\b&a\end{pmatrix}}.}
1857:, complex numbers can be represented in linear algebra. The real unit
1540:
6109:{\displaystyle i!=\Gamma (1+i)=i\Gamma (i)\approx 0.4980-0.1549\,i.}
3104:
As a complex number, the imaginary unit follows all of the rules of
4149:, this last equation can also apply to arbitrary complex values of
2591:
More generally, in the geometric algebra of any higher-dimensional
1608:
distinct solutions, which are equally valid and which happen to be
7488:; and "Decimal expansion of the negated argument of i!", Sequence
7484:(ed.). "Decimal expansion of the absolute value of i!", Sequence
4853:
4170:
1793:
38:
6759:{\displaystyle -{\tfrac {1}{\sqrt {2}}}-{\tfrac {1}{\sqrt {2}}}i}
5879:
treated as the same value; another is to take the domain to be a
2595:, a unit bivector of any arbitrary planar orientation squares to
6705:{\displaystyle {\tfrac {1}{\sqrt {2}}}+{\tfrac {1}{\sqrt {2}}}i}
3771:
repeat in a cycle expressible with the following pattern, where
1482:, with a zero real component and a unit imaginary component. In
209:
2303:, but the set of all real-coefficient polynomials divisible by
6973:
realized that this ratio could be interpreted as a bivector.
7457:(ed.). "Decimal expansion of the real part of i!", Sequence
6934:
Circles
Disturbed: The interplay of mathematics and narrative
2552:, and when multiplied by any vector leaves it unchanged (the
6937:(illustrated ed.). Princeton University Press. p.
5647:
with appropriate scaling, can be represented as an infinite
5592:{\displaystyle \exp i\varphi =\cos \varphi +i\sin \varphi .}
7489:
7485:
7462:
7458:
6565:
is a real number, this equation has two real solutions for
5872:, with complex values separated by any integer multiple of
5526:{\displaystyle \exp z=\cosh z+\sinh z=\cos(-iz)+i\sin(-iz)}
3195:
The imaginary unit can also be multiplied by any arbitrary
1737:
related to the convention of labelling orientations in the
1833:
that keep each real number fixed, namely the identity and
179:). Here, the term "imaginary" is used because there is no
87:
can be used to extend the real numbers to what are called
6645:. Substituting either of these results into the equation
1770:
A more formal expression of this indistinguishability of
1110:
is defined solely by the property that its square is β1:
954:{\displaystyle \ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}}
891:{\displaystyle \ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}}
714:{\displaystyle \ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}}
651:{\displaystyle \ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}}
317:
7402:
Mathematical
Methods for Optical Physics and Engineering
5401:, a real multiple of the lattice of imaginary integers.
3015:
are guaranteed to be valid for real, positive values of
1562:
is the point located one unit from the origin along the
30:"i (number)" redirects here. For internet numbers, see
7230:
6740:
6723:
6686:
6669:
6622:
6582:
6526:
6481:
6466:
6426:
6418:
6289:
are real parameters to be determined, or equivalently
5910:
5070:
5035:
5018:
4987:
4956:
4939:
4911:
4703:
4686:
4600:
4583:
4460:
4398:
4337:
4273:
4189:
4102:
4059:
4016:
3969:
3466:
3067:
3040:
2936:
2843:
2769:
2651:
2617:
2112:
2021:
1975:
1713:
1674:
1652:
arise from this labelling. For example, by convention
1519:
480:{\displaystyle \ i^{-3}={\phantom {-}}i{\phantom {1}}}
420:{\displaystyle \ i^{-4}={\phantom {-}}1{\phantom {i}}}
7576:. Mathematical Association of America. Archived from
7014:(illustrated ed.). Courier Corporation. p.
6718:
6667:
6611:
6574:
6337:
6126:
6041:
5947:
5896:
5852:
Because the exponential is periodic, its inverse the
5766:
5661:
5607:
5546:
5438:
5150:
4882:
4744:
4512:
4241:
3942:
3783:
3664:
3515:
3464:
3243:
3132:
2687:
2529:, the geometric product or quotient of two arbitrary
2482:
2414:
2397:{\displaystyle \mathbb {R} /\langle x^{2}+1\rangle .}
2349:
2309:
2272:
2231:
2088:
1963:
1817:
1708:
1669:
1517:
1195:
1116:
1084:
1027:
970:
907:
844:
787:
730:
667:
604:
550:
496:
436:
376:
350:
144:
122:
7168:
A first course in complex analysis with applications
6000:{\displaystyle \log _{i}x=-{\frac {2i\ln x}{\pi }}.}
5331:. The set of roots equals the corresponding set of
1753:-axis. Despite the signs written with them, neither
1635:, though it is inherently ambiguous which is which.
3088:. For a more thorough discussion, see the articles
2541:The quotient of a vector with itself is the scalar
2184:. Larger matrices could also be used; for example,
7513:
7354:
6855:
6758:
6704:
6653:in turn, we will get the corresponding result for
6637:
6597:
6553:
6513:
6452:
6404:
6266:To find such a number, one can solve the equation
6225:
6108:
5999:
5929:
5843:{\displaystyle x^{ni}=\cos(n\ln x)+i\sin(n\ln x).}
5842:
5746:
5620:
5591:
5525:
5305:
5109:
4844:
4728:
4494:
4219:
4131:
3923:of the unit complex numbers under multiplication.
3919:of order 4, a discrete subgroup of the continuous
3901:
3753:{\displaystyle i(a+bi)=-b+ai,\quad -i(a+bi)=b-ai.}
3752:
3646:
3482:
3437:
3174:
3080:
3053:
3007:
2922:
2827:
2661:
2633:
2599:, so can be taken to represent the imaginary unit
2506:
2420:
2396:
2328:
2291:
2251:
2151:
2060:
1825:
1729:
1687:
1532:
1431:
1138:
1093:
1067:
1010:
953:
890:
827:
770:
713:
650:
587:
533:
479:
419:
359:
155:
130:
2965:
2868:
1313:
7405:. Cambridge University Press. pp. 278β284.
5681:
7076:(Teachers' ed.). Henri Picciotto. p.
6118:The magnitude and argument of this number are:
3184:Thus, the imaginary unit is the generator of a
7201:"i to the i is a Real Number β Math Fun Facts"
7171:. Boston: Jones and Bartlett. pp. 24β25.
7165:Zill, Dennis G.; Shanahan, Patrick D. (2003).
6801:(3rd ed.). New York : Wiley. p. 49.
2533:is a sum of a scalar (real number) part and a
7369:Bulletin of the American Mathematical Society
6931:DoxiadΔs, Apostolos K.; Mazur, Barry (2012).
6799:Mathematical Methods in the Physical Sciences
5890:For example, if one chooses any branch where
5090:
5060:
5006:
4981:
4930:
4905:
4480:
4450:
4428:
4392:
4356:
4331:
4295:
4267:
4121:
4096:
4078:
4053:
4038:
4010:
3989:
3963:
8:
6822:Silver, Daniel S. (NovemberβDecember 2017).
2611:The imaginary unit was historically written
2388:
2369:
1068:{\displaystyle \ \ i^{7}\ =-i{\phantom {1}}}
1011:{\displaystyle \ \ i^{6}\ =-1{\phantom {i}}}
828:{\displaystyle \ \ i^{3}\ =-i{\phantom {1}}}
771:{\displaystyle \ \ i^{2}\ =-1{\phantom {i}}}
265:Square roots of negative numbers are called
235:, the imaginary unit is normally denoted by
6790:Elementary vectors for electrical engineers
5404:The complex exponential can be broken into
1865:can be represented by any pair of matrices
1837:. For more on this general phenomenon, see
1439:and so on, cycling through the four values
1152:defined this way, it follows directly from
227:is sometimes used instead. For example, in
7524:. Chichester: Princeton University Press.
7045:(4th ed.). Cengage Learning. p.
6638:{\displaystyle x=-{\tfrac {1}{\sqrt {2}}}}
6531:
6527:
6514:{\textstyle x^{2}-{\tfrac {1}{4}}x^{-2}=0}
5930:{\displaystyle \ln i={\tfrac {1}{2}}\pi i}
1312:
7496:On-Line Encyclopedia of Integer Sequences
7469:On-Line Encyclopedia of Integer Sequences
7380:
7325:
7304:
7283:
7262:
7249:
7229:
7145:University of Toronto Mathematics Network
6739:
6722:
6717:
6685:
6668:
6666:
6621:
6610:
6598:{\displaystyle x={\tfrac {1}{\sqrt {2}}}}
6581:
6573:
6539:
6525:
6496:
6480:
6471:
6465:
6441:
6425:
6417:
6359:
6346:
6338:
6336:
6194:
6158:
6150:
6127:
6125:
6099:
6040:
5970:
5952:
5946:
5909:
5895:
5771:
5765:
5720:
5714:
5700:
5684:
5660:
5612:
5606:
5545:
5437:
5266:
5215:
5180:
5171:
5149:
5089:
5088:
5069:
5065:
5059:
5058:
5053:
5034:
5017:
5005:
5004:
4986:
4980:
4979:
4974:
4955:
4938:
4929:
4928:
4910:
4904:
4903:
4898:
4888:
4883:
4881:
4818:
4788:
4779:
4754:
4743:
4702:
4685:
4677:
4665:
4659:
4658:
4626:
4599:
4582:
4574:
4573:
4568:
4556:
4550:
4549:
4517:
4513:
4511:
4479:
4478:
4459:
4455:
4449:
4448:
4443:
4438:
4427:
4426:
4409:
4397:
4391:
4390:
4385:
4371:
4355:
4354:
4336:
4330:
4329:
4324:
4319:
4304:
4300:
4294:
4293:
4284:
4272:
4266:
4265:
4260:
4246:
4242:
4240:
4207:
4200:
4188:
4120:
4119:
4101:
4095:
4094:
4086:
4077:
4076:
4058:
4052:
4051:
4046:
4037:
4036:
4015:
4009:
4008:
4003:
3994:
3988:
3987:
3968:
3962:
3961:
3956:
3947:
3941:
3872:
3840:
3811:
3788:
3782:
3663:
3625:
3612:
3593:
3585:
3559:
3546:
3527:
3519:
3514:
3465:
3463:
3244:
3242:
3188:under addition, specifically an infinite
3131:
3068:
3066:
3044:
3039:
2995:
2990:
2972:
2971:
2966:
2959:
2958:
2943:
2942:
2937:
2935:
2905:
2904:
2893:
2875:
2874:
2869:
2850:
2849:
2844:
2842:
2820:
2806:
2770:
2768:
2741:
2740:
2735:
2733:
2732:
2722:
2709:
2686:
2652:
2650:
2618:
2616:
2481:
2413:
2408:to the complex numbers, and the variable
2376:
2364:
2351:
2350:
2348:
2314:
2308:
2277:
2271:
2233:
2232:
2230:
2107:
2087:
2016:
1970:
1962:
1819:
1818:
1816:
1712:
1707:
1673:
1668:
1554:, which is a special interpretation of a
1518:
1516:
1392:
1372:
1355:
1340:
1295:
1278:
1221:
1204:
1196:
1194:
1121:
1115:
1083:
1057:
1056:
1038:
1026:
1000:
999:
981:
969:
943:
942:
931:
930:
918:
906:
880:
879:
868:
867:
855:
843:
817:
816:
798:
786:
760:
759:
741:
729:
703:
702:
691:
690:
678:
666:
640:
639:
628:
627:
615:
603:
588:{\displaystyle \ i^{-1}=-i{\phantom {1}}}
577:
576:
558:
549:
534:{\displaystyle \ i^{-2}=-1{\phantom {i}}}
523:
522:
504:
495:
469:
468:
457:
456:
444:
435:
409:
408:
397:
396:
384:
375:
349:
146:
145:
143:
124:
123:
121:
7108:" [the square root of minus one]
3112:Imaginary integers and imaginary numbers
331:
223:is ambiguous or problematic, the letter
7365:"Euler and his Work on Infinite Series"
6778:
6259:
4183:Just like all nonzero complex numbers,
1933:The most common choice is to represent
219:In contexts in which use of the letter
7520:[the square root of minus one]
7111:. Princeton University Press. p.
7042:Math for Electricity & Electronics
3452:When multiplied by the imaginary unit
2580:, is a unit bivector which squares to
2456:. In this representation, the numbers
1792:(as an extension of the real numbers)
1550:to this angle works as well.) In the
190:There are two complex square roots of
7070:Picciotto, Henri; Wah, Anita (1994).
3203:. These numbers can be pictured on a
7:
7442:10.4169/amer.math.monthly.120.07.660
7011:Mathematical Fallacies and Paradoxes
6987:Hermann GΓΌnther GraΓmann (1809β1877)
6876:participating institution membership
6453:{\textstyle y={\tfrac {1}{2}}x^{-1}}
6022:is most often given in terms of the
1807:isomorphism. That is, there are two
1730:{\displaystyle -{\tfrac {\pi }{2}},}
1458:. As with any non-zero real number,
293:used the term as early as 1670. The
3483:{\displaystyle {\tfrac {1}{2}}\pi }
2196:could be represented by any of the
1745:-axis with positive angles turning
1688:{\displaystyle +{\tfrac {\pi }{2}}}
6786:Stubbings, George Wilfred (1945).
6195:
6132:
6075:
6051:
5691:
5655:translated by imaginary integers:
4738:Squaring either expression yields
2760:
2757:
2754:
2751:
2748:
2745:
2742:
1543:. (Adding any integer multiple of
1533:{\displaystyle {\tfrac {\pi }{2}}}
25:
7104:An Imaginary Tale: The story of "
6824:"The New Language of Mathematics"
2837:Generally, the calculation rules
2070:Then an arbitrary complex number
1919:can be represented by the matrix
1749:in the direction of the positive
99:. A simple example of the use of
7551:"Imaginary Roots of Polynomials"
7516:An Imaginary Tale: The story of
7073:Algebra: Themes, tools, concepts
6794:. London: I. Pitman. p. 69.
6460:into the first equation, we get
3231:in the complex plane called the
2161:More generally, any real-valued
208:of every real number other than
204:, just as there are two complex
6554:{\textstyle \implies 4x^{4}=1.}
6187:
5367:representing multiplication by
5119:For a general positive integer
5052:
4973:
3867:
3835:
3806:
3707:
3578:
2819:
2009:
1702:is said to have an argument of
6528:
6210:
6198:
6151:
6147:
6135:
6128:
6084:
6078:
6066:
6054:
5834:
5819:
5804:
5789:
5688:
5649:partial fraction decomposition
5520:
5508:
5493:
5481:
5343:. These are the vertices of a
4655:
4643:
4546:
4534:
4504:In rectangular form, they are
3729:
3714:
3683:
3668:
3633:
3613:
3567:
3547:
3422:
3404:
3398:
3380:
3370:
3355:
3352:
3337:
3324:
3312:
3306:
3294:
3284:
3269:
3263:
3248:
3219:Integer sums of the real unit
3163:
3151:
2796:
2787:
2781:
2772:
2507:{\displaystyle z\mapsto az+b.}
2486:
2464:are at the same distance from
2428:expresses the imaginary unit.
2361:
2355:
2243:
2237:
1784:is that, although the complex
1399:
1393:
1323:
1314:
1247:
1238:
177:Fundamental theorem of algebra
1:
7429:American Mathematical Monthly
7382:10.1090/S0273-0979-07-01175-5
5347:inscribed within the complex
3926:Written as a special case of
3175:{\displaystyle ai+bi=(a+b)i.}
3120:times the imaginary unit, an
2252:{\displaystyle \mathbb {R} ,}
1638:The only differences between
1185:). Higher integral powers of
1167:are both square roots of β1.
261:Complex number Β§ History
156:{\displaystyle \mathbb {C} ,}
138:to the complex number system
7605:Quadratic irrational numbers
7137:"What is the square root of
6995:10.1007/978-94-015-8753-2_20
6657:. Thus, the square roots of
3911:Thus, under multiplication,
2973:
2944:
2906:
2876:
2851:
2645:. The radical sign notation
2188:could be represented by the
1826:{\displaystyle \mathbb {C} }
1628:) and the other is labelled
1566:(which is orthogonal to the
131:{\displaystyle \mathbb {R} }
7363:Varadarajan, V. S. (2007).
5941:is a positive real number,
5375:representing a rotation by
1507:(or magnitude) of 1 and an
308:is an undivided whole, and
297:notation was introduced by
273:, only what are now called
247:is commonly used to denote
233:control systems engineering
27:Principal square root of β1
7626:
6985:. In Schubring, G. (ed.).
5313:The value associated with
4235:. In polar form, they are
4220:{\textstyle i=e^{\pi i/2}}
2634:{\textstyle {\sqrt {-1}},}
2435:
258:
29:
7125:– via Google Books.
7090:– via Google Books.
7059:– via Google Books.
7028:– via Google Books.
6951:– via Google Books.
6863:Oxford English Dictionary
5355:Exponential and logarithm
5335:rotated by the principal
4141:With a careful choice of
3081:{\textstyle {\sqrt {-7}}}
3054:{\textstyle i{\sqrt {7}}}
2180:, so could be chosen for
1741:relative to the positive
1139:{\displaystyle i^{2}=-1.}
1094:{\displaystyle \ \vdots }
360:{\displaystyle \ \vdots }
285:is generally credited to
7534:– via Archive.org.
6899:A History of Mathematics
4858:The three cube roots of
4175:The two square roots of
2662:{\textstyle {\sqrt {x}}}
2221:are real numbers form a
2200:for spatial dimensions.
1597:, the defining equation
271:early-modern mathematics
7512:Nahin, Paul J. (1998).
7101:Nahin, Paul J. (2010).
7039:Kramer, Arthur (2012).
6965:in the foreword to his
6868:Oxford University Press
5421:trigonometric functions
4153:, including cases like
3223:and the imaginary unit
2554:identity transformation
2329:{\displaystyle x^{2}+1}
2292:{\displaystyle x^{2}+1}
2080:can be represented by:
1811:of the complex numbers
1614:multiplicative inverses
103:in a complex number is
69:) is a solution to the
7610:Mathematical constants
7356:
6842:10.1511/2017.105.6.364
6797:Boas, Mary L. (2006).
6760:
6706:
6639:
6599:
6555:
6515:
6454:
6406:
6227:
6110:
6018:of the imaginary unit
6001:
5931:
5844:
5748:
5719:
5622:
5593:
5527:
5307:
5111:
4863:
4846:
4730:
4496:
4221:
4180:
4133:
3903:
3754:
3648:
3484:
3439:
3176:
3082:
3055:
3009:
2924:
2829:
2663:
2635:
2508:
2432:Graphic representation
2422:
2404:This quotient ring is
2398:
2330:
2293:
2253:
2153:
2062:
1909:Then a complex number
1849:Using the concepts of
1827:
1731:
1689:
1534:
1490:can be represented as
1471:can be represented in
1433:
1140:
1095:
1069:
1012:
955:
892:
829:
772:
715:
652:
589:
535:
481:
421:
361:
229:electrical engineering
167:for every nonconstant
163:in which at least one
157:
132:
52:
7357:
7008:Bunch, Bryan (2012).
6904:John Wiley & Sons
6761:
6707:
6640:
6600:
6556:
6516:
6455:
6407:
6329:equating coefficients
6228:
6111:
6002:
5932:
5858:multi-valued function
5845:
5749:
5696:
5623:
5621:{\displaystyle i^{i}}
5594:
5528:
5308:
5112:
4857:
4847:
4731:
4497:
4222:
4174:
4134:
3904:
3755:
3656:In rectangular form,
3649:
3500:. When multiplied by
3485:
3448:Quarter-turn rotation
3440:
3177:
3083:
3056:
3010:
2925:
2830:
2664:
2636:
2509:
2423:
2399:
2331:
2294:
2254:
2154:
2063:
1855:matrix multiplication
1828:
1732:
1690:
1535:
1467:As a complex number,
1434:
1141:
1096:
1070:
1013:
956:
893:
830:
773:
716:
653:
590:
536:
482:
422:
362:
259:Further information:
158:
133:
79:Although there is no
61:unit imaginary number
42:
7228:
6980:"Grassmann's Vision"
6716:
6665:
6609:
6572:
6524:
6464:
6416:
6335:
6124:
6039:
5945:
5894:
5764:
5659:
5653:reciprocal functions
5605:
5544:
5436:
5410:hyperbolic functions
5148:
4880:
4862:in the complex plane
4742:
4579:
4510:
4239:
4187:
4179:in the complex plane
3940:
3915:is a generator of a
3781:
3662:
3513:
3462:
3241:
3130:
3065:
3038:
2981:
2952:
2934:
2914:
2884:
2859:
2841:
2685:
2649:
2615:
2480:
2412:
2347:
2340:, and so there is a
2307:
2270:
2229:
2192:identity matrix and
2086:
1961:
1815:
1706:
1667:
1591:quadratic polynomial
1515:
1193:
1114:
1082:
1062:
1025:
1005:
968:
948:
936:
905:
885:
873:
842:
822:
785:
765:
728:
708:
696:
665:
645:
633:
602:
582:
548:
528:
494:
474:
462:
434:
414:
402:
374:
348:
142:
120:
83:with this property,
7557:on 16 December 2019
6866:(Online ed.).
5361:complex exponential
4575:
2982:
2974:
2953:
2945:
2915:
2907:
2885:
2877:
2860:
2852:
2299:has no real-number
1861:and imaginary unit
1835:complex conjugation
1809:field automorphisms
1659:is said to have an
1106:The imaginary unit
1058:
1001:
944:
932:
881:
869:
818:
761:
704:
692:
641:
629:
578:
524:
470:
458:
410:
398:
43:The imaginary unit
7499:. OEIS Foundation.
7472:. OEIS Foundation.
7352:
6857:"imaginary number"
6829:American Scientist
6756:
6751:
6734:
6702:
6697:
6680:
6635:
6633:
6595:
6593:
6551:
6511:
6490:
6450:
6435:
6402:
6400:
6223:
6106:
5997:
5927:
5919:
5840:
5744:
5695:
5628:is a real number.
5618:
5589:
5523:
5303:
5107:
5079:
5044:
5029:
4996:
4965:
4950:
4920:
4864:
4842:
4726:
4724:
4714:
4697:
4611:
4594:
4492:
4490:
4469:
4407:
4346:
4282:
4217:
4181:
4129:
4111:
4068:
4025:
3978:
3899:
3750:
3644:
3480:
3475:
3435:
3433:
3172:
3106:complex arithmetic
3078:
3051:
3005:
2920:
2825:
2801:
2659:
2631:
2504:
2418:
2394:
2326:
2289:
2249:
2173:of one squares to
2149:
2140:
2058:
2049:
2000:
1823:
1727:
1722:
1685:
1683:
1530:
1528:
1429:
1427:
1136:
1091:
1065:
1008:
951:
888:
825:
768:
711:
648:
585:
531:
477:
417:
357:
214:double square root
183:having a negative
153:
128:
71:quadratic equation
53:
7600:Algebraic numbers
7412:978-0-511-91510-9
7341:
7320:
7299:
7278:
7257:
7122:978-1-4008-3029-9
7087:978-1-56107-252-1
7056:978-1-133-70753-0
7025:978-0-486-13793-3
6963:Hermann Grassmann
6948:978-0-691-14904-2
6917:978-0-471-54397-8
6874:(Subscription or
6750:
6749:
6733:
6732:
6696:
6695:
6679:
6678:
6632:
6631:
6592:
6591:
6489:
6434:
6176:
6175:
5992:
5918:
5854:complex logarithm
5739:
5680:
5397:for all integers
5290:
5239:
5194:
5188:
5078:
5043:
5028:
5024:
4995:
4964:
4949:
4945:
4919:
4893:
4831:
4813:
4772:
4771:
4713:
4709:
4696:
4692:
4670:
4631:
4610:
4606:
4593:
4589:
4561:
4533:
4522:
4468:
4406:
4376:
4345:
4281:
4251:
4233:additive inverses
4227:has two distinct
4110:
4067:
4024:
3977:
3474:
3233:Gaussian integers
3215:Gaussian integers
3122:imaginary integer
3076:
3049:
3003:
2985:
2956:
2918:
2888:
2863:
2823:
2811:
2799:
2765:
2730:
2717:
2657:
2626:
2523:geometric algebra
2517:Geometric algebra
2421:{\displaystyle x}
2190:4 Γ 4
2163:2 Γ 2
1943:2 Γ 2
1721:
1682:
1527:
1418:
1415:
1391:
1339:
1104:
1103:
1087:
1046:
1033:
1030:
989:
976:
973:
926:
913:
910:
863:
850:
847:
806:
793:
790:
749:
736:
733:
686:
673:
670:
623:
610:
607:
553:
499:
439:
379:
353:
173:Algebraic closure
114:Imaginary numbers
18:Square root of β1
16:(Redirected from
7617:
7581:
7580:on 13 July 2007.
7566:
7564:
7562:
7553:. Archived from
7535:
7523:
7519:
7500:
7482:Sloane, N. J. A.
7479:
7473:
7455:Sloane, N. J. A.
7453:
7423:
7417:
7416:
7393:
7387:
7386:
7384:
7361:
7359:
7358:
7353:
7342:
7340:
7326:
7321:
7319:
7305:
7300:
7298:
7284:
7279:
7277:
7263:
7258:
7250:
7222:
7216:
7215:
7213:
7211:
7197:
7191:
7190:
7162:
7156:
7155:
7153:
7151:
7140:
7133:
7127:
7126:
7107:
7098:
7092:
7091:
7067:
7061:
7060:
7036:
7030:
7029:
7005:
6999:
6998:
6984:
6971:William Clifford
6967:Ausdehnungslehre
6959:
6953:
6952:
6928:
6922:
6921:
6894:Merzbach, Uta C.
6886:
6880:
6879:
6871:
6859:
6852:
6846:
6845:
6819:
6813:
6812:
6795:
6793:
6783:
6767:
6765:
6763:
6762:
6757:
6752:
6745:
6741:
6735:
6728:
6724:
6711:
6709:
6708:
6703:
6698:
6691:
6687:
6681:
6674:
6670:
6661:are the numbers
6660:
6656:
6652:
6644:
6642:
6641:
6636:
6634:
6627:
6623:
6604:
6602:
6601:
6596:
6594:
6587:
6583:
6568:
6564:
6560:
6558:
6557:
6552:
6544:
6543:
6520:
6518:
6517:
6512:
6504:
6503:
6491:
6482:
6476:
6475:
6459:
6457:
6456:
6451:
6449:
6448:
6436:
6427:
6411:
6409:
6408:
6403:
6401:
6364:
6363:
6351:
6350:
6326:
6307:
6288:
6284:
6280:
6264:
6232:
6230:
6229:
6224:
6213:
6177:
6174:
6160:
6159:
6154:
6131:
6115:
6113:
6112:
6107:
6032:
6021:
6006:
6004:
6003:
5998:
5993:
5988:
5971:
5957:
5956:
5940:
5936:
5934:
5933:
5928:
5920:
5911:
5878:
5867:
5849:
5847:
5846:
5841:
5779:
5778:
5759:
5753:
5751:
5750:
5745:
5740:
5738:
5721:
5718:
5713:
5694:
5646:
5627:
5625:
5624:
5619:
5617:
5616:
5598:
5596:
5595:
5590:
5532:
5530:
5529:
5524:
5429:
5425:
5418:
5414:
5408:components, the
5400:
5396:
5389:
5385:
5378:
5374:
5370:
5366:
5342:
5338:
5330:
5326:
5319:
5312:
5310:
5309:
5304:
5299:
5295:
5291:
5289:
5281:
5267:
5248:
5244:
5240:
5238:
5230:
5216:
5200:
5196:
5195:
5190:
5189:
5181:
5172:
5144:
5133:
5127:
5122:
5116:
5114:
5113:
5108:
5094:
5093:
5087:
5080:
5071:
5064:
5063:
5057:
5045:
5036:
5030:
5020:
5019:
5010:
5009:
4997:
4988:
4985:
4984:
4978:
4966:
4957:
4951:
4941:
4940:
4934:
4933:
4921:
4912:
4909:
4908:
4902:
4894:
4892:
4884:
4873:
4861:
4851:
4849:
4848:
4843:
4832:
4827:
4819:
4814:
4809:
4789:
4784:
4783:
4778:
4774:
4773:
4767:
4766:
4755:
4735:
4733:
4732:
4727:
4725:
4715:
4705:
4704:
4698:
4688:
4687:
4678:
4673:
4671:
4666:
4664:
4663:
4632:
4627:
4612:
4602:
4601:
4595:
4585:
4584:
4581:
4580:
4569:
4564:
4562:
4557:
4555:
4554:
4531:
4523:
4518:
4501:
4499:
4498:
4493:
4491:
4484:
4483:
4477:
4470:
4461:
4454:
4453:
4447:
4439:
4434:
4432:
4431:
4416:
4408:
4399:
4396:
4395:
4389:
4377:
4372:
4360:
4359:
4347:
4338:
4335:
4334:
4328:
4320:
4315:
4313:
4312:
4308:
4299:
4298:
4291:
4283:
4274:
4271:
4270:
4264:
4252:
4247:
4226:
4224:
4223:
4218:
4216:
4215:
4211:
4178:
4162:
4152:
4147:principal values
4138:
4136:
4135:
4130:
4125:
4124:
4112:
4103:
4100:
4099:
4093:
4082:
4081:
4069:
4060:
4057:
4056:
4050:
4042:
4041:
4026:
4017:
4014:
4013:
4007:
3999:
3998:
3993:
3992:
3979:
3970:
3967:
3966:
3960:
3952:
3951:
3933:
3914:
3908:
3906:
3905:
3900:
3886:
3885:
3854:
3853:
3825:
3824:
3796:
3795:
3775:is any integer:
3774:
3770:
3759:
3757:
3756:
3751:
3653:
3651:
3650:
3645:
3640:
3639:
3629:
3601:
3600:
3574:
3573:
3563:
3535:
3534:
3506:
3495:
3491:
3489:
3487:
3486:
3481:
3476:
3467:
3455:
3444:
3442:
3441:
3436:
3434:
3226:
3222:
3201:imaginary number
3181:
3179:
3178:
3173:
3087:
3085:
3084:
3079:
3077:
3069:
3060:
3058:
3057:
3052:
3050:
3045:
3033:
3029:
3022:
3018:
3014:
3012:
3011:
3006:
3004:
2999:
2991:
2986:
2984:
2983:
2967:
2964:
2963:
2957:
2955:
2954:
2938:
2929:
2927:
2926:
2921:
2919:
2917:
2916:
2894:
2889:
2887:
2886:
2870:
2864:
2862:
2861:
2845:
2834:
2832:
2831:
2826:
2824:
2821:
2812:
2807:
2802:
2800:
2771:
2767:
2766:
2764:
2763:
2739:
2734:
2731:
2723:
2718:
2710:
2679:
2668:
2666:
2665:
2660:
2658:
2653:
2640:
2638:
2637:
2632:
2627:
2619:
2602:
2598:
2583:
2579:
2569:
2551:
2513:
2511:
2510:
2505:
2467:
2463:
2459:
2427:
2425:
2424:
2419:
2403:
2401:
2400:
2395:
2381:
2380:
2368:
2354:
2335:
2333:
2332:
2327:
2319:
2318:
2298:
2296:
2295:
2290:
2282:
2281:
2258:
2256:
2255:
2250:
2236:
2210:
2195:
2191:
2187:
2183:
2179:
2164:
2158:
2156:
2155:
2150:
2145:
2144:
2079:
2067:
2065:
2064:
2059:
2054:
2053:
2005:
2004:
1954:
1950:
1944:
1940:
1936:
1929:
1918:
1908:
1897:
1883:
1872:
1868:
1864:
1860:
1832:
1830:
1829:
1824:
1822:
1783:
1776:
1766:
1759:
1752:
1744:
1736:
1734:
1733:
1728:
1723:
1714:
1701:
1694:
1692:
1691:
1686:
1684:
1675:
1658:
1651:
1644:
1634:
1627:
1623:
1603:
1585:
1578:
1561:
1549:
1539:
1537:
1536:
1531:
1529:
1520:
1502:
1496:
1489:
1481:
1473:rectangular form
1470:
1464:
1457:
1450:
1446:
1442:
1438:
1436:
1435:
1430:
1428:
1416:
1413:
1406:
1389:
1382:
1377:
1376:
1360:
1359:
1337:
1330:
1305:
1300:
1299:
1283:
1282:
1254:
1231:
1226:
1225:
1209:
1208:
1188:
1184:
1180:
1174:
1166:
1159:
1151:
1145:
1143:
1142:
1137:
1126:
1125:
1109:
1100:
1098:
1097:
1092:
1085:
1074:
1072:
1071:
1066:
1064:
1063:
1044:
1043:
1042:
1031:
1028:
1017:
1015:
1014:
1009:
1007:
1006:
987:
986:
985:
974:
971:
960:
958:
957:
952:
950:
949:
938:
937:
924:
923:
922:
911:
908:
897:
895:
894:
889:
887:
886:
875:
874:
861:
860:
859:
848:
845:
834:
832:
831:
826:
824:
823:
804:
803:
802:
791:
788:
777:
775:
774:
769:
767:
766:
747:
746:
745:
734:
731:
720:
718:
717:
712:
710:
709:
698:
697:
684:
683:
682:
671:
668:
657:
655:
654:
649:
647:
646:
635:
634:
621:
620:
619:
608:
605:
594:
592:
591:
586:
584:
583:
566:
565:
551:
540:
538:
537:
532:
530:
529:
512:
511:
497:
486:
484:
483:
478:
476:
475:
464:
463:
452:
451:
437:
426:
424:
423:
418:
416:
415:
404:
403:
392:
391:
377:
366:
364:
363:
358:
351:
338:
332:
323:
296:
279:negative numbers
249:electric current
246:
242:
238:
226:
222:
203:
196:
193:
162:
160:
159:
154:
149:
137:
135:
134:
129:
127:
110:
102:
86:
78:
67:
46:
21:
7625:
7624:
7620:
7619:
7618:
7616:
7615:
7614:
7595:Complex numbers
7585:
7584:
7568:
7560:
7558:
7547:Euler, Leonhard
7545:
7542:
7532:
7517:
7511:
7508:
7506:Further reading
7503:
7480:
7476:
7425:
7424:
7420:
7413:
7395:
7394:
7390:
7362:
7330:
7309:
7288:
7267:
7226:
7225:
7223:
7219:
7209:
7207:
7199:
7198:
7194:
7179:
7164:
7163:
7159:
7149:
7147:
7138:
7135:
7134:
7130:
7123:
7105:
7100:
7099:
7095:
7088:
7069:
7068:
7064:
7057:
7038:
7037:
7033:
7026:
7007:
7006:
7002:
6982:
6976:Hestenes, David
6974:
6969:of 1844; later
6960:
6956:
6949:
6930:
6929:
6925:
6918:
6888:
6887:
6883:
6873:
6854:
6853:
6849:
6821:
6820:
6816:
6809:
6796:
6785:
6784:
6780:
6776:
6771:
6770:
6714:
6713:
6663:
6662:
6658:
6654:
6646:
6607:
6606:
6570:
6569:
6566:
6562:
6535:
6522:
6521:
6492:
6467:
6462:
6461:
6437:
6414:
6413:
6399:
6398:
6388:
6376:
6375:
6365:
6355:
6342:
6333:
6332:
6309:
6290:
6286:
6282:
6267:
6265:
6261:
6256:
6242:Hyperbolic unit
6238:
6164:
6122:
6121:
6037:
6036:
6027:
6019:
6012:
5972:
5948:
5943:
5942:
5938:
5892:
5891:
5881:Riemann surface
5873:
5861:
5767:
5762:
5761:
5757:
5725:
5657:
5656:
5632:
5608:
5603:
5602:
5542:
5541:
5536:Euler's formula
5434:
5433:
5427:
5423:
5416:
5412:
5398:
5391:
5387:
5380:
5376:
5372:
5368:
5364:
5357:
5345:regular polygon
5340:
5336:
5328:
5324:
5314:
5282:
5268:
5265:
5261:
5231:
5217:
5214:
5210:
5173:
5161:
5157:
5146:
5145:
5135:
5131:
5125:
5120:
4878:
4877:
4871:
4859:
4820:
4790:
4756:
4750:
4746:
4745:
4740:
4739:
4723:
4722:
4672:
4633:
4620:
4619:
4563:
4524:
4508:
4507:
4489:
4488:
4433:
4378:
4365:
4364:
4314:
4292:
4253:
4237:
4236:
4196:
4185:
4184:
4176:
4169:
4154:
4150:
3986:
3943:
3938:
3937:
3931:
3930:for an integer
3928:Euler's formula
3912:
3868:
3836:
3807:
3784:
3779:
3778:
3772:
3768:
3765:
3660:
3659:
3608:
3589:
3542:
3523:
3511:
3510:
3501:
3493:
3460:
3459:
3457:
3453:
3450:
3432:
3431:
3373:
3334:
3333:
3287:
3239:
3238:
3224:
3220:
3217:
3128:
3127:
3114:
3102:
3063:
3062:
3036:
3035:
3031:
3027:
3020:
3016:
2932:
2931:
2839:
2838:
2683:
2682:
2674:
2647:
2646:
2613:
2612:
2609:
2600:
2596:
2593:Euclidean space
2581:
2571:
2557:
2542:
2527:Euclidean plane
2519:
2478:
2477:
2465:
2461:
2457:
2448:Cartesian plane
2440:
2434:
2410:
2409:
2372:
2345:
2344:
2310:
2305:
2304:
2273:
2268:
2267:
2266:The polynomial
2227:
2226:
2212:
2205:
2193:
2189:
2185:
2181:
2174:
2162:
2139:
2138:
2133:
2127:
2126:
2118:
2108:
2084:
2083:
2071:
2048:
2047:
2042:
2036:
2035:
2027:
2017:
1999:
1998:
1993:
1987:
1986:
1981:
1971:
1959:
1958:
1952:
1951:and the matrix
1948:
1946:identity matrix
1942:
1938:
1934:
1920:
1910:
1899:
1884:
1874:
1870:
1866:
1862:
1858:
1847:
1813:
1812:
1803:unique up to a
1778:
1771:
1761:
1754:
1750:
1742:
1739:Cartesian plane
1704:
1703:
1696:
1665:
1664:
1653:
1646:
1639:
1629:
1625:
1618:
1598:
1587:
1580:
1574:
1559:
1556:Cartesian plane
1544:
1513:
1512:
1498:
1491:
1487:
1476:
1468:
1459:
1452:
1448:
1444:
1440:
1426:
1425:
1405:
1381:
1368:
1361:
1351:
1348:
1347:
1329:
1304:
1291:
1284:
1274:
1271:
1270:
1253:
1230:
1217:
1210:
1200:
1191:
1190:
1186:
1182:
1176:
1172:
1161:
1157:
1149:
1117:
1112:
1111:
1107:
1080:
1079:
1034:
1023:
1022:
977:
966:
965:
914:
903:
902:
851:
840:
839:
794:
783:
782:
737:
726:
725:
674:
663:
662:
611:
600:
599:
554:
546:
545:
500:
492:
491:
440:
432:
431:
380:
372:
371:
346:
345:
339:
336:
330:
321:
294:
263:
257:
244:
240:
236:
224:
220:
212:(which has one
198:
194:
191:
140:
139:
118:
117:
104:
100:
89:complex numbers
84:
73:
65:
44:
35:
28:
23:
22:
15:
12:
11:
5:
7623:
7621:
7613:
7612:
7607:
7602:
7597:
7587:
7586:
7583:
7582:
7574:mathdl.maa.org
7541:
7540:External links
7538:
7537:
7536:
7530:
7507:
7504:
7502:
7501:
7474:
7418:
7411:
7388:
7375:(4): 515β539.
7371:. New Series.
7351:
7348:
7345:
7339:
7336:
7333:
7329:
7324:
7318:
7315:
7312:
7308:
7303:
7297:
7294:
7291:
7287:
7282:
7276:
7273:
7270:
7266:
7261:
7256:
7253:
7248:
7245:
7242:
7239:
7236:
7233:
7217:
7192:
7177:
7157:
7128:
7121:
7093:
7086:
7062:
7055:
7031:
7024:
7000:
6954:
6947:
6923:
6916:
6890:Boyer, Carl B.
6881:
6847:
6836:(6): 364β371.
6814:
6807:
6777:
6775:
6772:
6769:
6768:
6755:
6748:
6744:
6738:
6731:
6727:
6721:
6701:
6694:
6690:
6684:
6677:
6673:
6630:
6626:
6620:
6617:
6614:
6590:
6586:
6580:
6577:
6550:
6547:
6542:
6538:
6534:
6530:
6510:
6507:
6502:
6499:
6495:
6488:
6485:
6479:
6474:
6470:
6447:
6444:
6440:
6433:
6430:
6424:
6421:
6397:
6394:
6391:
6389:
6387:
6384:
6381:
6378:
6377:
6374:
6371:
6368:
6366:
6362:
6358:
6354:
6349:
6345:
6341:
6340:
6258:
6257:
6255:
6252:
6251:
6250:
6249:in quaternions
6244:
6237:
6234:
6222:
6219:
6216:
6212:
6209:
6206:
6203:
6200:
6197:
6193:
6190:
6186:
6183:
6180:
6173:
6170:
6167:
6163:
6157:
6153:
6149:
6146:
6143:
6140:
6137:
6134:
6130:
6105:
6102:
6098:
6095:
6092:
6089:
6086:
6083:
6080:
6077:
6074:
6071:
6068:
6065:
6062:
6059:
6056:
6053:
6050:
6047:
6044:
6024:gamma function
6011:
6008:
5996:
5991:
5987:
5984:
5981:
5978:
5975:
5969:
5966:
5963:
5960:
5955:
5951:
5926:
5923:
5917:
5914:
5908:
5905:
5902:
5899:
5839:
5836:
5833:
5830:
5827:
5824:
5821:
5818:
5815:
5812:
5809:
5806:
5803:
5800:
5797:
5794:
5791:
5788:
5785:
5782:
5777:
5774:
5770:
5743:
5737:
5734:
5731:
5728:
5724:
5717:
5712:
5709:
5706:
5703:
5699:
5693:
5690:
5687:
5683:
5679:
5676:
5673:
5670:
5667:
5664:
5651:as the sum of
5615:
5611:
5588:
5585:
5582:
5579:
5576:
5573:
5570:
5567:
5564:
5561:
5558:
5555:
5552:
5549:
5522:
5519:
5516:
5513:
5510:
5507:
5504:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5471:
5468:
5465:
5462:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5356:
5353:
5333:roots of unity
5302:
5298:
5294:
5288:
5285:
5280:
5277:
5274:
5271:
5264:
5260:
5257:
5254:
5251:
5247:
5243:
5237:
5234:
5229:
5226:
5223:
5220:
5213:
5209:
5206:
5203:
5199:
5193:
5187:
5184:
5179:
5176:
5170:
5167:
5164:
5160:
5156:
5153:
5106:
5103:
5100:
5097:
5092:
5086:
5083:
5077:
5074:
5068:
5062:
5056:
5051:
5048:
5042:
5039:
5033:
5027:
5023:
5016:
5013:
5008:
5003:
5000:
4994:
4991:
4983:
4977:
4972:
4969:
4963:
4960:
4954:
4948:
4944:
4937:
4932:
4927:
4924:
4918:
4915:
4907:
4901:
4897:
4891:
4887:
4841:
4838:
4835:
4830:
4826:
4823:
4817:
4812:
4808:
4805:
4802:
4799:
4796:
4793:
4787:
4782:
4777:
4770:
4765:
4762:
4759:
4753:
4749:
4721:
4718:
4712:
4708:
4701:
4695:
4691:
4684:
4681:
4676:
4674:
4669:
4662:
4657:
4654:
4651:
4648:
4645:
4642:
4639:
4636:
4634:
4630:
4625:
4622:
4621:
4618:
4615:
4609:
4605:
4598:
4592:
4588:
4578:
4572:
4567:
4565:
4560:
4553:
4548:
4545:
4542:
4539:
4536:
4530:
4527:
4525:
4521:
4516:
4515:
4487:
4482:
4476:
4473:
4467:
4464:
4458:
4452:
4446:
4442:
4437:
4435:
4430:
4425:
4422:
4419:
4415:
4412:
4405:
4402:
4394:
4388:
4384:
4381:
4379:
4375:
4370:
4367:
4366:
4363:
4358:
4353:
4350:
4344:
4341:
4333:
4327:
4323:
4318:
4316:
4311:
4307:
4303:
4297:
4290:
4287:
4280:
4277:
4269:
4263:
4259:
4256:
4254:
4250:
4245:
4244:
4214:
4210:
4206:
4203:
4199:
4195:
4192:
4168:
4165:
4128:
4123:
4118:
4115:
4109:
4106:
4098:
4092:
4089:
4085:
4080:
4075:
4072:
4066:
4063:
4055:
4049:
4045:
4040:
4035:
4032:
4029:
4023:
4020:
4012:
4006:
4002:
3997:
3991:
3985:
3982:
3976:
3973:
3965:
3959:
3955:
3950:
3946:
3898:
3895:
3892:
3889:
3884:
3881:
3878:
3875:
3871:
3866:
3863:
3860:
3857:
3852:
3849:
3846:
3843:
3839:
3834:
3831:
3828:
3823:
3820:
3817:
3814:
3810:
3805:
3802:
3799:
3794:
3791:
3787:
3767:The powers of
3764:
3763:Integer powers
3761:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3706:
3703:
3700:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3643:
3638:
3635:
3632:
3628:
3624:
3621:
3618:
3615:
3611:
3607:
3604:
3599:
3596:
3592:
3588:
3584:
3581:
3577:
3572:
3569:
3566:
3562:
3558:
3555:
3552:
3549:
3545:
3541:
3538:
3533:
3530:
3526:
3522:
3518:
3479:
3473:
3470:
3449:
3446:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3374:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3345:
3342:
3339:
3336:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3288:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3246:
3229:square lattice
3216:
3213:
3209:imaginary axis
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3113:
3110:
3101:
3098:
3075:
3072:
3061:, rather than
3048:
3043:
3002:
2998:
2994:
2989:
2980:
2977:
2970:
2962:
2951:
2948:
2941:
2913:
2910:
2903:
2900:
2897:
2892:
2883:
2880:
2873:
2867:
2858:
2855:
2848:
2818:
2815:
2810:
2805:
2798:
2795:
2792:
2789:
2786:
2783:
2780:
2777:
2774:
2762:
2759:
2756:
2753:
2750:
2747:
2744:
2738:
2729:
2726:
2721:
2716:
2713:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2656:
2630:
2625:
2622:
2608:
2605:
2518:
2515:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2436:Main article:
2433:
2430:
2417:
2393:
2390:
2387:
2384:
2379:
2375:
2371:
2367:
2363:
2360:
2357:
2353:
2325:
2322:
2317:
2313:
2288:
2285:
2280:
2276:
2248:
2245:
2242:
2239:
2235:
2211:
2202:
2198:Dirac matrices
2169:of zero and a
2165:matrix with a
2148:
2143:
2137:
2134:
2132:
2129:
2128:
2125:
2122:
2119:
2117:
2114:
2113:
2111:
2106:
2103:
2100:
2097:
2094:
2091:
2057:
2052:
2046:
2043:
2041:
2038:
2037:
2034:
2031:
2028:
2026:
2023:
2022:
2020:
2015:
2012:
2008:
2003:
1997:
1994:
1992:
1989:
1988:
1985:
1982:
1980:
1977:
1976:
1974:
1969:
1966:
1846:
1843:
1821:
1806:
1802:
1726:
1720:
1717:
1711:
1681:
1678:
1672:
1607:
1586:
1572:
1564:imaginary axis
1526:
1523:
1511:(or angle) of
1505:absolute value
1424:
1421:
1412:
1409:
1407:
1404:
1401:
1398:
1395:
1388:
1385:
1383:
1380:
1375:
1371:
1367:
1364:
1362:
1358:
1354:
1350:
1349:
1346:
1343:
1336:
1333:
1331:
1328:
1325:
1322:
1319:
1316:
1311:
1308:
1306:
1303:
1298:
1294:
1290:
1287:
1285:
1281:
1277:
1273:
1272:
1269:
1266:
1263:
1260:
1257:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1232:
1229:
1224:
1220:
1216:
1213:
1211:
1207:
1203:
1199:
1198:
1135:
1132:
1129:
1124:
1120:
1102:
1101:
1090:
1076:
1075:
1061:
1055:
1052:
1049:
1041:
1037:
1019:
1018:
1004:
998:
995:
992:
984:
980:
962:
961:
947:
941:
935:
929:
921:
917:
899:
898:
884:
878:
872:
866:
858:
854:
836:
835:
821:
815:
812:
809:
801:
797:
779:
778:
764:
758:
755:
752:
744:
740:
722:
721:
707:
701:
695:
689:
681:
677:
659:
658:
644:
638:
632:
626:
618:
614:
596:
595:
581:
575:
572:
569:
564:
561:
557:
542:
541:
527:
521:
518:
515:
510:
507:
503:
488:
487:
473:
467:
461:
455:
450:
447:
443:
428:
427:
413:
407:
401:
395:
390:
387:
383:
368:
367:
356:
342:
341:
335:The powers of
329:
326:
316:is the number
299:Leonhard Euler
287:RenΓ© Descartes
256:
253:
152:
148:
126:
97:multiplication
57:imaginary unit
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7622:
7611:
7608:
7606:
7603:
7601:
7598:
7596:
7593:
7592:
7590:
7579:
7575:
7571:
7570:"Convergence"
7556:
7552:
7548:
7544:
7543:
7539:
7533:
7531:0-691-02795-1
7527:
7522:
7521:
7510:
7509:
7505:
7498:
7497:
7491:
7487:
7483:
7478:
7475:
7471:
7470:
7464:
7460:
7456:
7451:
7447:
7443:
7439:
7435:
7431:
7430:
7422:
7419:
7414:
7408:
7404:
7403:
7398:
7392:
7389:
7383:
7378:
7374:
7370:
7366:
7349:
7346:
7343:
7337:
7334:
7331:
7327:
7322:
7316:
7313:
7310:
7306:
7301:
7295:
7292:
7289:
7285:
7280:
7274:
7271:
7268:
7264:
7259:
7254:
7251:
7246:
7243:
7240:
7237:
7234:
7231:
7221:
7218:
7206:
7202:
7196:
7193:
7188:
7184:
7180:
7178:0-7637-1437-2
7174:
7170:
7169:
7161:
7158:
7146:
7142:
7132:
7129:
7124:
7118:
7114:
7110:
7109:
7097:
7094:
7089:
7083:
7079:
7075:
7074:
7066:
7063:
7058:
7052:
7048:
7044:
7043:
7035:
7032:
7027:
7021:
7017:
7013:
7012:
7004:
7001:
6996:
6992:
6988:
6981:
6977:
6972:
6968:
6964:
6958:
6955:
6950:
6944:
6940:
6936:
6935:
6927:
6924:
6919:
6913:
6909:
6905:
6901:
6900:
6895:
6891:
6885:
6882:
6877:
6869:
6865:
6864:
6858:
6851:
6848:
6843:
6839:
6835:
6831:
6830:
6825:
6818:
6815:
6810:
6808:0-471-19826-9
6804:
6800:
6792:
6791:
6782:
6779:
6773:
6753:
6746:
6742:
6736:
6729:
6725:
6719:
6699:
6692:
6688:
6682:
6675:
6671:
6650:
6628:
6624:
6618:
6615:
6612:
6588:
6584:
6578:
6575:
6548:
6545:
6540:
6536:
6532:
6508:
6505:
6500:
6497:
6493:
6486:
6483:
6477:
6472:
6468:
6445:
6442:
6438:
6431:
6428:
6422:
6419:
6412:Substituting
6395:
6392:
6390:
6385:
6382:
6379:
6372:
6369:
6367:
6360:
6356:
6352:
6347:
6343:
6330:
6324:
6320:
6316:
6312:
6305:
6301:
6297:
6293:
6279:
6275:
6271:
6263:
6260:
6253:
6248:
6245:
6243:
6240:
6239:
6235:
6233:
6220:
6217:
6214:
6207:
6204:
6201:
6191:
6188:
6184:
6181:
6178:
6171:
6168:
6165:
6161:
6155:
6144:
6141:
6138:
6119:
6116:
6103:
6100:
6096:
6093:
6090:
6087:
6081:
6072:
6069:
6063:
6060:
6057:
6048:
6045:
6042:
6034:
6031:
6026:evaluated at
6025:
6017:
6009:
6007:
5994:
5989:
5985:
5982:
5979:
5976:
5973:
5967:
5964:
5961:
5958:
5953:
5949:
5924:
5921:
5915:
5912:
5906:
5903:
5900:
5897:
5888:
5886:
5882:
5877:
5871:
5865:
5859:
5855:
5850:
5837:
5831:
5828:
5825:
5822:
5816:
5813:
5810:
5807:
5801:
5798:
5795:
5792:
5786:
5783:
5780:
5775:
5772:
5768:
5754:
5741:
5735:
5732:
5729:
5726:
5722:
5715:
5710:
5707:
5704:
5701:
5697:
5685:
5677:
5674:
5671:
5668:
5665:
5662:
5654:
5650:
5644:
5640:
5636:
5631:The quotient
5629:
5613:
5609:
5599:
5586:
5583:
5580:
5577:
5574:
5571:
5568:
5565:
5562:
5559:
5556:
5553:
5550:
5547:
5539:
5537:
5533:
5517:
5514:
5511:
5505:
5502:
5499:
5496:
5490:
5487:
5484:
5478:
5475:
5472:
5469:
5466:
5463:
5460:
5457:
5454:
5451:
5448:
5445:
5442:
5439:
5431:
5422:
5411:
5407:
5402:
5395:
5384:
5362:
5354:
5352:
5350:
5346:
5334:
5323:
5317:
5300:
5296:
5292:
5286:
5283:
5278:
5275:
5272:
5269:
5262:
5258:
5255:
5252:
5249:
5245:
5241:
5235:
5232:
5227:
5224:
5221:
5218:
5211:
5207:
5204:
5201:
5197:
5191:
5185:
5182:
5177:
5174:
5168:
5165:
5162:
5158:
5154:
5151:
5142:
5139:= 0, 1, ...,
5138:
5129:
5117:
5104:
5101:
5098:
5095:
5084:
5081:
5075:
5072:
5066:
5054:
5049:
5046:
5040:
5037:
5031:
5025:
5021:
5014:
5011:
5001:
4998:
4992:
4989:
4975:
4970:
4967:
4961:
4958:
4952:
4946:
4942:
4935:
4925:
4922:
4916:
4913:
4899:
4895:
4889:
4885:
4875:
4869:
4856:
4852:
4839:
4836:
4833:
4828:
4824:
4821:
4815:
4810:
4806:
4803:
4800:
4797:
4794:
4791:
4785:
4780:
4775:
4768:
4763:
4760:
4757:
4751:
4747:
4736:
4719:
4716:
4710:
4706:
4699:
4693:
4689:
4682:
4679:
4675:
4667:
4652:
4649:
4646:
4640:
4637:
4635:
4628:
4623:
4616:
4613:
4607:
4603:
4596:
4590:
4586:
4576:
4570:
4566:
4558:
4543:
4540:
4537:
4528:
4526:
4519:
4505:
4502:
4485:
4474:
4471:
4465:
4462:
4456:
4444:
4440:
4436:
4423:
4420:
4417:
4413:
4410:
4403:
4400:
4386:
4382:
4380:
4373:
4368:
4361:
4351:
4348:
4342:
4339:
4325:
4321:
4317:
4309:
4305:
4301:
4288:
4285:
4278:
4275:
4261:
4257:
4255:
4248:
4234:
4230:
4212:
4208:
4204:
4201:
4197:
4193:
4190:
4173:
4166:
4164:
4161:
4157:
4148:
4144:
4139:
4126:
4116:
4113:
4107:
4104:
4090:
4087:
4083:
4073:
4070:
4064:
4061:
4047:
4043:
4033:
4030:
4027:
4021:
4018:
4004:
4000:
3995:
3983:
3980:
3974:
3971:
3957:
3953:
3948:
3944:
3935:
3929:
3924:
3922:
3918:
3909:
3896:
3893:
3890:
3887:
3882:
3879:
3876:
3873:
3869:
3864:
3861:
3858:
3855:
3850:
3847:
3844:
3841:
3837:
3832:
3829:
3826:
3821:
3818:
3815:
3812:
3808:
3803:
3800:
3797:
3792:
3789:
3785:
3776:
3762:
3760:
3747:
3744:
3741:
3738:
3735:
3732:
3726:
3723:
3720:
3717:
3711:
3708:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3680:
3677:
3674:
3671:
3665:
3657:
3654:
3641:
3636:
3630:
3626:
3622:
3619:
3616:
3609:
3605:
3602:
3597:
3594:
3590:
3586:
3582:
3579:
3575:
3570:
3564:
3560:
3556:
3553:
3550:
3543:
3539:
3536:
3531:
3528:
3524:
3520:
3516:
3508:
3505:
3499:
3498:anticlockwise
3477:
3471:
3468:
3447:
3445:
3428:
3425:
3419:
3416:
3413:
3410:
3407:
3401:
3395:
3392:
3389:
3386:
3383:
3377:
3375:
3367:
3364:
3361:
3358:
3349:
3346:
3343:
3340:
3330:
3327:
3321:
3318:
3315:
3309:
3303:
3300:
3297:
3291:
3289:
3281:
3278:
3275:
3272:
3266:
3260:
3257:
3254:
3251:
3236:
3234:
3230:
3214:
3212:
3210:
3206:
3202:
3198:
3193:
3191:
3187:
3182:
3169:
3166:
3160:
3157:
3154:
3148:
3145:
3142:
3139:
3136:
3133:
3125:
3123:
3119:
3111:
3109:
3107:
3099:
3097:
3095:
3091:
3073:
3070:
3046:
3041:
3024:
3000:
2996:
2992:
2987:
2978:
2975:
2968:
2949:
2946:
2939:
2911:
2908:
2901:
2898:
2895:
2890:
2881:
2878:
2871:
2865:
2856:
2853:
2846:
2835:
2816:
2813:
2808:
2803:
2793:
2790:
2784:
2778:
2775:
2736:
2727:
2724:
2719:
2714:
2711:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2677:
2672:
2654:
2644:
2628:
2623:
2620:
2606:
2604:
2594:
2589:
2587:
2578:
2574:
2568:
2564:
2560:
2555:
2550:
2546:
2539:
2536:
2532:
2528:
2524:
2516:
2514:
2501:
2498:
2495:
2492:
2489:
2483:
2475:
2471:
2455:
2454:
2453:complex plane
2449:
2445:
2439:
2438:Complex plane
2431:
2429:
2415:
2407:
2391:
2385:
2382:
2377:
2373:
2365:
2358:
2343:
2342:quotient ring
2339:
2323:
2320:
2315:
2311:
2302:
2286:
2283:
2278:
2274:
2264:
2262:
2246:
2240:
2224:
2220:
2216:
2208:
2203:
2201:
2199:
2178:
2172:
2168:
2159:
2146:
2141:
2135:
2130:
2123:
2120:
2115:
2109:
2104:
2101:
2098:
2095:
2092:
2089:
2081:
2078:
2074:
2068:
2055:
2050:
2044:
2039:
2032:
2029:
2024:
2018:
2013:
2010:
2006:
2001:
1995:
1990:
1983:
1978:
1972:
1967:
1964:
1956:
1947:
1931:
1927:
1923:
1917:
1913:
1906:
1902:
1895:
1891:
1887:
1881:
1877:
1856:
1852:
1844:
1842:
1840:
1836:
1810:
1804:
1800:
1798:
1795:
1791:
1787:
1782:
1775:
1768:
1765:
1758:
1748:
1747:anticlockwise
1740:
1724:
1718:
1715:
1709:
1700:
1679:
1676:
1670:
1662:
1657:
1650:
1643:
1636:
1633:
1622:
1615:
1611:
1605:
1601:
1596:
1595:multiple root
1592:
1584:
1577:
1573:
1571:
1569:
1565:
1557:
1553:
1552:complex plane
1548:
1542:
1524:
1521:
1510:
1506:
1501:
1495:
1485:
1480:
1474:
1465:
1462:
1456:
1422:
1419:
1410:
1408:
1402:
1396:
1386:
1384:
1378:
1373:
1369:
1365:
1363:
1356:
1352:
1344:
1341:
1334:
1332:
1326:
1320:
1317:
1309:
1307:
1301:
1296:
1292:
1288:
1286:
1279:
1275:
1267:
1264:
1261:
1258:
1256:
1250:
1244:
1241:
1235:
1233:
1227:
1222:
1218:
1214:
1212:
1205:
1201:
1179:
1168:
1165:
1155:
1146:
1133:
1130:
1127:
1122:
1118:
1088:
1078:
1077:
1059:
1053:
1050:
1047:
1039:
1035:
1021:
1020:
1002:
996:
993:
990:
982:
978:
964:
963:
945:
939:
933:
927:
919:
915:
901:
900:
882:
876:
870:
864:
856:
852:
838:
837:
819:
813:
810:
807:
799:
795:
781:
780:
762:
756:
753:
750:
742:
738:
724:
723:
705:
699:
693:
687:
679:
675:
661:
660:
642:
636:
630:
624:
616:
612:
598:
597:
579:
573:
570:
567:
562:
559:
555:
544:
543:
525:
519:
516:
513:
508:
505:
501:
490:
489:
471:
465:
459:
453:
448:
445:
441:
430:
429:
411:
405:
399:
393:
388:
385:
381:
370:
369:
354:
344:
343:
334:
333:
327:
325:
319:
315:
311:
307:
302:
300:
292:
288:
284:
280:
276:
272:
268:
262:
254:
252:
250:
234:
230:
217:
215:
211:
207:
202:
188:
186:
182:
178:
174:
170:
166:
150:
115:
111:
108:
98:
94:
90:
82:
76:
72:
68:
62:
58:
50:
49:complex plane
41:
37:
33:
19:
7578:the original
7573:
7559:. Retrieved
7555:the original
7515:
7493:
7477:
7466:
7433:
7427:
7421:
7401:
7391:
7372:
7368:
7220:
7208:. Retrieved
7205:math.hmc.edu
7204:
7195:
7167:
7160:
7148:. Retrieved
7144:
7131:
7103:
7096:
7072:
7065:
7041:
7034:
7010:
7003:
6989:. Springer.
6986:
6966:
6957:
6933:
6926:
6898:
6884:
6861:
6850:
6833:
6827:
6817:
6798:
6789:
6781:
6648:
6322:
6318:
6314:
6310:
6303:
6299:
6295:
6291:
6277:
6273:
6269:
6262:
6247:Right versor
6120:
6117:
6035:
6029:
6013:
5889:
5875:
5863:
5851:
5755:
5642:
5638:
5634:
5630:
5600:
5540:
5534:
5432:
5406:even and odd
5403:
5393:
5382:
5358:
5339:-th root of
5327:-th root of
5315:
5140:
5136:
5118:
4876:
4865:
4737:
4506:
4503:
4229:square roots
4182:
4159:
4155:
4140:
3936:
3925:
3921:circle group
3917:cyclic group
3910:
3777:
3766:
3658:
3655:
3509:
3503:
3451:
3237:
3218:
3208:
3194:
3190:cyclic group
3183:
3126:
3121:
3115:
3103:
3094:Branch point
3025:
2836:
2822:(incorrect).
2675:
2670:
2610:
2590:
2576:
2572:
2566:
2562:
2558:
2548:
2544:
2540:
2520:
2451:
2441:
2265:
2219:coefficients
2213:
2206:
2176:
2160:
2082:
2076:
2072:
2069:
1957:
1932:
1925:
1921:
1915:
1911:
1904:
1900:
1893:
1889:
1885:
1879:
1875:
1848:
1839:Galois group
1780:
1773:
1769:
1763:
1756:
1698:
1655:
1648:
1641:
1637:
1631:
1620:
1599:
1588:
1582:
1575:
1546:
1499:
1493:
1478:
1466:
1460:
1454:
1177:
1169:
1163:
1147:
1105:
340:are cyclic:
313:
309:
305:
303:
291:Isaac Newton
282:
275:real numbers
266:
264:
218:
206:square roots
200:
189:
171:exists (see
112:
106:
74:
64:
60:
56:
54:
36:
7561:29 November
7436:: 662β665.
6906:. pp.
5349:unit circle
4143:branch cuts
3205:number line
3199:to form an
3197:real number
3090:Square root
2470:translation
2450:called the
2444:number line
2215:Polynomials
2171:determinant
1873:satisfying
1797:isomorphism
1624:(or simply
1503:), with an
314:unit number
269:because in
255:Terminology
239:instead of
181:real number
81:real number
7589:Categories
7397:Gbur, Greg
6878:required.)
6774:References
5937:then when
5885:branch cut
5760:power is:
5390:at points
5386:and image
4868:cube roots
4866:The three
4231:which are
3100:Properties
2607:Proper use
2586:isomorphic
2474:similarity
2406:isomorphic
2225:, denoted
1484:polar form
328:Definition
243:, because
169:polynomial
7347:⋯
7314:−
7272:−
7241:π
7238:
7232:π
7210:22 August
6737:−
6720:−
6619:−
6529:⟹
6498:−
6478:−
6443:−
6353:−
6218:−
6215:≈
6196:Γ
6192:
6179:≈
6172:π
6169:
6162:π
6133:Γ
6094:−
6088:≈
6076:Γ
6052:Γ
6016:factorial
6010:Factorial
5990:π
5983:
5968:−
5959:
5922:π
5901:
5829:
5817:
5799:
5787:
5708:−
5698:∑
5692:∞
5689:→
5672:π
5669:
5663:π
5584:φ
5581:
5569:φ
5566:
5557:φ
5551:
5512:−
5506:
5485:−
5479:
5467:
5455:
5443:
5322:principal
5293:π
5259:
5242:π
5208:
5166:π
5155:
5134:are, for
5128:-th roots
5099:−
5082:π
5067:−
5015:−
4999:π
4923:π
4804:−
4752:±
4700:−
4683:−
4641:−
4624:−
4577:−
4472:π
4457:−
4421:π
4418:−
4411:π
4369:−
4349:π
4286:π
4202:π
4117:π
4074:π
4031:π
3981:π
3891:−
3859:−
3739:−
3709:−
3690:−
3623:π
3620:−
3617:φ
3595:φ
3580:−
3557:π
3551:φ
3529:φ
3478:π
3390:−
3071:−
2899:⋅
2866:⋅
2791:−
2785:⋅
2776:−
2725:−
2720:⋅
2712:−
2701:⋅
2689:−
2621:−
2487:↦
2389:⟩
2370:⟨
2336:forms an
2121:−
2030:−
1716:π
1710:−
1677:π
1568:real axis
1522:π
1497:(or just
1318:−
1262:−
1242:−
1189:are thus
1171:treating
1131:−
1089:⋮
1051:−
994:−
934:−
871:−
811:−
754:−
694:−
631:−
571:−
560:−
517:−
506:−
460:−
446:−
400:−
386:−
355:⋮
283:imaginary
267:imaginary
7450:24405635
7399:(2011).
7187:50495529
7150:26 March
7141: ?"
6978:(1996).
6896:(1991).
6561:Because
6236:See also
5870:cylinder
2643:radicals
2582:−1
2535:bivector
2261:integers
2204:Root of
1851:matrices
1845:Matrices
1799:, it is
1661:argument
1610:additive
1593:with no
1589:Being a
1509:argument
93:addition
91:, using
77:+ 1 = 0.
32:i-number
7490:A212880
7486:A212879
7463:A212878
7459:A212877
6908:439β445
6221:0.3016.
5641:/ sinh
5637:= cosh
5419:or the
5320:is the
3490:radians
3227:form a
3118:integer
2531:vectors
2525:of the
2521:In the
1941:by the
1541:radians
1154:algebra
312:or the
47:in the
7528:
7448:
7409:
7185:
7175:
7119:
7084:
7053:
7022:
6945:
6914:
6805:
6321:= 0 +
6281:where
6182:0.5216
6097:0.1549
6091:0.4980
5123:, the
4532:
3207:, the
3023:only.
1805:unique
1790:unique
1451:, and
1417:
1414:
1390:
1338:
1086:
1045:
1032:
1029:
988:
975:
972:
925:
912:
909:
862:
849:
846:
805:
792:
789:
748:
735:
732:
685:
672:
669:
622:
609:
606:
552:
498:
438:
378:
352:
289:, and
185:square
7446:S2CID
7018:-34.
6983:(PDF)
6872:
6254:Notes
5856:is a
5633:coth
4167:Roots
3186:group
3026:When
2673:real
2338:ideal
2301:roots
2167:trace
1794:up to
1786:field
1477:0 + 1
1181:with
1156:that
1148:With
310:unity
105:2 + 3
7563:2012
7526:ISBN
7494:The
7467:The
7407:ISBN
7212:2024
7183:OCLC
7173:ISBN
7152:2007
7117:ISBN
7082:ISBN
7051:ISBN
7020:ISBN
6943:ISBN
6912:ISBN
6803:ISBN
6712:and
6605:and
6285:and
6276:) =
6166:sinh
6028:1 +
6014:The
5666:coth
5464:sinh
5452:cosh
5426:and
5417:sinh
5415:and
5413:cosh
5359:The
5143:β 1,
4874:are
4145:and
3092:and
3019:and
2930:and
2678:β₯ 0,
2671:only
2543:1 =
2460:and
2223:ring
1937:and
1898:and
1869:and
1853:and
1777:and
1760:nor
1695:and
1645:and
1612:and
1604:has
1602:= β1
1579:vs.
1492:1 Γ
1463:= 1.
1160:and
306:unit
231:and
210:zero
197:and
175:and
165:root
95:and
55:The
7567:at
7438:doi
7434:120
7377:doi
7235:cot
7078:424
6991:doi
6939:225
6838:doi
6834:105
6651:= 1
6327:By
6319:ixy
6317:+ 2
6296:ixy
6294:+ 2
6189:arg
5950:log
5814:sin
5784:cos
5682:lim
5578:sin
5563:cos
5548:exp
5503:sin
5476:cos
5440:exp
5428:sin
5424:cos
5394:kΟi
5318:= 0
5256:sin
5205:cos
5152:exp
5130:of
5055:exp
4976:exp
4900:exp
4870:of
4445:exp
4387:exp
4326:exp
4262:exp
4091:sin
4048:cos
4005:exp
3958:exp
3494:90Β°
3492:or
3030:or
2209:+ 1
1903:= β
1801:not
1788:is
1663:of
1606:two
1570:).
1475:as
324:).
318:one
216:).
192:β1:
59:or
7591::
7572:.
7549:.
7492:.
7465:.
7444:.
7432:.
7373:44
7367:.
7203:.
7181:.
7143:.
7115:.
7113:12
7080:.
7049:.
7047:81
7016:31
6941:.
6910:.
6902:.
6892:;
6860:.
6832:.
6826:.
6649:xy
6549:1.
6396:1.
6313:-
6302:=
6298:-
6274:iy
6272:+
6033::
5980:ln
5898:ln
5876:Οi
5864:Οi
5826:ln
5796:ln
5758:ni
5430::
5383:Οi
5351:.
4163:.
4158:=
3934:,
3496:)
3192:.
3108:.
3096:.
2603:.
2597:β1
2575:=
2573:Jv
2561:=
2263:.
2077:bi
2075:+
1955:,
1926:bJ
1924:+
1922:aI
1916:bi
1914:+
1892:=
1890:JI
1888:=
1886:IJ
1878:=
1841:.
1558:,
1486:,
1449:β1
1447:,
1443:,
1183:β1
1134:1.
304:A
301:.
251:.
187:.
7565:.
7518:i
7452:.
7440::
7415:.
7385:.
7379::
7350:.
7344:+
7338:2
7335:+
7332:z
7328:1
7323:+
7317:2
7311:z
7307:1
7302:+
7296:1
7293:+
7290:z
7286:1
7281:+
7275:1
7269:z
7265:1
7260:+
7255:z
7252:1
7247:=
7244:z
7214:.
7189:.
7154:.
7139:i
7106:i
6997:.
6993::
6920:.
6870:.
6844:.
6840::
6811:.
6766:.
6754:i
6747:2
6743:1
6730:2
6726:1
6700:i
6693:2
6689:1
6683:+
6676:2
6672:1
6659:i
6655:y
6647:2
6629:2
6625:1
6616:=
6613:x
6589:2
6585:1
6579:=
6576:x
6567:x
6563:x
6546:=
6541:4
6537:x
6533:4
6509:0
6506:=
6501:2
6494:x
6487:4
6484:1
6473:2
6469:x
6446:1
6439:x
6432:2
6429:1
6423:=
6420:y
6393:=
6386:y
6383:x
6380:2
6373:0
6370:=
6361:2
6357:y
6348:2
6344:x
6325:.
6323:i
6315:y
6311:x
6306:.
6304:i
6300:y
6292:x
6287:y
6283:x
6278:i
6270:x
6268:(
6211:)
6208:i
6205:+
6202:1
6199:(
6185:,
6156:=
6152:|
6148:)
6145:i
6142:+
6139:1
6136:(
6129:|
6104:.
6101:i
6085:)
6082:i
6079:(
6073:i
6070:=
6067:)
6064:i
6061:+
6058:1
6055:(
6049:=
6046:!
6043:i
6030:i
6020:i
5995:.
5986:x
5977:i
5974:2
5965:=
5962:x
5954:i
5939:x
5925:i
5916:2
5913:1
5907:=
5904:i
5874:2
5866:.
5862:2
5838:.
5835:)
5832:x
5823:n
5820:(
5811:i
5808:+
5805:)
5802:x
5793:n
5790:(
5781:=
5776:i
5773:n
5769:x
5742:.
5736:i
5733:k
5730:+
5727:z
5723:1
5716:n
5711:n
5705:=
5702:k
5686:n
5678:=
5675:z
5645:,
5643:z
5639:z
5635:z
5614:i
5610:i
5587:.
5575:i
5572:+
5560:=
5554:i
5521:)
5518:z
5515:i
5509:(
5500:i
5497:+
5494:)
5491:z
5488:i
5482:(
5473:=
5470:z
5461:+
5458:z
5449:=
5446:z
5399:k
5392:2
5388:1
5381:2
5377:1
5373:i
5369:e
5365:1
5341:i
5337:n
5329:i
5325:n
5316:k
5301:.
5297:)
5287:n
5284:2
5279:1
5276:+
5273:k
5270:4
5263:(
5253:i
5250:+
5246:)
5236:n
5233:2
5228:1
5225:+
5222:k
5219:4
5212:(
5202:=
5198:)
5192:n
5186:4
5183:1
5178:+
5175:k
5169:i
5163:2
5159:(
5141:n
5137:k
5132:i
5126:n
5121:n
5105:.
5102:i
5096:=
5091:)
5085:i
5076:2
5073:1
5061:(
5050:,
5047:i
5041:2
5038:1
5032:+
5026:2
5022:3
5012:=
5007:)
5002:i
4993:6
4990:5
4982:(
4971:,
4968:i
4962:2
4959:1
4953:+
4947:2
4943:3
4936:=
4931:)
4926:i
4917:6
4914:1
4906:(
4896:=
4890:3
4886:i
4872:i
4860:i
4840:.
4837:i
4834:=
4829:2
4825:i
4822:2
4816:=
4811:2
4807:1
4801:i
4798:2
4795:+
4792:1
4786:=
4781:2
4776:)
4769:2
4764:i
4761:+
4758:1
4748:(
4720:.
4717:i
4711:2
4707:2
4694:2
4690:2
4680:=
4668:2
4661:/
4656:)
4653:i
4650:+
4647:1
4644:(
4638:=
4629:i
4617:,
4614:i
4608:2
4604:2
4597:+
4591:2
4587:2
4571:=
4559:2
4552:/
4547:)
4544:i
4541:+
4538:1
4535:(
4529:=
4520:i
4486:.
4481:)
4475:i
4466:4
4463:3
4451:(
4441:=
4429:)
4424:i
4414:i
4404:4
4401:1
4393:(
4383:=
4374:i
4362:,
4357:)
4352:i
4343:4
4340:1
4332:(
4322:=
4310:2
4306:/
4302:1
4296:)
4289:i
4279:2
4276:1
4268:(
4258:=
4249:i
4213:2
4209:/
4205:i
4198:e
4194:=
4191:i
4177:i
4160:i
4156:n
4151:n
4127:.
4122:)
4114:n
4108:2
4105:1
4097:(
4088:i
4084:+
4079:)
4071:n
4065:2
4062:1
4054:(
4044:=
4039:)
4034:i
4028:n
4022:2
4019:1
4011:(
4001:=
3996:n
3990:)
3984:i
3975:2
3972:1
3964:(
3954:=
3949:n
3945:i
3932:n
3913:i
3897:.
3894:i
3888:=
3883:3
3880:+
3877:n
3874:4
3870:i
3865:,
3862:1
3856:=
3851:2
3848:+
3845:n
3842:4
3838:i
3833:,
3830:i
3827:=
3822:1
3819:+
3816:n
3813:4
3809:i
3804:,
3801:1
3798:=
3793:n
3790:4
3786:i
3773:n
3769:i
3748:.
3745:i
3742:a
3736:b
3733:=
3730:)
3727:i
3724:b
3721:+
3718:a
3715:(
3712:i
3705:,
3702:i
3699:a
3696:+
3693:b
3687:=
3684:)
3681:i
3678:b
3675:+
3672:a
3669:(
3666:i
3642:.
3637:i
3634:)
3631:2
3627:/
3614:(
3610:e
3606:r
3603:=
3598:i
3591:e
3587:r
3583:i
3576:,
3571:i
3568:)
3565:2
3561:/
3554:+
3548:(
3544:e
3540:r
3537:=
3532:i
3525:e
3521:r
3517:i
3504:i
3502:β
3472:2
3469:1
3458:(
3454:i
3429:.
3426:i
3423:)
3420:c
3417:b
3414:+
3411:d
3408:a
3405:(
3402:+
3399:)
3396:d
3393:b
3387:c
3384:a
3381:(
3378:=
3371:)
3368:i
3365:d
3362:+
3359:c
3356:(
3353:)
3350:i
3347:b
3344:+
3341:a
3338:(
3331:,
3328:i
3325:)
3322:d
3319:+
3316:b
3313:(
3310:+
3307:)
3304:c
3301:+
3298:a
3295:(
3292:=
3285:)
3282:i
3279:d
3276:+
3273:c
3270:(
3267:+
3264:)
3261:i
3258:b
3255:+
3252:a
3249:(
3225:i
3221:1
3170:.
3167:i
3164:)
3161:b
3158:+
3155:a
3152:(
3149:=
3146:i
3143:b
3140:+
3137:i
3134:a
3074:7
3047:7
3042:i
3032:y
3028:x
3021:y
3017:x
3001:y
2997:/
2993:x
2988:=
2979:y
2976:t
2969:y
2961:/
2950:y
2947:t
2940:x
2912:y
2909:t
2902:y
2896:x
2891:=
2882:y
2879:t
2872:y
2857:y
2854:t
2847:x
2817:1
2814:=
2809:1
2804:=
2797:)
2794:1
2788:(
2782:)
2779:1
2773:(
2761:y
2758:c
2755:a
2752:l
2749:l
2746:a
2743:f
2737:=
2728:1
2715:1
2707:=
2704:i
2698:i
2695:=
2692:1
2676:x
2655:x
2629:,
2624:1
2601:i
2577:u
2567:v
2565:/
2563:u
2559:J
2549:u
2547:/
2545:u
2502:.
2499:b
2496:+
2493:z
2490:a
2484:z
2466:0
2462:i
2458:1
2416:x
2392:.
2386:1
2383:+
2378:2
2374:x
2366:/
2362:]
2359:x
2356:[
2352:R
2324:1
2321:+
2316:2
2312:x
2287:1
2284:+
2279:2
2275:x
2247:,
2244:]
2241:x
2238:[
2234:R
2207:x
2194:i
2186:1
2182:J
2177:I
2175:β
2147:.
2142:)
2136:a
2131:b
2124:b
2116:a
2110:(
2105:=
2102:J
2099:b
2096:+
2093:I
2090:a
2073:a
2056:.
2051:)
2045:0
2040:1
2033:1
2025:0
2019:(
2014:=
2011:J
2007:,
2002:)
1996:1
1991:0
1984:0
1979:1
1973:(
1968:=
1965:I
1953:J
1949:I
1939:i
1935:1
1928:,
1912:a
1907:.
1905:I
1901:J
1896:,
1894:J
1882:,
1880:I
1876:I
1871:J
1867:I
1863:i
1859:1
1820:C
1781:i
1779:β
1774:i
1772:+
1764:i
1762:β
1757:i
1755:+
1751:y
1743:x
1725:,
1719:2
1699:i
1697:β
1680:2
1671:+
1656:i
1654:+
1649:i
1647:β
1642:i
1640:+
1632:i
1630:β
1626:i
1621:i
1619:+
1600:x
1583:i
1581:β
1576:i
1560:i
1547:Ο
1545:2
1525:2
1500:e
1494:e
1488:i
1479:i
1469:i
1461:i
1455:i
1453:β
1445:i
1441:1
1423:,
1420:i
1411:=
1403:i
1400:)
1397:1
1394:(
1387:=
1379:i
1374:4
1370:i
1366:=
1357:5
1353:i
1345:,
1342:1
1335:=
1327:i
1324:)
1321:i
1315:(
1310:=
1302:i
1297:3
1293:i
1289:=
1280:4
1276:i
1268:,
1265:i
1259:=
1251:i
1248:)
1245:1
1239:(
1236:=
1228:i
1223:2
1219:i
1215:=
1206:3
1202:i
1187:i
1178:i
1173:i
1164:i
1162:β
1158:i
1150:i
1128:=
1123:2
1119:i
1108:i
1060:1
1054:i
1048:=
1040:7
1036:i
1003:i
997:1
991:=
983:6
979:i
946:1
940:i
928:=
920:5
916:i
883:i
877:1
865:=
857:4
853:i
820:1
814:i
808:=
800:3
796:i
763:i
757:1
751:=
743:2
739:i
706:1
700:i
688:=
680:1
676:i
643:i
637:1
625:=
617:0
613:i
580:1
574:i
568:=
563:1
556:i
526:i
520:1
514:=
509:2
502:i
472:1
466:i
454:=
449:3
442:i
412:i
406:1
394:=
389:4
382:i
337:i
322:1
320:(
295:i
245:i
241:i
237:j
225:j
221:i
201:i
199:β
195:i
151:,
147:C
125:R
109:.
107:i
101:i
85:i
75:x
66:i
63:(
45:i
34:.
20:)
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