1890:"Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires."
1892:(Moreover, the true roots as well as the false are not always real; but sometimes only imaginary ; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this , x – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary .)
844:
768:
1241:
2928:
1011:
1236:{\displaystyle \textstyle {\sqrt {x\cdot y{\vphantom {t}}}}={\sqrt {(-x)\cdot (-y)}}\mathrel {\stackrel {\text{ (fallacy) }}{=}} {\sqrt {-x{\vphantom {ty}}}}\cdot {\sqrt {-y{\vphantom {ty}}}}=i{\sqrt {x{\vphantom {ty}}}}\cdot i{\sqrt {y{\vphantom {ty}}}}=-{\sqrt {x\cdot y{\vphantom {ty}}}}\,.}
874:-axis can be drawn with "positive" direction going up; "positive" imaginary numbers then increase in magnitude upwards, and "negative" imaginary numbers increase in magnitude downwards. This vertical axis is often called the "imaginary axis" and is denoted
798:. At the time, imaginary numbers and negative numbers were poorly understood and were regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including
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a real number, both causes a counterclockwise rotation about the origin by 90 degrees and scales the answer by a factor of
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948:
corresponds to a clockwise rotation of 90 degrees about the origin. Similarly, multiplying by a purely imaginary number
2400:
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as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of
2037:
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38:
866:
positively increasing in magnitude to the right and negatively increasing in magnitude to the left. At 0 on the
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But the result is clearly nonsense. The step where the square root was broken apart was illegitimate. (See
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63:
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2712:
2440:
2435:
2033:
Quaternions and
Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality
1599:
703:
820:(1777–1855). The geometric significance of complex numbers as points in a plane was first described by
1005:
are both positive real numbers, the following chain of equalities appears reasonable at first glance:
812:
and meant it to be derogatory. The use of imaginary numbers was not widely accepted until the work of
786:
is noted as the first to present a calculation involving the square root of a negative number, it was
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An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.
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in which three of the dimensions are analogous to the imaginary numbers in the complex field.
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extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of
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of 90 degrees about the origin, which is a quarter of a circle. Multiplication by
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A History of Non-Euclidean
Geometry: Evolution of the Concept of a Geometric Space
1917:, discusses ambiguities of meaning in imaginary expressions in historical context.
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to the real axis. One way of viewing imaginary numbers is to consider a standard
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Care must be used when working with imaginary numbers that are expressed as the
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1770:(illustrated ed.). Springer Science & Business Media. p. 121.
2070:
An
Imaginary Tale: The Story of "i" [the square root of minus one]
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938:
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Geometrically, imaginary numbers are found on the vertical axis of the
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in 1572. The concept had appeared in print earlier, such as in work by
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62:"Imaginary Numbers" redirects here. For the 2013 EP by The Maine, see
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2156:
1851:
Complex
Numbers: lattice simulation and zeta function applications
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2134:– an article that discusses the existence of imaginary numbers.
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1794:
Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009).
2412:
1904:
Negative Math: How
Mathematical Rules Can Be Positively Bent
1767:
Mathematical
Analysis: Approximation and Discrete Processes
1881:(Leiden, (Netherlands): Jan Maire, 1637), appended book:
688:
573:{\displaystyle \ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}}
510:{\displaystyle \ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}}
333:{\displaystyle \ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}}
270:{\displaystyle \ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}}
2132:
How can one show that imaginary numbers really do exist?
1993:
Webb, Stephen (2018). "5. Meaningless marks on paper".
1743:(Second ed.). Rastogi Publications. p. 11.2.
42:
2121:, explains many applications of imaginary expressions.
1995:
Clash of
Symbols – A Ride Through the Riches of Glyphs
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An
Imaginary Tale: the Story of the Square Root of −1
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790:who first set down the rules for multiplication of
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1959:Electric Power Systems – A Conceptual Introduction
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2153:Basic Explanation and Uses of Imaginary Numbers
1662:has other meanings (such as electrical current)
933:In this representation, multiplication by
1800:(6th ed.). Cengage Learning. p. 66.
1658:is usually used in engineering contexts where
2245:
2172:
1827:(2 ed.). World Scientific. p. 153.
1764:Giaquinta, Mariano; Modica, Giuseppe (2004).
691:is considered to be both real and imaginary.
8:
447:{\displaystyle \ \ i^{3}\ =-i{\phantom {1}}}
390:{\displaystyle \ \ i^{2}\ =-1{\phantom {i}}}
26:
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2073:. Princeton University Press. p. 12.
1691:. Cambridge University Press. p. 38.
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683:is an imaginary number, and its square is
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2109:. Princeton: Princeton University Press.
1906:, Princeton: Princeton University Press,
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207:{\displaystyle \ i^{-1}=-i{\phantom {1}}}
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153:{\displaystyle \ i^{-2}=-1{\phantom {i}}}
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59:Square root of a non-positive real number
70:
32:This is an accepted version of this page
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1648:
28:
1689:Fundamentals of Waves and Oscillations
7:
1927:Rozenfeld, Boris Abramovich (1988).
858:, which allows them to be presented
2441:Set-theoretically definable numbers
1740:A Text Book of Mathematics Class XI
654:, which is defined by its property
937:corresponds to a counterclockwise
57:
1797:College Algebra: Enhanced Edition
2926:
981:Square roots of negative numbers
1999:Springer Science+Business Media
1454:{\displaystyle :\;\mathbb {N} }
1416:{\displaystyle :\;\mathbb {Z} }
1378:{\displaystyle :\;\mathbb {Q} }
1340:{\displaystyle :\;\mathbb {R} }
1302:{\displaystyle :\;\mathbb {C} }
2420:{\displaystyle {\mathcal {P}}}
1848:Roy, Stephen Campbell (2007).
1069:
1060:
1054:
1045:
895:{\displaystyle i\mathbb {R} ,}
802:, who wrote about them in his
745:are called, respectively, the
719:can be added to a real number
1:
2775:Plane-based geometric algebra
1956:von Meier, Alexandra (2006).
919:{\displaystyle \mathbb {I} ,}
710:(in the early 19th century).
2735:{\displaystyle \mathbb {S} }
2658:{\displaystyle \mathbb {C} }
2619:{\displaystyle \mathbb {R} }
2581:{\displaystyle \mathbb {O} }
2553:{\displaystyle \mathbb {H} }
2525:{\displaystyle \mathbb {C} }
2497:{\displaystyle \mathbb {R} }
2385:{\displaystyle \mathbb {A} }
2352:{\displaystyle \mathbb {Q} }
2324:{\displaystyle \mathbb {Z} }
2296:{\displaystyle \mathbb {N} }
1902:Martinez, Albert A. (2006),
1214:
1181:
1154:
1127:
1100:
1029:
808:in which he coined the term
2007:10.1007/978-3-319-71350-2_5
847:90-degree rotations in the
2969:
2144:Why Use Imaginary Numbers?
2038:Princeton University Press
1821:Hargittai, István (1992).
763:History of complex numbers
760:
702:(in the 18th century) and
61:
2917:
2765:Algebra of physical space
2194:
1935:. Springer. p. 382.
737:, where the real numbers
599:{\displaystyle \ \vdots }
99:{\displaystyle \ \vdots }
2821:Extended complex numbers
2804:Extended natural numbers
839:Geometric interpretation
39:latest accepted revision
2067:Nahin, Paul J. (2010).
2030:Kuipers, J. B. (1999).
1683:Uno Ingard, K. (1988).
753:of the complex number.
665:of an imaginary number
2877:Transcendental numbers
2736:
2713:Hyperbolic quaternions
2659:
2620:
2582:
2554:
2526:
2498:
2421:
2386:
2353:
2325:
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1885:, book three, p. 380.
1879:Discours de la méthode
1854:. Horwood. p. 1.
1541:Dyadic (finite binary)
1455:
1417:
1379:
1341:
1303:
1237:
920:
896:
851:
833:quaternion imaginaries
829:William Rowan Hamilton
772:
623:
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574:
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448:
391:
334:
271:
208:
154:
100:
64:Imaginary Numbers (EP)
2809:Extended real numbers
2737:
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2630:Split-complex numbers
2621:
2583:
2555:
2527:
2499:
2422:
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2363:Constructible numbers
2354:
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2140:BBC Radio 4 programme
1964:John Wiley & Sons
1718:mathworld.wolfram.com
1456:
1418:
1380:
1342:
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1084: (fallacy)
921:
897:
846:
770:
704:Augustin-Louis Cauchy
624:
601:
575:
512:
449:
392:
335:
272:
209:
155:
101:
2841:Supernatural numbers
2751:Multicomplex numbers
2724:
2708:Dual-complex numbers
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2608:
2570:
2542:
2514:
2486:
2468:Composition algebras
2436:Arithmetical numbers
2407:
2374:
2341:
2313:
2285:
2138:5Numbers programme 4
2101:Nahin, Paul (1998).
2001:. pp. 204–205.
1737:Sinha, K.C. (2008).
1586:Algebraic irrational
1439:
1401:
1363:
1325:
1287:
1248:Mathematical fallacy
1222:
1189:
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856:complex number plane
818:Carl Friedrich Gauss
713:An imaginary number
708:Carl Friedrich Gauss
643:is the product of a
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221:
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147:
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2746:Split-biquaternions
2458:Eisenstein integers
2396:Closed-form numbers
2219:Unit complex number
1783:Extract of page 121
1712:Weisstein, Eric W.
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775:Although the Greek
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29:Page version status
2904:Profinite integers
2867:Irrational numbers
2732:
2655:
2616:
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2451:Gaussian rationals
2431:Computable numbers
2417:
2382:
2349:
2321:
2293:
2149:2019-08-25 at the
2086:Extract of page 12
2040:. pp. 10–11.
1966:. pp. 61–62.
1714:"Imaginary Number"
1451:
1413:
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1273:
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997:. For example, if
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267:
204:
150:
96:
35:
2940:
2939:
2851:Superreal numbers
2831:Levi-Civita field
2826:Hyperreal numbers
2770:Spacetime algebra
2756:Geometric algebra
2669:Bicomplex numbers
2635:Split-quaternions
2476:Division algebras
2446:Gaussian integers
2368:Algebraic numbers
2271:definable numbers
2227:
2226:
2199:Complex conjugate
2080:978-1-4008-3029-9
2016:978-3-319-71350-2
1861:978-1-904275-25-1
1824:Fivefold Symmetry
1807:978-1-4390-4379-0
1777:978-0-8176-4337-9
1750:978-81-7133-912-9
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622:{\displaystyle i}
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18:Imaginary Numbers
16:(Redirected from
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2760:Clifford algebra
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796:Gerolamo Cardano
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641:imaginary number
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388:
386:
385:
366:
365:
364:
353:
350:
339:
337:
336:
331:
329:
328:
317:
316:
303:
302:
301:
290:
287:
276:
274:
273:
268:
266:
265:
254:
253:
240:
239:
238:
227:
224:
213:
211:
210:
205:
203:
202:
185:
184:
170:
159:
157:
156:
151:
149:
148:
131:
130:
116:
105:
103:
102:
97:
90:
77:
71:
47:3 September 2024
21:
2968:
2967:
2963:
2962:
2961:
2959:
2958:
2957:
2953:Complex numbers
2943:
2942:
2941:
2936:
2913:
2892:
2882:
2855:
2846:Surreal numbers
2836:Ordinal numbers
2781:
2722:
2721:
2683:
2645:
2644:
2642:
2640:Split-octonions
2606:
2605:
2597:
2591:
2568:
2567:
2540:
2539:
2512:
2511:
2508:Complex numbers
2484:
2483:
2462:
2405:
2404:
2372:
2371:
2339:
2338:
2311:
2310:
2283:
2282:
2279:Natural numbers
2264:
2258:
2228:
2223:
2190:
2188:Complex numbers
2185:
2151:Wayback Machine
2128:
2117:
2100:
2097:
2092:
2091:
2081:
2066:
2065:
2061:
2052:
2050:
2048:
2029:
2028:
2024:
2017:
1992:
1991:
1987:
1978:
1976:
1974:
1955:
1954:
1950:
1943:
1926:
1925:
1921:
1914:
1901:
1900:
1896:
1875:Descartes, René
1873:
1869:
1862:
1847:
1846:
1842:
1835:
1820:
1819:
1815:
1808:
1793:
1792:
1788:
1778:
1763:
1762:
1758:
1751:
1736:
1735:
1731:
1722:
1720:
1711:
1710:
1706:
1699:
1682:
1681:
1677:
1672:
1667:
1666:
1659:
1655:
1654:
1650:
1645:
1437:
1436:
1399:
1398:
1361:
1360:
1323:
1322:
1285:
1284:
1256:
1010:
1009:
1002:
998:
983:
970:
968:
961:
957:
953:
949:
942:
934:
927:
903:
902:
876:
875:
871:
867:
841:
792:complex numbers
788:Rafael Bombelli
765:
759:
742:
738:
728:
720:
714:
684:
677:
676:. For example,
670:
666:
655:
651:
630:
611:
610:
585:
584:
533:
522:
521:
470:
459:
458:
413:
402:
401:
356:
345:
344:
293:
282:
281:
230:
219:
218:
173:
165:
164:
119:
111:
110:
85:
84:
78:
75:
67:
60:
55:
54:
53:
52:
51:
50:
34:
22:
15:
12:
11:
5:
2966:
2964:
2956:
2955:
2945:
2944:
2938:
2937:
2935:
2934:
2924:
2922:Classification
2918:
2915:
2914:
2912:
2911:
2909:Normal numbers
2906:
2901:
2879:
2874:
2869:
2863:
2861:
2857:
2856:
2854:
2853:
2848:
2843:
2838:
2833:
2828:
2823:
2818:
2817:
2816:
2806:
2801:
2795:
2793:
2791:infinitesimals
2783:
2782:
2780:
2779:
2778:
2777:
2772:
2767:
2753:
2748:
2743:
2730:
2715:
2710:
2705:
2700:
2694:
2692:
2685:
2684:
2682:
2681:
2676:
2671:
2666:
2653:
2637:
2632:
2627:
2614:
2601:
2599:
2593:
2592:
2590:
2589:
2576:
2561:
2548:
2533:
2520:
2505:
2492:
2472:
2470:
2464:
2463:
2461:
2460:
2455:
2454:
2453:
2443:
2438:
2433:
2428:
2414:
2398:
2393:
2380:
2365:
2360:
2347:
2332:
2319:
2304:
2291:
2275:
2273:
2266:
2265:
2259:
2257:
2256:
2249:
2242:
2234:
2225:
2224:
2222:
2221:
2216:
2211:
2206:
2201:
2195:
2192:
2191:
2186:
2184:
2183:
2176:
2169:
2161:
2155:
2154:
2141:
2135:
2127:
2126:External links
2124:
2123:
2122:
2115:
2096:
2093:
2090:
2089:
2079:
2059:
2046:
2022:
2015:
1985:
1972:
1948:
1941:
1919:
1912:
1894:
1887:From page 380:
1867:
1860:
1840:
1833:
1813:
1806:
1786:
1776:
1756:
1749:
1729:
1704:
1697:
1674:
1673:
1671:
1668:
1665:
1664:
1647:
1646:
1644:
1641:
1638:
1637:
1634:
1633:
1630:
1629:
1626:
1625:
1619:
1618:
1615:
1614:
1611:
1610:
1607:
1606:
1603:
1602:
1600:Transcendental
1596:
1595:
1589:
1588:
1579:
1569:
1568:
1565:
1564:
1561:
1560:
1557:
1556:
1553:
1552:
1550:
1544:
1543:
1537:
1536:
1534:Finite decimal
1527:
1517:
1516:
1513:
1512:
1509:
1508:
1502:
1501:
1498:
1497:
1494:
1493:
1487:
1486:
1480:
1479:
1472:
1471:
1461:
1449:
1444:
1423:
1411:
1406:
1385:
1373:
1368:
1347:
1335:
1330:
1309:
1297:
1292:
1275:Number systems
1263:
1262:
1255:
1252:
1244:
1243:
1231:
1221:
1218:
1211:
1208:
1205:
1200:
1197:
1188:
1185:
1178:
1173:
1170:
1161:
1158:
1151:
1146:
1143:
1134:
1131:
1124:
1121:
1116:
1107:
1104:
1097:
1094:
1080:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1042:
1033:
1026:
1023:
1020:
982:
979:
915:
911:
891:
887:
883:
840:
837:
814:Leonhard Euler
800:René Descartes
761:Main article:
758:
755:
751:imaginary part
725:complex number
700:Leonhard Euler
696:René Descartes
649:imaginary unit
635:
634:
618:
607:
606:
595:
581:
580:
566:
560:
554:
548:
540:
536:
518:
517:
503:
497:
491:
485:
477:
473:
455:
454:
440:
434:
431:
428:
420:
416:
398:
397:
383:
377:
374:
371:
363:
359:
341:
340:
326:
320:
314:
308:
300:
296:
278:
277:
263:
257:
251:
245:
237:
233:
215:
214:
200:
194:
191:
188:
183:
180:
176:
161:
160:
146:
140:
137:
134:
129:
126:
122:
107:
106:
95:
81:
80:
74:The powers of
58:
56:
36:
30:
27:
25:
24:
23:
14:
13:
10:
9:
6:
4:
3:
2:
2965:
2954:
2951:
2950:
2948:
2933:
2925:
2923:
2920:
2919:
2916:
2910:
2907:
2905:
2902:
2899:
2895:
2889:
2885:
2880:
2878:
2875:
2873:
2872:Fuzzy numbers
2870:
2868:
2865:
2864:
2862:
2858:
2852:
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2815:
2812:
2811:
2810:
2807:
2805:
2802:
2800:
2797:
2796:
2794:
2792:
2788:
2784:
2776:
2773:
2771:
2768:
2766:
2763:
2762:
2761:
2757:
2754:
2752:
2749:
2747:
2744:
2719:
2716:
2714:
2711:
2709:
2706:
2704:
2701:
2699:
2696:
2695:
2693:
2691:
2686:
2680:
2677:
2675:
2674:Biquaternions
2672:
2670:
2667:
2641:
2638:
2636:
2633:
2631:
2628:
2603:
2602:
2600:
2594:
2565:
2562:
2537:
2534:
2509:
2506:
2481:
2477:
2474:
2473:
2471:
2469:
2465:
2459:
2456:
2452:
2449:
2448:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2402:
2399:
2397:
2394:
2369:
2366:
2364:
2361:
2336:
2333:
2308:
2305:
2280:
2277:
2276:
2274:
2272:
2267:
2262:
2255:
2250:
2248:
2243:
2241:
2236:
2235:
2232:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2204:Complex plane
2202:
2200:
2197:
2196:
2193:
2189:
2182:
2177:
2175:
2170:
2168:
2163:
2162:
2159:
2152:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2129:
2125:
2118:
2116:0-691-02795-1
2112:
2107:
2106:
2099:
2098:
2094:
2087:
2082:
2076:
2072:
2071:
2063:
2060:
2049:
2047:0-691-10298-8
2043:
2039:
2035:
2034:
2026:
2023:
2018:
2012:
2008:
2004:
2000:
1996:
1989:
1986:
1975:
1973:0-471-17859-4
1969:
1965:
1961:
1960:
1952:
1949:
1944:
1942:0-387-96458-4
1938:
1934:
1930:
1923:
1920:
1915:
1913:0-691-12309-8
1909:
1905:
1898:
1895:
1891:
1888:
1884:
1880:
1876:
1871:
1868:
1863:
1857:
1853:
1852:
1844:
1841:
1836:
1834:981-02-0600-3
1830:
1826:
1825:
1817:
1814:
1809:
1803:
1799:
1798:
1790:
1787:
1784:
1779:
1773:
1769:
1768:
1760:
1757:
1752:
1746:
1742:
1741:
1733:
1730:
1719:
1715:
1708:
1705:
1700:
1698:0-521-33957-X
1694:
1690:
1686:
1679:
1676:
1669:
1652:
1649:
1642:
1624:
1621:
1620:
1601:
1598:
1597:
1594:
1591:
1590:
1587:
1584:
1583:
1580:
1578:
1575:
1574:
1571:
1570:
1551:
1549:
1546:
1545:
1542:
1539:
1538:
1535:
1532:
1531:
1528:
1526:
1523:
1522:
1519:
1518:
1507:
1504:
1503:
1492:
1489:
1488:
1485:
1484:Prime numbers
1482:
1481:
1477:
1474:
1473:
1469:
1466:
1465:
1462:
1442:
1435:
1432:
1431:
1428:
1427:
1424:
1404:
1397:
1394:
1393:
1390:
1389:
1386:
1366:
1359:
1356:
1355:
1352:
1351:
1348:
1328:
1321:
1318:
1317:
1314:
1313:
1310:
1290:
1283:
1280:
1279:
1276:
1271:
1270:
1267:
1266:
1261:
1258:
1257:
1253:
1251:
1249:
1229:
1219:
1216:
1209:
1206:
1203:
1198:
1195:
1186:
1183:
1176:
1171:
1168:
1159:
1156:
1149:
1144:
1141:
1132:
1129:
1122:
1119:
1114:
1105:
1102:
1095:
1092:
1078:
1066:
1063:
1057:
1051:
1048:
1040:
1031:
1024:
1021:
1018:
1008:
1007:
1006:
996:
992:
988:
980:
978:
973:
964:
946:
940:
931:
913:
889:
881:
865:
861:
860:perpendicular
857:
850:
849:complex plane
845:
838:
836:
834:
830:
825:
824:(1745–1818).
823:
822:Caspar Wessel
819:
815:
811:
807:
806:
801:
797:
793:
789:
785:
782:
778:
777:mathematician
769:
764:
756:
754:
752:
748:
735:
731:
726:
717:
711:
709:
705:
701:
697:
692:
690:
687:. The number
681:
674:
664:
658:
650:
646:
642:
633:
632:root of unity
616:
609:
608:
593:
583:
582:
564:
558:
552:
546:
538:
534:
520:
519:
501:
495:
489:
483:
475:
471:
457:
456:
438:
432:
429:
426:
418:
414:
400:
399:
381:
375:
372:
369:
361:
357:
343:
342:
324:
318:
312:
306:
298:
294:
280:
279:
261:
255:
249:
243:
235:
231:
217:
216:
198:
192:
189:
186:
181:
178:
174:
163:
162:
144:
138:
135:
132:
127:
124:
120:
109:
108:
93:
83:
82:
73:
72:
69:
65:
48:
44:
40:
33:
19:
2893:
2883:
2698:Dual numbers
2690:hypercomplex
2480:Real numbers
2208:
2104:
2095:Bibliography
2069:
2062:
2051:. Retrieved
2032:
2025:
1994:
1988:
1977:. Retrieved
1958:
1951:
1932:
1929:"Chapter 10"
1922:
1903:
1897:
1889:
1883:La Géométrie
1882:
1878:
1870:
1850:
1843:
1823:
1816:
1796:
1789:
1766:
1759:
1739:
1732:
1721:. Retrieved
1717:
1707:
1688:
1678:
1651:
1622:
1245:
991:square roots
984:
971:
962:
944:
932:
853:
826:
809:
805:La Géométrie
803:
774:
750:
746:
733:
729:
727:of the form
715:
712:
693:
679:
672:
656:
640:
638:
79:are cyclic:
68:
46:
37:This is the
31:
2860:Other types
2679:Bioctonions
2536:Quaternions
2214:Real number
1685:"Chapter 2"
864:number line
645:real number
2814:Projective
2787:Infinities
2053:2022-01-13
1979:2022-01-13
1723:2020-08-10
1670:References
1577:Irrational
723:to form a
2898:solenoids
2718:Sedenions
2564:Octonions
1623:Imaginary
1207:⋅
1199:−
1169:⋅
1120:−
1115:⋅
1093:−
1064:−
1058:⋅
1049:−
1022:⋅
870:-axis, a
827:In 1843,
810:imaginary
747:real part
594:⋮
553:−
490:−
430:−
373:−
313:−
250:−
190:−
179:−
136:−
125:−
94:⋮
2947:Category
2307:Integers
2269:Sets of
2147:Archived
1525:Fraction
1358:Rational
1254:See also
960:. When
939:rotation
781:engineer
749:and the
647:and the
629:is a 4th
43:reviewed
2888:numbers
2720: (
2566: (
2538: (
2510: (
2482: (
2403: (
2401:Periods
2370: (
2337: (
2309: (
2281: (
2263:systems
1434:Natural
1396:Integer
1282:Complex
989:of the
952:, with
757:History
2688:Other
2261:Number
2113:
2077:
2044:
2013:
1970:
1939:
1910:
1858:
1831:
1804:
1774:
1747:
1695:
975:|
969:|
965:< 0
663:square
661:. The
591:
544:
531:
528:
481:
468:
465:
424:
411:
408:
367:
354:
351:
304:
291:
288:
241:
228:
225:
171:
117:
91:
2896:-adic
2886:-adic
2643:Over
2604:Over
2598:types
2596:Split
1643:Notes
2932:List
2789:and
2111:ISBN
2075:ISBN
2042:ISBN
2011:ISBN
1968:ISBN
1937:ISBN
1908:ISBN
1856:ISBN
1829:ISBN
1802:ISBN
1772:ISBN
1745:ISBN
1693:ISBN
1478:: 1
1470:: 0
1468:Zero
1320:Real
1001:and
779:and
741:and
706:and
689:zero
659:= −1
2003:doi
1476:One
1250:.)
993:of
926:or
685:−25
669:is
639:An
45:on
2949::
2478::
2036:.
2009:.
1997:.
1962:.
1931:.
1877:,
1716:.
1687:.
1260:−1
977:.
950:bi
930:.
734:bi
732:+
716:bi
667:bi
41:,
2900:)
2894:p
2890:(
2884:p
2758:/
2742:)
2729:S
2665::
2652:C
2626::
2613:R
2588:)
2575:O
2560:)
2547:H
2532:)
2519:C
2504:)
2491:R
2427:)
2413:P
2392:)
2379:A
2359:)
2346:Q
2331:)
2318:Z
2303:)
2290:N
2253:e
2246:t
2239:v
2180:e
2173:t
2166:v
2119:.
2083:.
2056:.
2019:.
2005::
1982:.
1945:.
1864:.
1837:.
1810:.
1780:.
1753:.
1726:.
1701:.
1660:i
1656:j
1448:N
1443::
1410:Z
1405::
1372:Q
1367::
1334:R
1329::
1296:C
1291::
1230:.
1220:y
1217:t
1210:y
1204:x
1196:=
1187:y
1184:t
1177:y
1172:i
1160:y
1157:t
1150:x
1145:i
1142:=
1133:y
1130:t
1123:y
1106:y
1103:t
1096:x
1079:=
1070:)
1067:y
1061:(
1055:)
1052:x
1046:(
1041:=
1032:t
1025:y
1019:x
1003:y
999:x
972:b
963:b
958:b
954:b
945:i
943:−
935:i
928:ℑ
914:,
910:I
890:,
886:R
882:i
872:y
868:x
743:b
739:a
730:a
721:a
680:i
678:5
673:b
671:−
657:i
652:i
617:i
565:1
559:i
547:=
539:5
535:i
502:i
496:1
484:=
476:4
472:i
439:1
433:i
427:=
419:3
415:i
382:i
376:1
370:=
362:2
358:i
325:1
319:i
307:=
299:1
295:i
262:i
256:1
244:=
236:0
232:i
199:1
193:i
187:=
182:1
175:i
145:i
139:1
133:=
128:2
121:i
76:i
66:.
49:.
20:)
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