7532:
38:
3505:
5448:
8220:
5467:, such that any grid point corresponds to a rational number. This method, however, exhibits a form of redundancy, as several different grid points will correspond to the same rational number; these are highlighted in red on the provided graphic. An obvious example can be seen in the line going diagonally towards the bottom right; such ratios will always equal 1, as any
4971:
4763:
5043:
other than the identity. (A field automorphism must fix 0 and 1; as it must fix the sum and the difference of two fixed elements, it must fix every integer; as it must fix the quotient of two fixed elements, it must fix every rational number, and is thus the identity.)
3033:
4592:
1473:
On the other hand, if either denominator is negative, then each fraction with a negative denominator must first be converted into an equivalent form with a positive denominator—by changing the signs of both its numerator and denominator.
3896:
4030:
2209:
6066:
4134:
3734:
677:
4966:{\displaystyle {\begin{aligned}&(n_{1}n_{2}>0\quad {\text{and}}\quad m_{1}n_{2}\leq n_{1}m_{2})\\&\qquad {\text{or}}\\&(n_{1}n_{2}<0\quad {\text{and}}\quad m_{1}n_{2}\geq n_{1}m_{2}).\end{aligned}}}
2684:
5426:
2506:
2419:
1740:
3581:
1652:
1558:
4752:
2002:
4367:
1858:
6355:
4768:
2596:
3410:
3315:
5322:
1433:
1330:
1252:
2836:
2765:
4436:
4293:
6224:
6143:
332:
3458:
2115:
6266:
3223:
4673:
4189:
3182:
2543:
2349:
1202:
1135:
1073:
6556:
6518:
6480:
6442:
6404:
5965:
5873:
4470:
3141:
3074:
2863:
2718:
2309:
2276:
2243:
2069:
2036:
1928:
1892:
1786:
711:
286:
235:
5485:
As the set of all rational numbers is countable, and the set of all real numbers (as well as the set of irrational numbers) is uncountable, the set of rational numbers is a
6177:
5640:
5613:
5266:
5233:
5170:
5105:
386:
8032:
7955:
7916:
7878:
7850:
7822:
7794:
7707:
7674:
7646:
7618:
6299:
5800:
5756:
5697:
5197:
5133:
5068:
5035:
5003:
4061:
883:
810:
776:
179:
150:
121:
92:
63:
5459:, as is illustrated in the figure to the right. As a rational number can be expressed as a ratio of two integers, it is possible to assign two integers to any point on a
4480:
480:
5933:
1468:
3491:
6339:
3258:
1284:
7187:
5351:
1382:
1356:
3355:
7043:
7352:
3745:
8256:
3902:
921:
may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between "
7568:
2141:
5281:
set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that
5975:
4074:
3610:
553:
2614:
7255:
7225:
6917:
6815:
6785:
6758:
5359:
2439:
5141:
that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield
2358:
1679:
5513:; every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with
3532:
1582:
1488:
4689:
7068:
1946:
7155:
6946:
3414:
4304:
1804:
4138:
equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any
8327:
846:
7426:
8317:
8105:
7345:
8249:
8337:
8066:
7318:
933:
is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a
8322:
7561:
4599:
2553:
3365:
3268:
7717:
7440:
7313:
5464:
5287:
1398:
4039:, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers
7338:
5664:
1295:
1217:
893:
of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective
8332:
8242:
8100:
8056:
830:
493:
31:
2785:
8223:
8095:
6274:
5077:
961:
2727:
7554:
5668:
4378:
4235:
1148:
742:, and is contained in any field containing the integers. In other words, the field of rational numbers is a
1005:"avoided heresy by forbidding themselves from thinking of those lengths as numbers". So such lengths were
8183:
7712:
6270:
6188:
6077:
2314:
1933:
1028:
992:, which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use of
781:
204:
5479:
294:
8296:
8168:
8004:
7732:
7727:
7387:
6934:
6689:
3420:
3028:{\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},}
2078:
822:
429:
7531:
6239:
5475:
5474:
It is possible to generate all of the rational numbers without such redundancies: examples include the
5007:
of all rational numbers, together with the addition and multiplication operations shown above, forms a
3198:
4646:
4162:
3155:
2516:
2322:
1175:
1108:
1046:
7921:
7679:
7382:
7308:
6535:
6497:
6459:
6421:
6383:
5940:
5822:
5656:
5076:, which is a field that has no subfield other than itself. The rationals are the smallest field with
4631:
4446:
3601:
3117:
3050:
2694:
2285:
2252:
2219:
2045:
2012:
1904:
1868:
1762:
1039:
914:
687:
544:
262:
211:
4587:{\displaystyle \cdots ={\frac {-2m}{-2n}}={\frac {-m}{-n}}={\frac {m}{n}}={\frac {2m}{2n}}=\cdots .}
8132:
8042:
7999:
7981:
7759:
6157:
5620:
5558:
5552:
5246:
5213:
5150:
5085:
5008:
4036:
3089:
747:
735:
521:
366:
8015:
7938:
7899:
7861:
7833:
7805:
7777:
7690:
7657:
7629:
7601:
6282:
5783:
5704:
5680:
5180:
5116:
5051:
5018:
4986:
4044:
940:
a curve defined over the rationals, but a curve which can be parameterized by rational functions.
866:
793:
759:
162:
133:
104:
75:
46:
8037:
7749:
7536:
7402:
7189:
Any two countable densely ordered sets without endpoints are isomorphic - a formal proof with KIV
6349:
5518:
5202:
5040:
4143:
3262:
2848:
1746:
461:
6344:
5895:
4683:
may be defined on the rational numbers, that extends the natural order of the integers. One has
1438:
8195:
8158:
8122:
8061:
8047:
7742:
7722:
7517:
7508:
7499:
7397:
7372:
7251:
7221:
7151:
7145:
6942:
6913:
6822:
6811:
6792:
6781:
6754:
6673:
6644:
6334:
5645:
5544:
5436:
5238:
3513:
3359:
3192:
3188:
1002:
926:
922:
785:
525:
448:
437:
409:
397:
337:
7262:
5778:
In addition to the absolute value metric mentioned above, there are other metrics which turn
8142:
8117:
8051:
7960:
7926:
7767:
7737:
7684:
7587:
7494:
7489:
7484:
7479:
7474:
7407:
7392:
7361:
7232:
6712:
6682:
6602:
6587:
5494:
5468:
5435:
set which is countable, dense (in the above sense), and has no least or greatest element is
5271:
1791:
1662:
1568:
1086:
815:
452:
3234:
1257:
8286:
8090:
7994:
7377:
6637:
6319:
5649:
5432:
5278:
4139:
3109:
1392:
If both denominators are positive (particularly if both fractions are in canonical form):
834:
751:
510:
401:
359:
155:
97:
5330:
3468:
1361:
1335:
3504:
1657:
If both fractions are in canonical form, the result is in canonical form if and only if
1563:
If both fractions are in canonical form, the result is in canonical form if and only if
8137:
8127:
8112:
7931:
7799:
7595:
6630:
6530:
6378:
6324:
5660:
5540:
5460:
3462:
1097:
934:
902:
731:
37:
3322:
829:
of the real numbers. The real numbers can be constructed from the rational numbers by
8311:
8200:
8173:
8082:
7450:
7431:
7421:
6834:
6371:
6304:
6229:
5770:
5138:
3891:{\displaystyle (m_{1},n_{1})+(m_{2},n_{2})\equiv (m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}),}
3147:
506:
17:
6859:
8291:
8281:
8163:
7965:
7460:
7455:
7412:
7325:
6580:
6149:
5812:
5672:
5548:
5510:
4066:
3517:
984:
for qualifying numbers appeared almost a century earlier, in 1570. This meaning of
838:
498:
68:
7275:
5080:
zero. Every field of characteristic zero contains a unique subfield isomorphic to
7245:
7215:
6805:
6775:
3508:
A diagram showing a representation of the equivalent classes of pairs of integers
7989:
7771:
7250:(2nd, illustrated ed.). Springer Science & Business Media. p. 26.
6416:
6329:
5447:
5142:
5073:
4680:
4025:{\displaystyle (m_{1},n_{1})\times (m_{2},n_{2})\equiv (m_{1}m_{2},n_{1}n_{2}).}
3228:
3079:
906:
898:
743:
444:
433:
393:
251:
188:
7970:
7827:
7069:"How a Mathematical Superstition Stultified Algebra for Over a Thousand Years"
7039:
5533:
5514:
5490:
4637:
The integers may be considered to be rational numbers identifying the integer
4229:
belong to the same equivalence class (that is are equivalent) if and only if
2204:{\displaystyle {\frac {\,{\dfrac {a}{b}}\,}{\dfrac {c}{d}}}={\frac {ad}{bc}}.}
930:
514:
980:
with its modern meaning was attested in
English about 1660, while the use of
5653:
5506:
5456:
826:
244:
6061:{\displaystyle \left|{\frac {a}{b}}\right|_{p}={\frac {|a|_{p}}{|b|_{p}}}.}
7120:
4129:{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \backslash \{0\}))/\sim ,}
3729:{\displaystyle (m_{1},n_{1})\sim (m_{2},n_{2})\iff m_{1}n_{2}=m_{2}n_{1}.}
2040:
is in canonical form, then the canonical form of its reciprocal is either
672:{\displaystyle (p_{1},q_{1})\sim (p_{2},q_{2})\iff p_{1}q_{2}=p_{2}q_{1}.}
8078:
8009:
7855:
6621:
5529:
5525:
5486:
727:
413:
200:
126:
5648:. The rational numbers are an important example of a space which is not
5543:. The rational numbers, as a subspace of the real numbers, also carry a
2679:{\displaystyle \left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.}
8276:
7623:
6492:
5883:
5421:{\displaystyle {\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}.}
5206:
4620:
3521:
2501:{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}.}
842:
739:
421:
240:
750:
if and only if it contains the rational numbers as a subfield. Finite
8213:
7577:
196:
7330:
6941:(6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244.
2414:{\displaystyle {\frac {ad}{bc}}={\frac {a}{b}}\cdot {\frac {d}{c}}.}
1735:{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.}
8234:
3739:
Addition and multiplication can be defined by the following rules:
6753:(6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160.
5446:
3576:{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \setminus \{0\}))}
3503:
1647:{\displaystyle {\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.}
1553:{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}
889:
425:
36:
7546:
5270:
i.e. the field of roots of rational polynomials, is the field of
4747:{\displaystyle {\frac {m_{1}}{n_{1}}}\leq {\frac {m_{2}}{n_{2}}}}
6564:
1997:{\displaystyle \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}.}
1158:, changing the sign of the resulting numerator and denominator.
8238:
7550:
7334:
6356:
Divine
Proportions: Rational Trigonometry to Universal Geometry
6572:
5652:. The rationals are characterized topologically as the unique
4362:{\displaystyle {\frac {m_{1}}{n_{1}}}={\frac {m_{2}}{n_{2}}}}
1853:{\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}.}
1038:
Every rational number may be expressed in a unique way as an
1014:
993:
5644:
All three topologies coincide and turn the rationals into a
4097:
7044:"Does rational come from ratio or ratio come from rational"
6303:
is equivalent to either the usual real absolute value or a
486:
6780:(illustrated ed.). Courier Corporation. p. 382.
5451:
Illustration of the countability of the positive rationals
2722:
is in canonical form, the canonical form of the result is
909:(i.e., a point whose coordinates are rational numbers); a
7326:"Rational Number" From MathWorld – A Wolfram Web Resource
7175:. Springer Science & Business Media. p. A.VII.5.
3496:
are different ways to represent the same rational value.
3078:
can be represented as a finite continued fraction, whose
1896:
is in canonical form, the same is true for its opposite.
2780:
is even. Otherwise, the canonical form of the result is
2511:
The result is in canonical form if the same is true for
1749:—even if both original fractions are in canonical form.
290:
is a rational number, as is every integer (for example,
6340:
Naive height—height of a rational number in lowest term
2987:
2975:
2961:
2949:
2928:
2916:
2895:
2883:
1029:
Fraction (mathematics) § Arithmetic with fractions
887:
refers to the fact that a rational number represents a
4651:
4451:
4167:
3391:
3376:
3292:
3279:
3163:
3122:
3055:
2990:
2978:
2964:
2952:
2931:
2919:
2898:
2886:
2790:
2732:
2699:
2521:
2327:
2290:
2257:
2224:
2083:
2050:
2017:
1909:
1873:
1767:
1180:
1113:
1051:
692:
308:
267:
216:
8018:
7941:
7902:
7864:
7836:
7808:
7780:
7693:
7660:
7632:
7604:
6538:
6500:
6462:
6424:
6386:
6285:
6242:
6191:
6160:
6080:
5978:
5943:
5898:
5825:
5786:
5707:
5683:
5623:
5561:
5362:
5333:
5290:
5249:
5216:
5183:
5153:
5119:
5088:
5054:
5021:
4989:
4766:
4692:
4649:
4483:
4449:
4381:
4307:
4238:
4165:
4077:
4047:
3905:
3748:
3613:
3535:
3471:
3423:
3368:
3325:
3271:
3237:
3201:
3158:
3120:
3053:
2866:
2788:
2730:
2697:
2617:
2556:
2519:
2442:
2361:
2325:
2288:
2255:
2222:
2163:
2150:
2144:
2081:
2048:
2015:
1949:
1907:
1871:
1807:
1765:
1682:
1585:
1491:
1441:
1401:
1364:
1338:
1298:
1260:
1220:
1178:
1111:
1049:
869:
796:
762:
690:
556:
464:
396:. The real numbers that are rational are those whose
369:
297:
265:
214:
165:
136:
107:
78:
49:
8151:
8077:
7979:
7887:
7758:
7585:
4474:may be represented by infinitely many pairs, since
8026:
7949:
7910:
7872:
7844:
7816:
7788:
7701:
7668:
7640:
7612:
6912:. India: Oxford University Press. pp. 75–78.
6550:
6512:
6474:
6436:
6398:
6293:
6260:
6218:
6171:
6137:
6060:
5959:
5927:
5867:
5794:
5750:
5691:
5634:
5607:
5528:of the real numbers, the rationals are neither an
5420:
5345:
5316:
5260:
5227:
5191:
5164:
5127:
5099:
5062:
5029:
4997:
4965:
4746:
4667:
4603:. The canonical representative is the unique pair
4586:
4464:
4430:
4361:
4287:
4183:
4128:
4055:
4024:
3890:
3728:
3575:
3485:
3452:
3404:
3349:
3309:
3252:
3217:
3176:
3135:
3068:
3027:
2830:
2759:
2712:
2678:
2591:{\displaystyle \left({\frac {a}{b}}\right)^{0}=1.}
2590:
2537:
2500:
2413:
2343:
2303:
2270:
2237:
2203:
2109:
2063:
2030:
1996:
1922:
1886:
1852:
1780:
1734:
1646:
1552:
1462:
1427:
1376:
1350:
1324:
1278:
1246:
1206:which is its canonical form as a rational number.
1196:
1129:
1067:
877:
804:
770:
705:
671:
474:
380:
326:
280:
229:
173:
144:
115:
86:
57:
5539:By virtue of their order, the rationals carry an
3405:{\displaystyle 2+{\tfrac {1}{2}}+{\tfrac {1}{6}}}
3310:{\displaystyle 2+{\tfrac {1}{1+{\tfrac {1}{2}}}}}
1001:This unusual history originated in the fact that
438:Repeating decimal § Extension to other bases
7186:Giese, Martin; Schönegge, Arno (December 1995).
7147:Encyclopedic Dictionary of Mathematics, Volume 1
5471:number divided by itself will always equal one.
5317:{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}
1428:{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}
1289:If both fractions are in canonical form, then:
30:"Rationals" redirects here. For other uses, see
7220:(2nd, revised ed.). CRC Press. p. 1.
1139:its canonical form may be obtained by dividing
7015:(2nd ed.). Oxford University Press. 1989.
6988:(2nd ed.). Oxford University Press. 1989.
6965:(2nd ed.). Oxford University Press. 1989.
340:of all rational numbers, also referred to as "
8250:
7562:
7346:
5493:real numbers are irrational, in the sense of
1325:{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}
1247:{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}
8:
7247:A First Course in Discrete Dynamical Systems
4106:
4100:
3564:
3558:
3319:continued fraction in abbreviated notation:
7150:. London, England: MIT Press. p. 578.
7090:The Nature and Growth of Modern Mathematics
6228:is not complete, and its completion is the
400:either terminates after a finite number of
8257:
8243:
8235:
8219:
7569:
7555:
7547:
7353:
7339:
7331:
6810:. American Mathematical Soc. p. 104.
6542:
6504:
6466:
6428:
6390:
3679:
3675:
2831:{\displaystyle {\tfrac {-b^{n}}{-a^{n}}}.}
622:
618:
8020:
8019:
8017:
7943:
7942:
7940:
7904:
7903:
7901:
7866:
7865:
7863:
7838:
7837:
7835:
7810:
7809:
7807:
7782:
7781:
7779:
7695:
7694:
7692:
7662:
7661:
7659:
7634:
7633:
7631:
7606:
7605:
7603:
7092:. Princeton University Press. p. 28.
6751:Discrete Mathematics and its Applications
6544:
6543:
6537:
6506:
6505:
6499:
6468:
6467:
6461:
6430:
6429:
6423:
6392:
6391:
6385:
6287:
6286:
6284:
6249:
6245:
6244:
6241:
6207:
6196:
6195:
6190:
6162:
6161:
6159:
6129:
6124:
6109:
6085:
6079:
6046:
6041:
6032:
6024:
6019:
6010:
6007:
5998:
5984:
5977:
5944:
5942:
5913:
5908:
5899:
5897:
5853:
5840:
5835:
5826:
5824:
5788:
5787:
5785:
5743:
5729:
5706:
5685:
5684:
5682:
5625:
5624:
5622:
5597:
5583:
5560:
5405:
5376:
5363:
5361:
5332:
5304:
5291:
5289:
5251:
5250:
5248:
5218:
5217:
5215:
5185:
5184:
5182:
5155:
5154:
5152:
5121:
5120:
5118:
5090:
5089:
5087:
5056:
5055:
5053:
5023:
5022:
5020:
4991:
4990:
4988:
4947:
4937:
4924:
4914:
4904:
4891:
4881:
4862:
4847:
4837:
4824:
4814:
4804:
4791:
4781:
4767:
4765:
4736:
4726:
4720:
4709:
4699:
4693:
4691:
4650:
4648:
4597:Each equivalence class contains a unique
4555:
4542:
4519:
4490:
4482:
4450:
4448:
4419:
4409:
4396:
4386:
4380:
4351:
4341:
4335:
4324:
4314:
4308:
4306:
4276:
4266:
4253:
4243:
4237:
4166:
4164:
4115:
4093:
4092:
4082:
4081:
4076:
4049:
4048:
4046:
4010:
4000:
3987:
3977:
3958:
3945:
3926:
3913:
3904:
3876:
3866:
3853:
3843:
3830:
3820:
3801:
3788:
3769:
3756:
3747:
3717:
3707:
3694:
3684:
3666:
3653:
3634:
3621:
3612:
3551:
3550:
3540:
3539:
3534:
3470:
3441:
3428:
3422:
3390:
3375:
3367:
3324:
3291:
3278:
3270:
3236:
3205:
3200:
3162:
3157:
3121:
3119:
3054:
3052:
2996:
2991:
2979:
2972:
2965:
2953:
2946:
2937:
2932:
2920:
2913:
2904:
2899:
2887:
2880:
2871:
2865:
2815:
2800:
2789:
2787:
2748:
2738:
2731:
2729:
2698:
2696:
2665:
2655:
2649:
2637:
2623:
2616:
2576:
2562:
2555:
2520:
2518:
2487:
2477:
2471:
2462:
2448:
2441:
2398:
2385:
2362:
2360:
2326:
2324:
2289:
2287:
2256:
2254:
2223:
2221:
2178:
2161:
2149:
2148:
2145:
2143:
2082:
2080:
2049:
2047:
2016:
2014:
1981:
1969:
1955:
1948:
1908:
1906:
1872:
1870:
1832:
1815:
1806:
1766:
1764:
1709:
1696:
1683:
1681:
1612:
1599:
1586:
1584:
1518:
1505:
1492:
1490:
1440:
1415:
1402:
1400:
1363:
1337:
1312:
1299:
1297:
1259:
1234:
1221:
1219:
1179:
1177:
1112:
1110:
1050:
1048:
871:
870:
868:
798:
797:
795:
764:
763:
761:
691:
689:
660:
650:
637:
627:
609:
596:
577:
564:
555:
465:
463:
371:
370:
368:
307:
296:
266:
264:
215:
213:
167:
166:
164:
138:
137:
135:
109:
108:
106:
80:
79:
77:
51:
50:
48:
7121:"Fraction - Encyclopedia of Mathematics"
6777:Elements of Pure and Applied Mathematics
1170:can be expressed as the rational number
505:). Since the set of rational numbers is
6744:
6742:
6738:
5501:Real numbers and topological properties
3555:
2760:{\displaystyle {\tfrac {b^{n}}{a^{n}}}}
4431:{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}
4288:{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}
988:came from the mathematical meaning of
715:then denotes the equivalence class of
420:). This statement is true not only in
7214:Anthony Vazzana; David Garth (2015).
7115:
7113:
7111:
7109:
7107:
7105:
7103:
7101:
7099:
6903:
6901:
6899:
6807:The Collected Works of Julia Robinson
5667:. The rational numbers do not form a
3512:The rational numbers may be built as
901:are rational numbers. For example, a
7:
6897:
6895:
6893:
6891:
6889:
6887:
6885:
6883:
6881:
6879:
6219:{\displaystyle (\mathbb {Q} ,d_{p})}
6138:{\displaystyle d_{p}(x,y)=|x-y|_{p}}
5615:and this yields a third topology on
3045:are integers. Every rational number
1482:Two fractions are added as follows:
1013:, that is "not to be spoken about" (
7733:Set-theoretically definable numbers
5455:The set of all rational numbers is
4611:in the equivalence class such that
2135:are nonzero, the division rule is
327:{\displaystyle -5={\tfrac {-5}{1}}}
3453:{\displaystyle 2^{3}\times 3^{-1}}
3088:can be determined by applying the
2110:{\displaystyle {\tfrac {-b}{-a}},}
424:, but also in every other integer
416:of digits over and over (example:
25:
6261:{\displaystyle \mathbb {Q} _{p}.}
3218:{\displaystyle 2.{\overline {6}}}
2843:Continued fraction representation
509:, and the set of real numbers is
451:. Irrational numbers include the
8218:
7530:
4668:{\displaystyle {\tfrac {n}{1}}.}
4600:canonical representative element
4184:{\displaystyle {\tfrac {m}{n}}.}
4149:The equivalence class of a pair
3177:{\displaystyle 2{\tfrac {2}{3}}}
2538:{\displaystyle {\tfrac {a}{b}}.}
2433:is a non-negative integer, then
2344:{\displaystyle {\tfrac {c}{d}}:}
1673:The rule for multiplication is:
1197:{\displaystyle {\tfrac {n}{1}},}
1130:{\displaystyle {\tfrac {a}{b}},}
1103:Starting from a rational number
1068:{\displaystyle {\tfrac {a}{b}},}
847:Construction of the real numbers
6551:{\displaystyle :\;\mathbb {N} }
6513:{\displaystyle :\;\mathbb {Z} }
6475:{\displaystyle :\;\mathbb {Q} }
6437:{\displaystyle :\;\mathbb {R} }
6399:{\displaystyle :\;\mathbb {C} }
5960:{\displaystyle {\frac {a}{b}},}
5868:{\displaystyle |a|_{p}=p_{-n},}
4909:
4903:
4861:
4809:
4803:
4465:{\displaystyle {\tfrac {m}{n}}}
4035:This equivalence relation is a
3136:{\displaystyle {\tfrac {8}{3}}}
3069:{\displaystyle {\tfrac {a}{b}}}
2713:{\displaystyle {\tfrac {a}{b}}}
2425:Exponentiation to integer power
2304:{\displaystyle {\tfrac {a}{b}}}
2271:{\displaystyle {\tfrac {c}{d}}}
2238:{\displaystyle {\tfrac {a}{b}}}
2064:{\displaystyle {\tfrac {b}{a}}}
2031:{\displaystyle {\tfrac {a}{b}}}
1923:{\displaystyle {\tfrac {a}{b}}}
1887:{\displaystyle {\tfrac {a}{b}}}
1781:{\displaystyle {\tfrac {a}{b}}}
726:Rational numbers together with
706:{\displaystyle {\tfrac {p}{q}}}
447:that is not rational is called
352:is usually denoted by boldface
281:{\displaystyle {\tfrac {3}{7}}}
230:{\displaystyle {\tfrac {p}{q}}}
7067:Coolman, Robert (2016-01-29).
6213:
6192:
6125:
6110:
6103:
6091:
6042:
6033:
6020:
6011:
5909:
5900:
5836:
5827:
5744:
5730:
5723:
5711:
5598:
5584:
5577:
5565:
5547:. The rational numbers form a
5111:With the order defined above,
4953:
4874:
4853:
4774:
4632:representation in lowest terms
4112:
4109:
4089:
4078:
4069:by this equivalence relation,
4016:
3970:
3964:
3938:
3932:
3906:
3882:
3813:
3807:
3781:
3775:
3749:
3676:
3672:
3646:
3640:
3614:
3570:
3567:
3547:
3536:
3344:
3326:
3265:using traditional typography:
3247:
3241:
825:, the rational numbers form a
619:
615:
589:
583:
557:
125:, while rationals include the
1:
8067:Plane-based geometric algebra
7217:Introduction to Number Theory
6172:{\displaystyle \mathbb {Q} .}
5815:and for any non-zero integer
5635:{\displaystyle \mathbb {Q} .}
5608:{\displaystyle d(x,y)=|x-y|,}
5261:{\displaystyle \mathbb {Q} ,}
5228:{\displaystyle \mathbb {Z} .}
5165:{\displaystyle \mathbb {Q} .}
5100:{\displaystyle \mathbb {Q} .}
2280:is equivalent to multiplying
517:real numbers are irrational.
381:{\displaystyle \mathbb {Q} .}
199:that can be expressed as the
8027:{\displaystyle \mathbb {S} }
7950:{\displaystyle \mathbb {C} }
7911:{\displaystyle \mathbb {R} }
7873:{\displaystyle \mathbb {O} }
7845:{\displaystyle \mathbb {H} }
7817:{\displaystyle \mathbb {C} }
7789:{\displaystyle \mathbb {R} }
7702:{\displaystyle \mathbb {A} }
7669:{\displaystyle \mathbb {Q} }
7641:{\displaystyle \mathbb {Z} }
7613:{\displaystyle \mathbb {N} }
7244:Richard A. Holmgren (2012).
6294:{\displaystyle \mathbb {Q} }
6273:states that any non-trivial
5795:{\displaystyle \mathbb {Q} }
5751:{\displaystyle d(x,y)=|x-y|}
5692:{\displaystyle \mathbb {Q} }
5192:{\displaystyle \mathbb {Q} }
5128:{\displaystyle \mathbb {Q} }
5063:{\displaystyle \mathbb {Q} }
5030:{\displaystyle \mathbb {Q} }
4998:{\displaystyle \mathbb {Q} }
4056:{\displaystyle \mathbb {Q} }
3210:
878:{\displaystyle \mathbb {Q} }
805:{\displaystyle \mathbb {Q} }
771:{\displaystyle \mathbb {Q} }
174:{\displaystyle \mathbb {N} }
154:, which in turn include the
145:{\displaystyle \mathbb {Z} }
116:{\displaystyle \mathbb {C} }
96:, which are included in the
87:{\displaystyle \mathbb {R} }
58:{\displaystyle \mathbb {Q} }
7441:Quadratic irrational number
7427:Pisot–Vijayaraghavan number
7314:Encyclopedia of Mathematics
5519:regular continued fractions
5465:Cartesian coordinate system
3604:is defined on this set by
1096:. This is often called the
475:{\displaystyle {\sqrt {2}}}
408:), or eventually begins to
8354:
7173:Algebra II: Chapters 4 - 7
6939:Elements of Modern Algebra
5928:{\displaystyle |0|_{p}=0.}
5804:into a topological field:
5768:
2846:
1899:A nonzero rational number
1745:where the result may be a
1026:
1015:
994:
29:
8272:
8209:
8057:Algebra of physical space
7526:
7368:
7013:Oxford English Dictionary
6986:Oxford English Dictionary
6963:Oxford English Dictionary
6908:Biggs, Norman L. (2002).
5439:to the rational numbers.
4641:with the rational number
3415:prime power decomposition
2857:is an expression such as
2855:finite continued fraction
2119:depending on the sign of
1463:{\displaystyle ad<bc.}
968:. On the contrary, it is
905:is a point with rational
897:sometimes means that the
350:field of rational numbers
32:Rational (disambiguation)
8113:Extended complex numbers
8096:Extended natural numbers
7144:Sūgakkai, Nihon (1993).
6804:Robinson, Julia (1996).
6277:on the rational numbers
5935:For any rational number
5879:is the highest power of
4634:of the rational number.
4441:Every equivalence class
3585:be the set of the pairs
3227:repeating decimal using
1100:of the rational number.
952:are defined in terms of
861:in reference to the set
520:Rational numbers can be
27:Quotient of two integers
8328:Fractions (mathematics)
6839:Encyclopedia Britannica
6749:Rosen, Kenneth (2007).
5353:are positive), we have
1149:greatest common divisor
917:of rational numbers; a
782:algebraic number fields
392:A rational number is a
8318:Elementary mathematics
8169:Transcendental numbers
8028:
8005:Hyperbolic quaternions
7951:
7912:
7874:
7846:
7818:
7790:
7703:
7670:
7642:
7614:
7537:Mathematics portal
7125:encyclopediaofmath.org
6638:Dyadic (finite binary)
6552:
6514:
6476:
6438:
6400:
6295:
6262:
6220:
6173:
6139:
6062:
5961:
5929:
5869:
5796:
5752:
5693:
5675:are the completion of
5636:
5609:
5452:
5422:
5347:
5318:
5262:
5229:
5193:
5166:
5129:
5101:
5064:
5031:
4999:
4967:
4748:
4669:
4588:
4466:
4432:
4363:
4289:
4185:
4130:
4065:is the defined as the
4057:
4026:
3892:
3730:
3577:
3509:
3487:
3454:
3406:
3351:
3311:
3254:
3219:
3178:
3137:
3070:
3029:
2832:
2761:
2714:
2680:
2592:
2539:
2502:
2415:
2345:
2305:
2272:
2239:
2205:
2111:
2065:
2032:
1998:
1934:multiplicative inverse
1924:
1888:
1854:
1782:
1757:Every rational number
1736:
1648:
1554:
1464:
1429:
1378:
1352:
1326:
1280:
1248:
1198:
1131:
1069:
879:
806:
772:
707:
673:
476:
382:
328:
282:
231:
184:
175:
146:
117:
88:
59:
8297:Superparticular ratio
8101:Extended real numbers
8029:
7952:
7922:Split-complex numbers
7913:
7875:
7847:
7819:
7791:
7704:
7680:Constructible numbers
7671:
7643:
7615:
7171:Bourbaki, N. (2003).
7088:Kramer, Edna (1983).
6553:
6515:
6477:
6439:
6401:
6296:
6263:
6221:
6174:
6140:
6063:
5962:
5930:
5870:
5797:
5753:
5694:
5669:complete metric space
5637:
5610:
5450:
5423:
5348:
5319:
5263:
5230:
5194:
5167:
5130:
5102:
5065:
5032:
5000:
4968:
4749:
4670:
4589:
4467:
4433:
4364:
4290:
4186:
4131:
4058:
4027:
3893:
3731:
3578:
3507:
3488:
3455:
3407:
3352:
3312:
3255:
3253:{\displaystyle 2.(6)}
3220:
3179:
3138:
3104:Other representations
3071:
3030:
2833:
2762:
2715:
2681:
2593:
2540:
2503:
2416:
2346:
2306:
2273:
2240:
2206:
2112:
2066:
2033:
1999:
1925:
1889:
1855:
1783:
1737:
1649:
1555:
1465:
1430:
1379:
1353:
1327:
1281:
1279:{\displaystyle ad=bc}
1249:
1199:
1162:Embedding of integers
1132:
1070:
972:that is derived from
880:
823:mathematical analysis
807:
773:
708:
674:
528:of pairs of integers
477:
383:
329:
283:
232:
176:
147:
118:
89:
60:
41:The rational numbers
40:
18:Rational number field
8338:Sets of real numbers
8133:Supernatural numbers
8043:Multicomplex numbers
8016:
8000:Dual-complex numbers
7939:
7900:
7862:
7834:
7806:
7778:
7760:Composition algebras
7728:Arithmetical numbers
7691:
7658:
7630:
7602:
7383:Constructible number
6910:Discrete Mathematics
6774:Lass, Harry (2009).
6683:Algebraic irrational
6536:
6498:
6460:
6422:
6384:
6283:
6240:
6189:
6158:
6078:
5976:
5941:
5896:
5823:
5784:
5705:
5681:
5665:totally disconnected
5663:. The space is also
5621:
5559:
5505:The rationals are a
5360:
5331:
5288:
5277:The rationals are a
5247:
5214:
5181:
5151:
5117:
5086:
5052:
5019:
4987:
4764:
4690:
4647:
4481:
4447:
4379:
4305:
4236:
4163:
4075:
4045:
3903:
3746:
3611:
3602:equivalence relation
3533:
3527:More precisely, let
3469:
3421:
3366:
3323:
3269:
3235:
3199:
3156:
3118:
3051:
2864:
2786:
2728:
2695:
2615:
2554:
2517:
2440:
2359:
2323:
2286:
2253:
2220:
2142:
2079:
2046:
2013:
1947:
1905:
1869:
1805:
1763:
1680:
1583:
1489:
1439:
1399:
1362:
1336:
1296:
1258:
1218:
1176:
1109:
1047:
1040:irreducible fraction
1034:Irreducible fraction
867:
794:
760:
688:
554:
547:defined as follows:
545:equivalence relation
462:
418:9/44 = 0.20454545...
367:
295:
263:
212:
163:
134:
105:
76:
67:are included in the
47:
8323:Field (mathematics)
8038:Split-biquaternions
7750:Eisenstein integers
7713:Closed-form numbers
7509:Supersilver ratio (
7500:Supergolden ratio (
7274:Weisstein, Eric W.
6858:Weisstein, Eric W.
6823:Extract of page 104
6793:Extract of page 382
6374:
6271:Ostrowski's theorem
5553:absolute difference
5346:{\displaystyle b,d}
4630:. It is called the
4037:congruence relation
3514:equivalence classes
3500:Formal construction
3486:{\displaystyle 3'6}
3090:Euclidean algorithm
2989:
2977:
2963:
2951:
2930:
2918:
2897:
2885:
1794:, often called its
1377:{\displaystyle b=d}
1351:{\displaystyle a=c}
976:: the first use of
923:rational expression
919:rational polynomial
748:characteristic zero
738:which contains the
526:equivalence classes
8196:Profinite integers
8159:Irrational numbers
8024:
7947:
7908:
7870:
7842:
7814:
7786:
7743:Gaussian rationals
7723:Computable numbers
7699:
7666:
7638:
7610:
7403:Eisenstein integer
7263:Extract of page 26
7195:(Technical report)
6548:
6510:
6472:
6434:
6396:
6370:
6350:Rational data type
6291:
6258:
6233:-adic number field
6216:
6169:
6135:
6058:
5957:
5925:
5865:
5792:
5748:
5689:
5632:
5605:
5453:
5418:
5343:
5314:
5258:
5225:
5203:field of fractions
5189:
5162:
5125:
5097:
5060:
5041:field automorphism
5027:
4995:
4963:
4961:
4744:
4665:
4660:
4584:
4462:
4460:
4428:
4359:
4285:
4181:
4176:
4144:field of fractions
4126:
4053:
4022:
3888:
3726:
3573:
3510:
3483:
3450:
3402:
3400:
3385:
3347:
3307:
3305:
3301:
3263:continued fraction
3250:
3215:
3174:
3172:
3133:
3131:
3066:
3064:
3025:
3018:
3013:
3008:
3003:
2984:
2958:
2925:
2892:
2849:Continued fraction
2828:
2823:
2757:
2755:
2710:
2708:
2676:
2588:
2535:
2530:
2498:
2411:
2341:
2336:
2301:
2299:
2268:
2266:
2235:
2233:
2201:
2172:
2159:
2107:
2102:
2061:
2059:
2028:
2026:
1994:
1936:, also called its
1920:
1918:
1884:
1882:
1850:
1778:
1776:
1747:reducible fraction
1732:
1644:
1550:
1460:
1425:
1374:
1348:
1322:
1276:
1244:
1194:
1189:
1127:
1122:
1065:
1060:
1009:, in the sense of
948:Although nowadays
875:
802:
768:
746:, and a field has
703:
701:
669:
472:
378:
346:field of rationals
324:
322:
278:
276:
227:
225:
185:
171:
142:
113:
84:
55:
8305:
8304:
8232:
8231:
8143:Superreal numbers
8123:Levi-Civita field
8118:Hyperreal numbers
8062:Spacetime algebra
8048:Geometric algebra
7961:Bicomplex numbers
7927:Split-quaternions
7768:Division algebras
7738:Gaussian integers
7685:Algebraic numbers
7588:definable numbers
7544:
7543:
7518:Twelfth root of 2
7398:Doubling the cube
7388:Conway's constant
7373:Algebraic integer
7362:Algebraic numbers
7309:"Rational number"
7280:Wolfram MathWorld
7257:978-1-4419-8732-7
7233:Extract of page 1
7227:978-1-4987-1752-6
6933:Gilbert, Jimmie;
6919:978-0-19-871369-2
6864:Wolfram MathWorld
6860:"Rational Number"
6835:"Rational number"
6817:978-0-8218-0575-6
6787:978-0-486-47186-0
6760:978-0-07-288008-3
6730:
6729:
6726:
6725:
6722:
6721:
6718:
6717:
6707:
6706:
6703:
6702:
6699:
6698:
6695:
6694:
6664:
6663:
6660:
6659:
6656:
6655:
6652:
6651:
6645:Repeating decimal
6612:
6611:
6608:
6607:
6603:Negative integers
6597:
6596:
6593:
6592:
6588:Composite numbers
6335:Gaussian rational
6183:The metric space
6053:
5992:
5952:
5701:under the metric
5646:topological field
5545:subspace topology
5480:Stern–Brocot tree
5413:
5400:
5371:
5312:
5299:
5272:algebraic numbers
5239:algebraic closure
4907:
4865:
4807:
4742:
4715:
4659:
4573:
4550:
4537:
4514:
4459:
4357:
4330:
4298:This means that
4175:
4142:and produces its
3593:of integers such
3399:
3384:
3360:Egyptian fraction
3304:
3300:
3213:
3189:repeating decimal
3171:
3130:
3063:
3020:
3015:
3010:
3005:
2988:
2976:
2962:
2950:
2929:
2917:
2896:
2884:
2822:
2754:
2707:
2671:
2631:
2570:
2529:
2493:
2456:
2406:
2393:
2380:
2335:
2298:
2265:
2232:
2196:
2173:
2171:
2158:
2101:
2058:
2025:
1989:
1963:
1917:
1881:
1845:
1823:
1775:
1727:
1704:
1691:
1639:
1607:
1594:
1545:
1513:
1500:
1423:
1410:
1320:
1307:
1242:
1229:
1188:
1121:
1059:
927:rational function
816:algebraic numbers
786:algebraic closure
700:
470:
398:decimal expansion
321:
275:
224:
16:(Redirected from
8345:
8333:Rational numbers
8266:Rational numbers
8259:
8252:
8245:
8236:
8222:
8221:
8189:
8179:
8091:Cardinal numbers
8052:Clifford algebra
8033:
8031:
8030:
8025:
8023:
7995:Dual quaternions
7956:
7954:
7953:
7948:
7946:
7917:
7915:
7914:
7909:
7907:
7879:
7877:
7876:
7871:
7869:
7851:
7849:
7848:
7843:
7841:
7823:
7821:
7820:
7815:
7813:
7795:
7793:
7792:
7787:
7785:
7708:
7706:
7705:
7700:
7698:
7675:
7673:
7672:
7667:
7665:
7652:Rational numbers
7647:
7645:
7644:
7639:
7637:
7619:
7617:
7616:
7611:
7609:
7571:
7564:
7557:
7548:
7535:
7534:
7512:
7503:
7495:Square root of 7
7490:Square root of 6
7485:Square root of 5
7480:Square root of 3
7475:Square root of 2
7468:
7464:
7435:
7416:
7408:Gaussian integer
7393:Cyclotomic field
7355:
7348:
7341:
7332:
7322:
7290:
7289:
7287:
7286:
7271:
7265:
7261:
7241:
7235:
7231:
7211:
7205:
7204:
7202:
7200:
7194:
7183:
7177:
7176:
7168:
7162:
7161:
7141:
7135:
7134:
7132:
7131:
7117:
7094:
7093:
7085:
7079:
7078:
7076:
7075:
7064:
7058:
7057:
7055:
7054:
7036:
7030:
7016:
7009:
7003:
6989:
6982:
6976:
6966:
6959:
6953:
6952:
6930:
6924:
6923:
6905:
6874:
6873:
6871:
6870:
6855:
6849:
6848:
6846:
6845:
6831:
6825:
6821:
6801:
6795:
6791:
6771:
6765:
6764:
6746:
6679:
6678:
6670:
6669:
6627:
6626:
6618:
6617:
6561:
6560:
6557:
6555:
6554:
6549:
6547:
6527:
6526:
6523:
6522:
6519:
6517:
6516:
6511:
6509:
6489:
6488:
6485:
6484:
6481:
6479:
6478:
6473:
6471:
6451:
6450:
6447:
6446:
6443:
6441:
6440:
6435:
6433:
6413:
6412:
6409:
6408:
6405:
6403:
6402:
6397:
6395:
6375:
6369:
6366:
6365:
6362:
6361:
6310:absolute value.
6307:
6302:
6300:
6298:
6297:
6292:
6290:
6269:
6267:
6265:
6264:
6259:
6254:
6253:
6248:
6232:
6227:
6225:
6223:
6222:
6217:
6212:
6211:
6199:
6180:
6178:
6176:
6175:
6170:
6165:
6144:
6142:
6141:
6136:
6134:
6133:
6128:
6113:
6090:
6089:
6067:
6065:
6064:
6059:
6054:
6052:
6051:
6050:
6045:
6036:
6030:
6029:
6028:
6023:
6014:
6008:
6003:
6002:
5997:
5993:
5985:
5968:
5966:
5964:
5963:
5958:
5953:
5945:
5934:
5932:
5931:
5926:
5918:
5917:
5912:
5903:
5892:In addition set
5888:
5882:
5878:
5874:
5872:
5871:
5866:
5861:
5860:
5845:
5844:
5839:
5830:
5818:
5810:
5803:
5801:
5799:
5798:
5793:
5791:
5773:
5757:
5755:
5754:
5749:
5747:
5733:
5700:
5698:
5696:
5695:
5690:
5688:
5657:metrizable space
5643:
5641:
5639:
5638:
5633:
5628:
5614:
5612:
5611:
5606:
5601:
5587:
5495:Lebesgue measure
5476:Calkin–Wilf tree
5437:order isomorphic
5427:
5425:
5424:
5419:
5414:
5406:
5401:
5399:
5388:
5377:
5372:
5364:
5352:
5350:
5349:
5344:
5323:
5321:
5320:
5315:
5313:
5305:
5300:
5292:
5269:
5267:
5265:
5264:
5259:
5254:
5236:
5234:
5232:
5231:
5226:
5221:
5200:
5198:
5196:
5195:
5190:
5188:
5173:
5171:
5169:
5168:
5163:
5158:
5136:
5134:
5132:
5131:
5126:
5124:
5108:
5106:
5104:
5103:
5098:
5093:
5071:
5069:
5067:
5066:
5061:
5059:
5038:
5036:
5034:
5033:
5028:
5026:
5006:
5004:
5002:
5001:
4996:
4994:
4972:
4970:
4969:
4964:
4962:
4952:
4951:
4942:
4941:
4929:
4928:
4919:
4918:
4908:
4905:
4896:
4895:
4886:
4885:
4870:
4866:
4863:
4859:
4852:
4851:
4842:
4841:
4829:
4828:
4819:
4818:
4808:
4805:
4796:
4795:
4786:
4785:
4770:
4753:
4751:
4750:
4745:
4743:
4741:
4740:
4731:
4730:
4721:
4716:
4714:
4713:
4704:
4703:
4694:
4676:
4674:
4672:
4671:
4666:
4661:
4652:
4640:
4629:
4618:
4614:
4610:
4593:
4591:
4590:
4585:
4574:
4572:
4564:
4556:
4551:
4543:
4538:
4536:
4528:
4520:
4515:
4513:
4502:
4491:
4473:
4471:
4469:
4468:
4463:
4461:
4452:
4437:
4435:
4434:
4429:
4424:
4423:
4414:
4413:
4401:
4400:
4391:
4390:
4368:
4366:
4365:
4360:
4358:
4356:
4355:
4346:
4345:
4336:
4331:
4329:
4328:
4319:
4318:
4309:
4294:
4292:
4291:
4286:
4281:
4280:
4271:
4270:
4258:
4257:
4248:
4247:
4228:
4210:
4192:
4190:
4188:
4187:
4182:
4177:
4168:
4156:
4137:
4135:
4133:
4132:
4127:
4119:
4096:
4085:
4064:
4062:
4060:
4059:
4054:
4052:
4031:
4029:
4028:
4023:
4015:
4014:
4005:
4004:
3992:
3991:
3982:
3981:
3963:
3962:
3950:
3949:
3931:
3930:
3918:
3917:
3897:
3895:
3894:
3889:
3881:
3880:
3871:
3870:
3858:
3857:
3848:
3847:
3835:
3834:
3825:
3824:
3806:
3805:
3793:
3792:
3774:
3773:
3761:
3760:
3735:
3733:
3732:
3727:
3722:
3721:
3712:
3711:
3699:
3698:
3689:
3688:
3671:
3670:
3658:
3657:
3639:
3638:
3626:
3625:
3599:
3592:
3584:
3582:
3580:
3579:
3574:
3554:
3543:
3492:
3490:
3489:
3484:
3479:
3459:
3457:
3456:
3451:
3449:
3448:
3433:
3432:
3411:
3409:
3408:
3403:
3401:
3392:
3386:
3377:
3356:
3354:
3353:
3350:{\displaystyle }
3348:
3316:
3314:
3313:
3308:
3306:
3303:
3302:
3293:
3280:
3259:
3257:
3256:
3251:
3224:
3222:
3221:
3216:
3214:
3206:
3185:
3183:
3181:
3180:
3175:
3173:
3164:
3144:
3142:
3140:
3139:
3134:
3132:
3123:
3099:
3087:
3077:
3075:
3073:
3072:
3067:
3065:
3056:
3044:
3034:
3032:
3031:
3026:
3021:
3019:
3017:
3016:
3014:
3012:
3011:
3009:
3007:
3006:
3004:
3002:
3001:
3000:
2985:
2983:
2973:
2959:
2957:
2947:
2942:
2941:
2926:
2924:
2914:
2909:
2908:
2893:
2891:
2881:
2876:
2875:
2839:
2837:
2835:
2834:
2829:
2824:
2821:
2820:
2819:
2806:
2805:
2804:
2791:
2779:
2775:
2768:
2766:
2764:
2763:
2758:
2756:
2753:
2752:
2743:
2742:
2733:
2721:
2719:
2717:
2716:
2711:
2709:
2700:
2685:
2683:
2682:
2677:
2672:
2670:
2669:
2660:
2659:
2650:
2645:
2644:
2636:
2632:
2624:
2607:
2597:
2595:
2594:
2589:
2581:
2580:
2575:
2571:
2563:
2547:In particular,
2546:
2544:
2542:
2541:
2536:
2531:
2522:
2507:
2505:
2504:
2499:
2494:
2492:
2491:
2482:
2481:
2472:
2467:
2466:
2461:
2457:
2449:
2432:
2420:
2418:
2417:
2412:
2407:
2399:
2394:
2386:
2381:
2379:
2371:
2363:
2352:
2350:
2348:
2347:
2342:
2337:
2328:
2312:
2310:
2308:
2307:
2302:
2300:
2291:
2279:
2277:
2275:
2274:
2269:
2267:
2258:
2246:
2244:
2242:
2241:
2236:
2234:
2225:
2210:
2208:
2207:
2202:
2197:
2195:
2187:
2179:
2174:
2164:
2162:
2160:
2151:
2146:
2134:
2122:
2118:
2116:
2114:
2113:
2108:
2103:
2100:
2092:
2084:
2072:
2070:
2068:
2067:
2062:
2060:
2051:
2039:
2037:
2035:
2034:
2029:
2027:
2018:
2003:
2001:
2000:
1995:
1990:
1982:
1977:
1976:
1968:
1964:
1956:
1931:
1929:
1927:
1926:
1921:
1919:
1910:
1895:
1893:
1891:
1890:
1885:
1883:
1874:
1859:
1857:
1856:
1851:
1846:
1841:
1833:
1828:
1824:
1816:
1792:additive inverse
1789:
1787:
1785:
1784:
1779:
1777:
1768:
1741:
1739:
1738:
1733:
1728:
1726:
1718:
1710:
1705:
1697:
1692:
1684:
1663:coprime integers
1660:
1653:
1651:
1650:
1645:
1640:
1638:
1630:
1613:
1608:
1600:
1595:
1587:
1569:coprime integers
1566:
1559:
1557:
1556:
1551:
1546:
1544:
1536:
1519:
1514:
1506:
1501:
1493:
1469:
1467:
1466:
1461:
1434:
1432:
1431:
1426:
1424:
1416:
1411:
1403:
1383:
1381:
1380:
1375:
1357:
1355:
1354:
1349:
1331:
1329:
1328:
1323:
1321:
1313:
1308:
1300:
1285:
1283:
1282:
1277:
1253:
1251:
1250:
1245:
1243:
1235:
1230:
1222:
1205:
1203:
1201:
1200:
1195:
1190:
1181:
1169:
1157:
1146:
1142:
1138:
1136:
1134:
1133:
1128:
1123:
1114:
1095:
1087:coprime integers
1084:
1080:
1076:
1074:
1072:
1071:
1066:
1061:
1052:
1018:
1017:
997:
996:
950:rational numbers
886:
884:
882:
881:
876:
874:
835:Cauchy sequences
814:is the field of
813:
811:
809:
808:
803:
801:
779:
777:
775:
774:
769:
767:
722:
714:
712:
710:
709:
704:
702:
693:
678:
676:
675:
670:
665:
664:
655:
654:
642:
641:
632:
631:
614:
613:
601:
600:
582:
581:
569:
568:
542:
535:
504:
496:
489:
485:
483:
481:
479:
478:
473:
471:
466:
453:square root of 2
419:
412:the same finite
407:
389:
387:
385:
384:
379:
374:
357:
335:
333:
331:
330:
325:
323:
317:
309:
289:
287:
285:
284:
279:
277:
268:
256:
249:
238:
236:
234:
233:
228:
226:
217:
182:
180:
178:
177:
172:
170:
153:
151:
149:
148:
143:
141:
124:
122:
120:
119:
114:
112:
95:
93:
91:
90:
85:
83:
66:
64:
62:
61:
56:
54:
21:
8353:
8352:
8348:
8347:
8346:
8344:
8343:
8342:
8308:
8307:
8306:
8301:
8287:Dyadic rational
8268:
8263:
8233:
8228:
8205:
8184:
8174:
8147:
8138:Surreal numbers
8128:Ordinal numbers
8073:
8014:
8013:
7975:
7937:
7936:
7934:
7932:Split-octonions
7898:
7897:
7889:
7883:
7860:
7859:
7832:
7831:
7804:
7803:
7800:Complex numbers
7776:
7775:
7754:
7689:
7688:
7656:
7655:
7628:
7627:
7600:
7599:
7596:Natural numbers
7581:
7575:
7545:
7540:
7529:
7522:
7510:
7501:
7469:
7466:
7462:
7446:Rational number
7433:
7432:Plastic ratio (
7414:
7378:Chebyshev nodes
7364:
7359:
7307:
7304:
7299:
7294:
7293:
7284:
7282:
7276:"p-adic Number"
7273:
7272:
7268:
7258:
7243:
7242:
7238:
7228:
7213:
7212:
7208:
7198:
7196:
7192:
7185:
7184:
7180:
7170:
7169:
7165:
7158:
7143:
7142:
7138:
7129:
7127:
7119:
7118:
7097:
7087:
7086:
7082:
7073:
7071:
7066:
7065:
7061:
7052:
7050:
7038:
7037:
7033:
7011:
7010:
7006:
6984:
6983:
6979:
6961:
6960:
6956:
6949:
6932:
6931:
6927:
6920:
6907:
6906:
6877:
6868:
6866:
6857:
6856:
6852:
6843:
6841:
6833:
6832:
6828:
6818:
6803:
6802:
6798:
6788:
6773:
6772:
6768:
6761:
6748:
6747:
6740:
6735:
6534:
6533:
6496:
6495:
6458:
6457:
6420:
6419:
6382:
6381:
6345:Niven's theorem
6320:Dyadic rational
6316:
6305:
6281:
6280:
6278:
6243:
6238:
6237:
6235:
6230:
6203:
6187:
6186:
6184:
6156:
6155:
6153:
6123:
6081:
6076:
6075:
6040:
6031:
6018:
6009:
5980:
5979:
5974:
5973:
5939:
5938:
5936:
5907:
5894:
5893:
5886:
5880:
5876:
5849:
5834:
5821:
5820:
5816:
5808:
5782:
5781:
5779:
5776:
5771:
5767:
5703:
5702:
5679:
5678:
5676:
5661:isolated points
5650:locally compact
5619:
5618:
5616:
5557:
5556:
5503:
5445:
5433:totally ordered
5389:
5378:
5358:
5357:
5329:
5328:
5286:
5285:
5279:densely ordered
5245:
5244:
5242:
5212:
5211:
5209:
5179:
5178:
5176:
5149:
5148:
5146:
5115:
5114:
5112:
5084:
5083:
5081:
5050:
5049:
5047:
5017:
5016:
5014:
4985:
4984:
4982:
4979:
4960:
4959:
4943:
4933:
4920:
4910:
4887:
4877:
4868:
4867:
4857:
4856:
4843:
4833:
4820:
4810:
4787:
4777:
4762:
4761:
4732:
4722:
4705:
4695:
4688:
4687:
4645:
4644:
4642:
4638:
4624:
4616:
4612:
4604:
4565:
4557:
4529:
4521:
4503:
4492:
4479:
4478:
4445:
4444:
4442:
4415:
4405:
4392:
4382:
4377:
4376:
4372:if and only if
4347:
4337:
4320:
4310:
4303:
4302:
4272:
4262:
4249:
4239:
4234:
4233:
4226:
4219:
4212:
4208:
4201:
4194:
4161:
4160:
4158:
4150:
4140:integral domain
4073:
4072:
4070:
4043:
4042:
4040:
4006:
3996:
3983:
3973:
3954:
3941:
3922:
3909:
3901:
3900:
3872:
3862:
3849:
3839:
3826:
3816:
3797:
3784:
3765:
3752:
3744:
3743:
3713:
3703:
3690:
3680:
3662:
3649:
3630:
3617:
3609:
3608:
3594:
3586:
3531:
3530:
3528:
3502:
3472:
3467:
3466:
3437:
3424:
3419:
3418:
3364:
3363:
3321:
3320:
3284:
3267:
3266:
3233:
3232:
3197:
3196:
3154:
3153:
3151:
3116:
3115:
3113:
3110:common fraction
3106:
3093:
3086:
3082:
3049:
3048:
3046:
3043:
3039:
2992:
2986:
2974:
2960:
2948:
2933:
2927:
2915:
2900:
2894:
2882:
2867:
2862:
2861:
2851:
2845:
2811:
2807:
2796:
2792:
2784:
2783:
2781:
2777:
2770:
2744:
2734:
2726:
2725:
2723:
2693:
2692:
2690:
2661:
2651:
2619:
2618:
2613:
2612:
2602:
2558:
2557:
2552:
2551:
2515:
2514:
2512:
2483:
2473:
2444:
2443:
2438:
2437:
2430:
2427:
2372:
2364:
2357:
2356:
2321:
2320:
2318:
2284:
2283:
2281:
2251:
2250:
2248:
2218:
2217:
2215:
2214:Thus, dividing
2188:
2180:
2147:
2140:
2139:
2132:
2129:
2120:
2093:
2085:
2077:
2076:
2074:
2044:
2043:
2041:
2011:
2010:
2008:
1951:
1950:
1945:
1944:
1903:
1902:
1900:
1867:
1866:
1864:
1834:
1811:
1803:
1802:
1761:
1760:
1758:
1755:
1719:
1711:
1678:
1677:
1671:
1658:
1631:
1614:
1581:
1580:
1577:
1564:
1537:
1520:
1487:
1486:
1480:
1437:
1436:
1435:if and only if
1397:
1396:
1390:
1360:
1359:
1334:
1333:
1332:if and only if
1294:
1293:
1256:
1255:
1254:if and only if
1216:
1215:
1212:
1174:
1173:
1171:
1167:
1164:
1152:
1144:
1140:
1107:
1106:
1104:
1090:
1082:
1078:
1045:
1044:
1042:
1036:
1031:
1025:
946:
911:rational matrix
865:
864:
862:
855:
792:
791:
789:
758:
757:
755:
716:
686:
685:
683:
656:
646:
633:
623:
605:
592:
573:
560:
552:
551:
537:
529:
502:
492:
487:
460:
459:
457:
455:
417:
405:
365:
364:
362:
360:blackboard bold
353:
310:
293:
292:
291:
261:
260:
258:
257:. For example,
254:
250:and a non-zero
247:
210:
209:
207:
193:rational number
161:
160:
158:
156:natural numbers
132:
131:
129:
103:
102:
100:
98:complex numbers
74:
73:
71:
45:
44:
42:
35:
28:
23:
22:
15:
12:
11:
5:
8351:
8349:
8341:
8340:
8335:
8330:
8325:
8320:
8310:
8309:
8303:
8302:
8300:
8299:
8294:
8289:
8284:
8279:
8273:
8270:
8269:
8264:
8262:
8261:
8254:
8247:
8239:
8230:
8229:
8227:
8226:
8216:
8214:Classification
8210:
8207:
8206:
8204:
8203:
8201:Normal numbers
8198:
8193:
8171:
8166:
8161:
8155:
8153:
8149:
8148:
8146:
8145:
8140:
8135:
8130:
8125:
8120:
8115:
8110:
8109:
8108:
8098:
8093:
8087:
8085:
8083:infinitesimals
8075:
8074:
8072:
8071:
8070:
8069:
8064:
8059:
8045:
8040:
8035:
8022:
8007:
8002:
7997:
7992:
7986:
7984:
7977:
7976:
7974:
7973:
7968:
7963:
7958:
7945:
7929:
7924:
7919:
7906:
7893:
7891:
7885:
7884:
7882:
7881:
7868:
7853:
7840:
7825:
7812:
7797:
7784:
7764:
7762:
7756:
7755:
7753:
7752:
7747:
7746:
7745:
7735:
7730:
7725:
7720:
7715:
7710:
7697:
7682:
7677:
7664:
7649:
7636:
7621:
7608:
7592:
7590:
7583:
7582:
7576:
7574:
7573:
7566:
7559:
7551:
7542:
7541:
7527:
7524:
7523:
7521:
7520:
7515:
7506:
7497:
7492:
7487:
7482:
7477:
7472:
7465:
7461:Silver ratio (
7458:
7453:
7448:
7443:
7438:
7429:
7424:
7419:
7413:Golden ratio (
7410:
7405:
7400:
7395:
7390:
7385:
7380:
7375:
7369:
7366:
7365:
7360:
7358:
7357:
7350:
7343:
7335:
7329:
7328:
7323:
7303:
7302:External links
7300:
7298:
7295:
7292:
7291:
7266:
7256:
7236:
7226:
7206:
7178:
7163:
7156:
7136:
7095:
7080:
7059:
7048:Stack Exchange
7042:(2017-05-09).
7031:
7004:
6977:
6954:
6947:
6935:Linda, Gilbert
6925:
6918:
6875:
6850:
6826:
6816:
6796:
6786:
6766:
6759:
6737:
6736:
6734:
6731:
6728:
6727:
6724:
6723:
6720:
6719:
6716:
6715:
6709:
6708:
6705:
6704:
6701:
6700:
6697:
6696:
6693:
6692:
6690:Transcendental
6686:
6685:
6676:
6666:
6665:
6662:
6661:
6658:
6657:
6654:
6653:
6650:
6649:
6647:
6641:
6640:
6634:
6633:
6631:Finite decimal
6624:
6614:
6613:
6610:
6609:
6606:
6605:
6599:
6598:
6595:
6594:
6591:
6590:
6584:
6583:
6577:
6576:
6569:
6568:
6558:
6546:
6541:
6520:
6508:
6503:
6482:
6470:
6465:
6444:
6432:
6427:
6406:
6394:
6389:
6372:Number systems
6360:
6359:
6352:
6347:
6342:
6337:
6332:
6327:
6325:Floating point
6322:
6315:
6312:
6289:
6275:absolute value
6257:
6252:
6247:
6215:
6210:
6206:
6202:
6198:
6194:
6168:
6164:
6146:
6145:
6132:
6127:
6122:
6119:
6116:
6112:
6108:
6105:
6102:
6099:
6096:
6093:
6088:
6084:
6069:
6068:
6057:
6049:
6044:
6039:
6035:
6027:
6022:
6017:
6013:
6006:
6001:
5996:
5991:
5988:
5983:
5956:
5951:
5948:
5924:
5921:
5916:
5911:
5906:
5902:
5864:
5859:
5856:
5852:
5848:
5843:
5838:
5833:
5829:
5790:
5769:Main article:
5766:
5760:
5746:
5742:
5739:
5736:
5732:
5728:
5725:
5722:
5719:
5716:
5713:
5710:
5687:
5631:
5627:
5604:
5600:
5596:
5593:
5590:
5586:
5582:
5579:
5576:
5573:
5570:
5567:
5564:
5541:order topology
5517:expansions as
5502:
5499:
5461:square lattice
5444:
5441:
5429:
5428:
5417:
5412:
5409:
5404:
5398:
5395:
5392:
5387:
5384:
5381:
5375:
5370:
5367:
5342:
5339:
5336:
5325:
5324:
5311:
5308:
5303:
5298:
5295:
5257:
5253:
5224:
5220:
5187:
5161:
5157:
5123:
5096:
5092:
5078:characteristic
5058:
5025:
4993:
4978:
4975:
4974:
4973:
4958:
4955:
4950:
4946:
4940:
4936:
4932:
4927:
4923:
4917:
4913:
4902:
4899:
4894:
4890:
4884:
4880:
4876:
4873:
4871:
4869:
4860:
4858:
4855:
4850:
4846:
4840:
4836:
4832:
4827:
4823:
4817:
4813:
4802:
4799:
4794:
4790:
4784:
4780:
4776:
4773:
4771:
4769:
4755:
4754:
4739:
4735:
4729:
4725:
4719:
4712:
4708:
4702:
4698:
4664:
4658:
4655:
4595:
4594:
4583:
4580:
4577:
4571:
4568:
4563:
4560:
4554:
4549:
4546:
4541:
4535:
4532:
4527:
4524:
4518:
4512:
4509:
4506:
4501:
4498:
4495:
4489:
4486:
4458:
4455:
4439:
4438:
4427:
4422:
4418:
4412:
4408:
4404:
4399:
4395:
4389:
4385:
4370:
4369:
4354:
4350:
4344:
4340:
4334:
4327:
4323:
4317:
4313:
4296:
4295:
4284:
4279:
4275:
4269:
4265:
4261:
4256:
4252:
4246:
4242:
4224:
4217:
4206:
4199:
4180:
4174:
4171:
4125:
4122:
4118:
4114:
4111:
4108:
4105:
4102:
4099:
4095:
4091:
4088:
4084:
4080:
4051:
4033:
4032:
4021:
4018:
4013:
4009:
4003:
3999:
3995:
3990:
3986:
3980:
3976:
3972:
3969:
3966:
3961:
3957:
3953:
3948:
3944:
3940:
3937:
3934:
3929:
3925:
3921:
3916:
3912:
3908:
3898:
3887:
3884:
3879:
3875:
3869:
3865:
3861:
3856:
3852:
3846:
3842:
3838:
3833:
3829:
3823:
3819:
3815:
3812:
3809:
3804:
3800:
3796:
3791:
3787:
3783:
3780:
3777:
3772:
3768:
3764:
3759:
3755:
3751:
3737:
3736:
3725:
3720:
3716:
3710:
3706:
3702:
3697:
3693:
3687:
3683:
3678:
3674:
3669:
3665:
3661:
3656:
3652:
3648:
3645:
3642:
3637:
3633:
3629:
3624:
3620:
3616:
3572:
3569:
3566:
3563:
3560:
3557:
3553:
3549:
3546:
3542:
3538:
3501:
3498:
3494:
3493:
3482:
3478:
3475:
3463:quote notation
3460:
3447:
3444:
3440:
3436:
3431:
3427:
3412:
3398:
3395:
3389:
3383:
3380:
3374:
3371:
3357:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3317:
3299:
3296:
3290:
3287:
3283:
3277:
3274:
3260:
3249:
3246:
3243:
3240:
3225:
3212:
3209:
3204:
3186:
3170:
3167:
3161:
3145:
3129:
3126:
3105:
3102:
3084:
3062:
3059:
3041:
3036:
3035:
3024:
2999:
2995:
2982:
2971:
2968:
2956:
2945:
2940:
2936:
2923:
2912:
2907:
2903:
2890:
2879:
2874:
2870:
2847:Main article:
2844:
2841:
2827:
2818:
2814:
2810:
2803:
2799:
2795:
2751:
2747:
2741:
2737:
2706:
2703:
2687:
2686:
2675:
2668:
2664:
2658:
2654:
2648:
2643:
2640:
2635:
2630:
2627:
2622:
2599:
2598:
2587:
2584:
2579:
2574:
2569:
2566:
2561:
2534:
2528:
2525:
2509:
2508:
2497:
2490:
2486:
2480:
2476:
2470:
2465:
2460:
2455:
2452:
2447:
2426:
2423:
2422:
2421:
2410:
2405:
2402:
2397:
2392:
2389:
2384:
2378:
2375:
2370:
2367:
2340:
2334:
2331:
2297:
2294:
2264:
2261:
2231:
2228:
2212:
2211:
2200:
2194:
2191:
2186:
2183:
2177:
2170:
2167:
2157:
2154:
2128:
2125:
2106:
2099:
2096:
2091:
2088:
2057:
2054:
2024:
2021:
2005:
2004:
1993:
1988:
1985:
1980:
1975:
1972:
1967:
1962:
1959:
1954:
1916:
1913:
1880:
1877:
1861:
1860:
1849:
1844:
1840:
1837:
1831:
1827:
1822:
1819:
1814:
1810:
1774:
1771:
1754:
1751:
1743:
1742:
1731:
1725:
1722:
1717:
1714:
1708:
1703:
1700:
1695:
1690:
1687:
1670:
1669:Multiplication
1667:
1655:
1654:
1643:
1637:
1634:
1629:
1626:
1623:
1620:
1617:
1611:
1606:
1603:
1598:
1593:
1590:
1576:
1573:
1561:
1560:
1549:
1543:
1540:
1535:
1532:
1529:
1526:
1523:
1517:
1512:
1509:
1504:
1499:
1496:
1479:
1476:
1471:
1470:
1459:
1456:
1453:
1450:
1447:
1444:
1422:
1419:
1414:
1409:
1406:
1389:
1386:
1385:
1384:
1373:
1370:
1367:
1347:
1344:
1341:
1319:
1316:
1311:
1306:
1303:
1287:
1286:
1275:
1272:
1269:
1266:
1263:
1241:
1238:
1233:
1228:
1225:
1211:
1208:
1193:
1187:
1184:
1163:
1160:
1126:
1120:
1117:
1098:canonical form
1064:
1058:
1055:
1035:
1032:
1024:
1021:
1003:ancient Greeks
945:
942:
935:rational curve
903:rational point
873:
854:
851:
841:, or infinite
800:
766:
732:multiplication
699:
696:
680:
679:
668:
663:
659:
653:
649:
645:
640:
636:
630:
626:
621:
617:
612:
608:
604:
599:
595:
591:
588:
585:
580:
576:
572:
567:
563:
559:
469:
428:, such as the
377:
373:
320:
316:
313:
306:
303:
300:
274:
271:
223:
220:
169:
140:
111:
82:
53:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8350:
8339:
8336:
8334:
8331:
8329:
8326:
8324:
8321:
8319:
8316:
8315:
8313:
8298:
8295:
8293:
8290:
8288:
8285:
8283:
8280:
8278:
8275:
8274:
8271:
8267:
8260:
8255:
8253:
8248:
8246:
8241:
8240:
8237:
8225:
8217:
8215:
8212:
8211:
8208:
8202:
8199:
8197:
8194:
8191:
8187:
8181:
8177:
8172:
8170:
8167:
8165:
8164:Fuzzy numbers
8162:
8160:
8157:
8156:
8154:
8150:
8144:
8141:
8139:
8136:
8134:
8131:
8129:
8126:
8124:
8121:
8119:
8116:
8114:
8111:
8107:
8104:
8103:
8102:
8099:
8097:
8094:
8092:
8089:
8088:
8086:
8084:
8080:
8076:
8068:
8065:
8063:
8060:
8058:
8055:
8054:
8053:
8049:
8046:
8044:
8041:
8039:
8036:
8011:
8008:
8006:
8003:
8001:
7998:
7996:
7993:
7991:
7988:
7987:
7985:
7983:
7978:
7972:
7969:
7967:
7966:Biquaternions
7964:
7962:
7959:
7933:
7930:
7928:
7925:
7923:
7920:
7895:
7894:
7892:
7886:
7857:
7854:
7829:
7826:
7801:
7798:
7773:
7769:
7766:
7765:
7763:
7761:
7757:
7751:
7748:
7744:
7741:
7740:
7739:
7736:
7734:
7731:
7729:
7726:
7724:
7721:
7719:
7716:
7714:
7711:
7686:
7683:
7681:
7678:
7653:
7650:
7625:
7622:
7597:
7594:
7593:
7591:
7589:
7584:
7579:
7572:
7567:
7565:
7560:
7558:
7553:
7552:
7549:
7539:
7538:
7533:
7525:
7519:
7516:
7514:
7507:
7505:
7498:
7496:
7493:
7491:
7488:
7486:
7483:
7481:
7478:
7476:
7473:
7471:
7459:
7457:
7454:
7452:
7451:Root of unity
7449:
7447:
7444:
7442:
7439:
7437:
7430:
7428:
7425:
7423:
7422:Perron number
7420:
7418:
7411:
7409:
7406:
7404:
7401:
7399:
7396:
7394:
7391:
7389:
7386:
7384:
7381:
7379:
7376:
7374:
7371:
7370:
7367:
7363:
7356:
7351:
7349:
7344:
7342:
7337:
7336:
7333:
7327:
7324:
7320:
7316:
7315:
7310:
7306:
7305:
7301:
7296:
7281:
7277:
7270:
7267:
7264:
7259:
7253:
7249:
7248:
7240:
7237:
7234:
7229:
7223:
7219:
7218:
7210:
7207:
7191:
7190:
7182:
7179:
7174:
7167:
7164:
7159:
7157:0-2625-9020-4
7153:
7149:
7148:
7140:
7137:
7126:
7122:
7116:
7114:
7112:
7110:
7108:
7106:
7104:
7102:
7100:
7096:
7091:
7084:
7081:
7070:
7063:
7060:
7049:
7045:
7041:
7035:
7032:
7028:
7024:
7020:
7014:
7008:
7005:
7001:
6997:
6993:
6987:
6981:
6978:
6974:
6970:
6964:
6958:
6955:
6950:
6948:0-534-40264-X
6944:
6940:
6936:
6929:
6926:
6921:
6915:
6911:
6904:
6902:
6900:
6898:
6896:
6894:
6892:
6890:
6888:
6886:
6884:
6882:
6880:
6876:
6865:
6861:
6854:
6851:
6840:
6836:
6830:
6827:
6824:
6819:
6813:
6809:
6808:
6800:
6797:
6794:
6789:
6783:
6779:
6778:
6770:
6767:
6762:
6756:
6752:
6745:
6743:
6739:
6732:
6714:
6711:
6710:
6691:
6688:
6687:
6684:
6681:
6680:
6677:
6675:
6672:
6671:
6668:
6667:
6648:
6646:
6643:
6642:
6639:
6636:
6635:
6632:
6629:
6628:
6625:
6623:
6620:
6619:
6616:
6615:
6604:
6601:
6600:
6589:
6586:
6585:
6582:
6581:Prime numbers
6579:
6578:
6574:
6571:
6570:
6566:
6563:
6562:
6559:
6539:
6532:
6529:
6528:
6525:
6524:
6521:
6501:
6494:
6491:
6490:
6487:
6486:
6483:
6463:
6456:
6453:
6452:
6449:
6448:
6445:
6425:
6418:
6415:
6414:
6411:
6410:
6407:
6387:
6380:
6377:
6376:
6373:
6368:
6367:
6364:
6363:
6358:
6357:
6353:
6351:
6348:
6346:
6343:
6341:
6338:
6336:
6333:
6331:
6328:
6326:
6323:
6321:
6318:
6317:
6313:
6311:
6309:
6276:
6272:
6255:
6250:
6234:
6208:
6204:
6200:
6181:
6166:
6151:
6130:
6120:
6117:
6114:
6106:
6100:
6097:
6094:
6086:
6082:
6074:
6073:
6072:
6055:
6047:
6037:
6025:
6015:
6004:
5999:
5994:
5989:
5986:
5981:
5972:
5971:
5970:
5954:
5949:
5946:
5922:
5919:
5914:
5904:
5890:
5885:
5862:
5857:
5854:
5850:
5846:
5841:
5831:
5814:
5805:
5775:
5765:-adic numbers
5764:
5761:
5759:
5740:
5737:
5734:
5726:
5720:
5717:
5714:
5708:
5674:
5670:
5666:
5662:
5658:
5655:
5651:
5647:
5629:
5602:
5594:
5591:
5588:
5580:
5574:
5571:
5568:
5562:
5554:
5551:by using the
5550:
5546:
5542:
5537:
5535:
5531:
5527:
5524:In the usual
5522:
5520:
5516:
5512:
5508:
5500:
5498:
5496:
5492:
5488:
5483:
5481:
5477:
5472:
5470:
5466:
5462:
5458:
5449:
5442:
5440:
5438:
5434:
5415:
5410:
5407:
5402:
5396:
5393:
5390:
5385:
5382:
5379:
5373:
5368:
5365:
5356:
5355:
5354:
5340:
5337:
5334:
5309:
5306:
5301:
5296:
5293:
5284:
5283:
5282:
5280:
5275:
5273:
5255:
5240:
5222:
5208:
5204:
5174:
5159:
5144:
5140:
5139:ordered field
5109:
5094:
5079:
5075:
5045:
5042:
5012:
5010:
4976:
4956:
4948:
4944:
4938:
4934:
4930:
4925:
4921:
4915:
4911:
4900:
4897:
4892:
4888:
4882:
4878:
4872:
4848:
4844:
4838:
4834:
4830:
4825:
4821:
4815:
4811:
4800:
4797:
4792:
4788:
4782:
4778:
4772:
4760:
4759:
4758:
4737:
4733:
4727:
4723:
4717:
4710:
4706:
4700:
4696:
4686:
4685:
4684:
4682:
4677:
4662:
4656:
4653:
4635:
4633:
4627:
4622:
4608:
4602:
4601:
4581:
4578:
4575:
4569:
4566:
4561:
4558:
4552:
4547:
4544:
4539:
4533:
4530:
4525:
4522:
4516:
4510:
4507:
4504:
4499:
4496:
4493:
4487:
4484:
4477:
4476:
4475:
4456:
4453:
4425:
4420:
4416:
4410:
4406:
4402:
4397:
4393:
4387:
4383:
4375:
4374:
4373:
4352:
4348:
4342:
4338:
4332:
4325:
4321:
4315:
4311:
4301:
4300:
4299:
4282:
4277:
4273:
4267:
4263:
4259:
4254:
4250:
4244:
4240:
4232:
4231:
4230:
4223:
4216:
4205:
4198:
4178:
4172:
4169:
4154:
4147:
4145:
4141:
4123:
4120:
4116:
4103:
4086:
4068:
4038:
4019:
4011:
4007:
4001:
3997:
3993:
3988:
3984:
3978:
3974:
3967:
3959:
3955:
3951:
3946:
3942:
3935:
3927:
3923:
3919:
3914:
3910:
3899:
3885:
3877:
3873:
3867:
3863:
3859:
3854:
3850:
3844:
3840:
3836:
3831:
3827:
3821:
3817:
3810:
3802:
3798:
3794:
3789:
3785:
3778:
3770:
3766:
3762:
3757:
3753:
3742:
3741:
3740:
3723:
3718:
3714:
3708:
3704:
3700:
3695:
3691:
3685:
3681:
3667:
3663:
3659:
3654:
3650:
3643:
3635:
3631:
3627:
3622:
3618:
3607:
3606:
3605:
3603:
3597:
3590:
3561:
3544:
3525:
3523:
3519:
3518:ordered pairs
3515:
3506:
3499:
3497:
3480:
3476:
3473:
3464:
3461:
3445:
3442:
3438:
3434:
3429:
3425:
3416:
3413:
3396:
3393:
3387:
3381:
3378:
3372:
3369:
3361:
3358:
3341:
3338:
3335:
3332:
3329:
3318:
3297:
3294:
3288:
3285:
3281:
3275:
3272:
3264:
3261:
3244:
3238:
3230:
3226:
3207:
3202:
3194:
3190:
3187:
3168:
3165:
3159:
3149:
3148:mixed numeral
3146:
3127:
3124:
3111:
3108:
3107:
3103:
3101:
3097:
3091:
3081:
3060:
3057:
3022:
2997:
2993:
2980:
2969:
2966:
2954:
2943:
2938:
2934:
2921:
2910:
2905:
2901:
2888:
2877:
2872:
2868:
2860:
2859:
2858:
2856:
2850:
2842:
2840:
2825:
2816:
2812:
2808:
2801:
2797:
2793:
2773:
2749:
2745:
2739:
2735:
2704:
2701:
2673:
2666:
2662:
2656:
2652:
2646:
2641:
2638:
2633:
2628:
2625:
2620:
2611:
2610:
2609:
2605:
2585:
2582:
2577:
2572:
2567:
2564:
2559:
2550:
2549:
2548:
2532:
2526:
2523:
2495:
2488:
2484:
2478:
2474:
2468:
2463:
2458:
2453:
2450:
2445:
2436:
2435:
2434:
2424:
2408:
2403:
2400:
2395:
2390:
2387:
2382:
2376:
2373:
2368:
2365:
2355:
2354:
2353:
2338:
2332:
2329:
2316:
2295:
2292:
2262:
2259:
2229:
2226:
2198:
2192:
2189:
2184:
2181:
2175:
2168:
2165:
2155:
2152:
2138:
2137:
2136:
2126:
2124:
2104:
2097:
2094:
2089:
2086:
2055:
2052:
2022:
2019:
1991:
1986:
1983:
1978:
1973:
1970:
1965:
1960:
1957:
1952:
1943:
1942:
1941:
1939:
1935:
1914:
1911:
1897:
1878:
1875:
1847:
1842:
1838:
1835:
1829:
1825:
1820:
1817:
1812:
1808:
1801:
1800:
1799:
1797:
1793:
1772:
1769:
1752:
1750:
1748:
1729:
1723:
1720:
1715:
1712:
1706:
1701:
1698:
1693:
1688:
1685:
1676:
1675:
1674:
1668:
1666:
1664:
1641:
1635:
1632:
1627:
1624:
1621:
1618:
1615:
1609:
1604:
1601:
1596:
1591:
1588:
1579:
1578:
1574:
1572:
1570:
1547:
1541:
1538:
1533:
1530:
1527:
1524:
1521:
1515:
1510:
1507:
1502:
1497:
1494:
1485:
1484:
1483:
1477:
1475:
1457:
1454:
1451:
1448:
1445:
1442:
1420:
1417:
1412:
1407:
1404:
1395:
1394:
1393:
1387:
1371:
1368:
1365:
1345:
1342:
1339:
1317:
1314:
1309:
1304:
1301:
1292:
1291:
1290:
1273:
1270:
1267:
1264:
1261:
1239:
1236:
1231:
1226:
1223:
1214:
1213:
1209:
1207:
1191:
1185:
1182:
1161:
1159:
1155:
1150:
1124:
1118:
1115:
1101:
1099:
1093:
1088:
1062:
1056:
1053:
1041:
1033:
1030:
1022:
1020:
1012:
1008:
1004:
999:
991:
987:
983:
979:
975:
971:
967:
963:
959:
955:
951:
943:
941:
939:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
891:
860:
852:
850:
848:
844:
840:
839:Dedekind cuts
836:
832:
828:
824:
819:
817:
787:
783:
753:
749:
745:
741:
737:
733:
729:
724:
720:
697:
694:
682:The fraction
666:
661:
657:
651:
647:
643:
638:
634:
628:
624:
610:
606:
602:
597:
593:
586:
578:
574:
570:
565:
561:
550:
549:
548:
546:
540:
533:
527:
523:
518:
516:
512:
508:
500:
495:
490:
467:
454:
450:
446:
441:
439:
435:
431:
427:
423:
415:
411:
403:
399:
395:
390:
375:
361:
356:
351:
347:
343:
342:the rationals
339:
318:
314:
311:
304:
301:
298:
272:
269:
253:
246:
242:
221:
218:
206:
202:
198:
194:
190:
157:
128:
99:
70:
39:
33:
19:
8292:Half-integer
8282:Dedekind cut
8265:
8185:
8175:
7990:Dual numbers
7982:hypercomplex
7772:Real numbers
7651:
7528:
7456:Salem number
7445:
7312:
7283:. Retrieved
7279:
7269:
7246:
7239:
7216:
7209:
7197:. Retrieved
7188:
7181:
7172:
7166:
7146:
7139:
7128:. Retrieved
7124:
7089:
7083:
7072:. Retrieved
7062:
7051:. Retrieved
7047:
7034:
7026:
7022:
7018:
7012:
7007:
7002:, sense 5.a.
6999:
6995:
6991:
6985:
6980:
6975:, sense 2.a.
6972:
6968:
6962:
6957:
6938:
6928:
6909:
6867:. Retrieved
6863:
6853:
6842:. Retrieved
6838:
6829:
6806:
6799:
6776:
6769:
6750:
6454:
6354:
6330:Ford circles
6182:
6147:
6070:
5891:
5813:prime number
5806:
5777:
5774:-adic number
5762:
5673:real numbers
5549:metric space
5538:
5523:
5511:real numbers
5507:dense subset
5504:
5484:
5473:
5454:
5443:Countability
5430:
5326:
5276:
5175:
5110:
5046:
5013:
4980:
4756:
4678:
4636:
4625:
4606:
4598:
4596:
4440:
4371:
4297:
4221:
4214:
4203:
4196:
4152:
4148:
4067:quotient set
4034:
3738:
3595:
3588:
3526:
3511:
3495:
3095:
3080:coefficients
3037:
2854:
2852:
2771:
2688:
2603:
2600:
2510:
2428:
2213:
2130:
2006:
1937:
1898:
1862:
1795:
1756:
1744:
1672:
1656:
1562:
1481:
1472:
1391:
1288:
1166:Any integer
1165:
1153:
1102:
1091:
1037:
1010:
1006:
1000:
989:
985:
981:
977:
973:
969:
965:
957:
953:
949:
947:
937:
918:
910:
899:coefficients
894:
888:
858:
856:
827:dense subset
820:
725:
718:
681:
543:, using the
538:
531:
519:
499:golden ratio
442:
391:
354:
349:
345:
341:
192:
186:
69:real numbers
8152:Other types
7971:Bioctonions
7828:Quaternions
7040:Shor, Peter
5489:, that is,
5074:prime field
4681:total order
4157:is denoted
3229:parentheses
1575:Subtraction
1019:in Greek).
956:, the term
907:coordinates
853:Terminology
780:are called
744:prime field
524:defined as
511:uncountable
445:real number
434:hexadecimal
394:real number
252:denominator
189:mathematics
8312:Categories
8106:Projective
8079:Infinities
7285:2021-08-17
7130:2021-08-17
7074:2021-03-20
7053:2021-03-19
7029:, sense 3.
7019:irrational
6869:2020-08-11
6844:2020-08-11
6733:References
6674:Irrational
6148:defines a
5671:, and the
5534:closed set
5491:almost all
5143:isomorphic
4977:Properties
4193:Two pairs
2315:reciprocal
1938:reciprocal
1151:, and, if
1027:See also:
1023:Arithmetic
1007:irrational
990:irrational
962:derivation
931:polynomial
831:completion
784:, and the
752:extensions
515:almost all
497:, and the
449:irrational
436:ones (see
406:3/4 = 0.75
404:(example:
8190:solenoids
8010:Sedenions
7856:Octonions
7319:EMS Press
7199:17 August
6996:a. (adv.)
6713:Imaginary
6118:−
5855:−
5738:−
5654:countable
5592:−
5457:countable
4931:≥
4831:≤
4718:≤
4579:⋯
4531:−
4523:−
4505:−
4494:−
4485:⋯
4121:∼
4098:∖
4087:×
3968:≡
3936:×
3811:≡
3677:⟺
3644:∼
3556:∖
3545:×
3443:−
3435:×
3211:¯
2967:⋱
2809:−
2794:−
2639:−
2396:⋅
2095:−
2087:−
1971:−
1836:−
1809:−
1694:⋅
1622:−
1597:−
1147:by their
1011:illogical
960:is not a
944:Etymology
857:The term
620:⟺
587:∼
507:countable
312:−
299:−
245:numerator
7624:Integers
7586:Sets of
6992:rational
6937:(2005).
6622:Fraction
6455:Rational
6314:See also
5969:we set
5884:dividing
5659:without
5530:open set
5526:topology
5487:null set
5469:non-zero
5463:as in a
5207:integers
4981:The set
3522:integers
3477:′
3193:vinculum
3191:using a
2127:Division
1796:opposite
1478:Addition
1388:Ordering
1210:Equality
986:rational
982:rational
974:rational
958:rational
895:rational
859:rational
843:decimals
833:, using
740:integers
728:addition
522:formally
414:sequence
241:integers
205:fraction
201:quotient
127:integers
8277:Integer
8180:numbers
8012: (
7858: (
7830: (
7802: (
7774: (
7718:Periods
7687: (
7654: (
7626: (
7598: (
7580:systems
7321:, 2001
6531:Natural
6493:Integer
6379:Complex
6301:
6279:
6268:
6236:
6226:
6185:
6179:
6154:
5967:
5937:
5802:
5780:
5758:above.
5699:
5677:
5642:
5617:
5555:metric
5509:of the
5327:(where
5268:
5243:
5235:
5210:
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5201:is the
5199:
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2545:
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2311:
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2133:b, c, d
2117:
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1790:has an
1788:
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1204:
1172:
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1043:
925:" and "
885:
863:
812:
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778:
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734:form a
713:
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422:base 10
388:
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348:or the
344:", the
288:
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239:of two
237:
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7980:Other
7578:Number
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6990:Entry
6967:Entry
6945:
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6150:metric
6071:Then
5875:where
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5532:nor a
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4628:> 0
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1077:where
1016:ἄλογος
995:ἄλογος
954:ratios
938:is not
915:matrix
430:binary
410:repeat
402:digits
197:number
8188:-adic
8178:-adic
7935:Over
7896:Over
7890:types
7888:Split
7297:Notes
7193:(PDF)
6969:ratio
6308:-adic
5811:be a
5072:is a
5009:field
3600:. An
978:ratio
970:ratio
966:ratio
929:" (a
913:is a
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845:(see
736:field
536:with
358:, or
195:is a
8224:List
8081:and
7252:ISBN
7222:ISBN
7201:2021
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7025:and
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6943:ISBN
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6575:: 1
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1992:.
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