7528:
27:
3494:
5437:
8241:
5456:, 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
4960:
4752:
5032:
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.)
3022:
4581:
1462:
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.
3885:
4019:
2198:
6055:
4123:
3723:
666:
4955:{\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}}}
2673:
5415:
2495:
2408:
1729:
3570:
1641:
1547:
4741:
1991:
4356:
1847:
4757:
2585:
3399:
3304:
5311:
1422:
1319:
1241:
2825:
2754:
4425:
4282:
6213:
6132:
321:
6344:
3447:
2104:
6255:
3212:
4662:
4178:
3171:
2532:
2338:
1191:
1124:
1062:
6545:
6507:
6469:
6431:
6393:
5954:
5862:
4459:
3130:
3063:
2852:
2707:
2298:
2265:
2232:
2058:
2025:
1917:
1881:
1775:
700:
275:
224:
7738:
5474:
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
6166:
5629:
5602:
5255:
5222:
5159:
5094:
375:
8053:
7976:
7937:
7899:
7871:
7843:
7815:
7703:
7670:
7642:
7614:
6288:
5789:
5745:
5686:
5186:
5122:
5057:
5024:
4992:
4050:
872:
799:
765:
168:
139:
110:
81:
52:
5448:, 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
4469:
469:
5922:
1457:
3480:
3403:
3247:
1273:
7183:
5340:
1371:
1345:
3344:
7039:
3734:
3891:
910:
may be a polynomial with rational coefficients, although the term "polynomial over the rationals" is generally preferred, to avoid confusion between "
2130:
6328:
5270:
set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that
5964:
4063:
3599:
542:
2603:
5348:
2428:
5130:
that has no subfield other than itself, and is the smallest ordered field, in the sense that every ordered field contains a unique subfield
2347:
1668:
7348:
5502:; every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with
3521:
1571:
1477:
8277:
4678:
7064:
1935:
7564:
4293:
1793:
4127:
equipped with the addition and the multiplication induced by the above operations. (This construction can be carried out with any
7251:
7221:
6913:
6811:
6781:
6754:
7753:
1017:
922:
is a rational expression and defines a rational function, even if its coefficients are not rational numbers). However, a
7748:
2542:
3354:
3257:
5276:
1387:
7151:
6942:
4028:, which means that it is compatible with the addition and multiplication defined above; the set of rational numbers
8348:
1284:
1206:
835:
7708:
882:
of two integers. In mathematics, "rational" is often used as a noun abbreviating "rational number". The adjective
8338:
8126:
7422:
7341:
8204:
2774:
8270:
426:
8358:
8087:
7314:
2716:
4367:
4224:
731:, and is contained in any field containing the integers. In other words, the field of rational numbers is a
8343:
7557:
4588:
994:"avoided heresy by forbidding themselves from thinking of those lengths as numbers". So such lengths were
6177:
6066:
981:, which was first used in 1551, and it was used in "translations of Euclid (following his peculiar use of
7713:
7436:
7309:
6678:
5453:
283:
3409:
3017:{\displaystyle a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\ddots +{\cfrac {1}{a_{n}}}}}}}}},}
2067:
7334:
6228:
5653:
5463:
It is possible to generate all of the rational numbers without such redundancies: examples include the
4996:
of all rational numbers, together with the addition and multiplication operations shown above, forms a
3187:
4635:
4151:
3144:
2505:
2311:
1164:
1097:
1035:
8353:
8263:
8121:
8077:
7304:
6524:
6486:
6448:
6410:
6372:
5929:
5811:
5065:, which is a field that has no subfield other than itself. The rationals are the smallest field with
4435:
3106:
3039:
2683:
2274:
2241:
2208:
2034:
2001:
1893:
1857:
1751:
819:
676:
482:
251:
200:
20:
7719:
4576:{\displaystyle \cdots ={\frac {-2m}{-2n}}={\frac {-m}{-n}}={\frac {m}{n}}={\frac {2m}{2n}}=\cdots .}
8244:
8116:
6263:
6146:
5609:
5547:
5235:
5202:
5139:
5074:
5066:
950:
355:
8036:
7959:
7920:
7882:
7854:
7826:
7798:
7686:
7653:
7625:
7597:
6271:
5772:
5693:
5669:
5169:
5105:
5040:
5007:
4975:
4033:
929:
a curve defined over the rationals, but a curve which can be parameterized by rational functions.
855:
782:
748:
510:
151:
122:
93:
64:
35:
7550:
7185:
Any two countable densely ordered sets without endpoints are isomorphic - a formal proof with KIV
5657:
1137:
450:
5884:
4672:
may be defined on the rational numbers, that extends the natural order of the integers. One has
6259:
2303:
1922:
1427:
770:
193:
7141:
6818:
6788:
8317:
8189:
8025:
7383:
7258:
6930:
6685:
5767:
In addition to the absolute value metric mentioned above, there are other metrics which turn
5468:
811:
418:
7228:
5424:
set which is countable, dense (in the above sense), and has no least or greatest element is
7942:
7675:
7378:
5645:
5464:
4620:
3590:
3223:
1246:
1028:
903:
533:
1381:
If both denominators are positive (particularly if both fractions are in canonical form):
8:
8153:
8063:
8020:
8002:
7780:
5541:
5319:
4997:
4025:
3457:
3078:
1350:
1324:
736:
724:
1646:
If both fractions are in canonical form, the result is in canonical form if and only if
1552:
If both fractions are in canonical form, the result is in canonical form if and only if
8058:
7770:
7532:
7398:
6338:
6333:
5507:
5191:
5029:
4132:
3251:
2837:
1735:
3311:
818:
of the real numbers. The real numbers can be constructed from the rational numbers by
8216:
8179:
8143:
8082:
8068:
7763:
7743:
7527:
7513:
7504:
7495:
7393:
7368:
7247:
7217:
7147:
6938:
6909:
6830:
6807:
6777:
6750:
6662:
6633:
6323:
5634:
5533:
5425:
5227:
3880:{\displaystyle (m_{1},n_{1})+(m_{2},n_{2})\equiv (m_{1}n_{2}+n_{1}m_{2},n_{1}n_{2}),}
3502:
3348:
3181:
3177:
991:
915:
911:
774:
514:
437:
398:
386:
326:
6855:
8234:
8163:
8138:
8072:
7981:
7947:
7788:
7758:
7680:
7583:
7490:
7485:
7480:
7475:
7470:
7403:
7388:
7357:
7321:
6708:
6671:
6591:
6576:
5483:
5457:
5260:
1780:
1651:
1557:
1075:
973:
for qualifying numbers appeared almost a century earlier, in 1570. This meaning of
804:
441:
7271:
5069:
zero. Every field of characteristic zero contains a unique subfield isomorphic to
8307:
8111:
8015:
7373:
7241:
7211:
6801:
6771:
6626:
6308:
5638:
5421:
5267:
4128:
3497:
A diagram showing a representation of the equivalent classes of pairs of integers
3098:
823:
740:
499:
390:
348:
144:
86:
7246:(2nd, illustrated ed.). Springer Science & Business Media. p. 26.
4014:{\displaystyle (m_{1},n_{1})\times (m_{2},n_{2})\equiv (m_{1}m_{2},n_{1}n_{2}).}
8158:
8148:
8133:
7952:
7820:
7591:
6619:
6519:
6367:
6313:
5649:
5529:
5449:
3451:
1086:
923:
891:
720:
7065:"How a Mathematical Superstition Stultified Algebra for Over a Thousand Years"
4626:
The integers may be considered to be rational numbers identifying the integer
4218:
belong to the same equivalence class (that is are equivalent) if and only if
2193:{\displaystyle {\frac {\,{\dfrac {a}{b}}\,}{\dfrac {c}{d}}}={\frac {ad}{bc}}.}
8332:
8221:
8194:
8103:
7446:
7427:
7417:
6360:
6293:
6218:
5759:
5127:
3136:
969:
with its modern meaning was attested in
English about 1660, while the use of
495:
8312:
8302:
8184:
7986:
7456:
7451:
7408:
6569:
6138:
6050:{\displaystyle \left|{\frac {a}{b}}\right|_{p}={\frac {|a|_{p}}{|b|_{p}}}.}
5801:
5661:
5537:
5499:
4055:
3506:
827:
487:
57:
7116:
4118:{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \backslash \{0\}))/\sim ,}
3718:{\displaystyle (m_{1},n_{1})\sim (m_{2},n_{2})\iff m_{1}n_{2}=m_{2}n_{1}.}
3493:
2029:
is in canonical form, then the canonical form of its reciprocal is either
661:{\displaystyle (p_{1},q_{1})\sim (p_{2},q_{2})\iff p_{1}q_{2}=p_{2}q_{1}.}
8010:
7792:
6405:
6318:
5131:
5062:
4669:
3217:
3068:
895:
887:
732:
433:
422:
382:
240:
177:
26:
5637:. The rational numbers are an important example of a space which is not
5532:. The rational numbers, as a subspace of the real numbers, also carry a
2668:{\displaystyle \left({\frac {a}{b}}\right)^{-n}={\frac {b^{n}}{a^{n}}}.}
7991:
7848:
7035:
5522:
5503:
5479:
5410:{\displaystyle {\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}.}
2490:{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}.}
919:
503:
739:
if and only if it contains the rational numbers as a subfield. Finite
5642:
5495:
5445:
815:
233:
7326:
6937:(6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244.
5436:
2403:{\displaystyle {\frac {ad}{bc}}={\frac {a}{b}}\cdot {\frac {d}{c}}.}
1724:{\displaystyle {\frac {a}{b}}\cdot {\frac {c}{d}}={\frac {ac}{bd}}.}
8255:
8099:
8030:
7876:
6610:
5518:
5514:
5475:
3728:
Addition and multiplication can be defined by the following rules:
716:
402:
189:
115:
6749:(6th ed.). New York, NY: McGraw-Hill. pp. 105, 158–160.
3565:{\displaystyle (\mathbb {Z} \times (\mathbb {Z} \setminus \{0\}))}
1636:{\displaystyle {\frac {a}{b}}-{\frac {c}{d}}={\frac {ad-bc}{bd}}.}
1542:{\displaystyle {\frac {a}{b}}+{\frac {c}{d}}={\frac {ad+bc}{bd}}.}
8297:
7619:
7542:
6481:
5872:
5259:
i.e. the field of roots of rational polynomials, is the field of
5195:
4736:{\displaystyle {\frac {m_{1}}{n_{1}}}\leq {\frac {m_{2}}{n_{2}}}}
4609:
3510:
831:
728:
410:
229:
1986:{\displaystyle \left({\frac {a}{b}}\right)^{-1}={\frac {b}{a}}.}
1147:, changing the sign of the resulting numerator and denominator.
7573:
6345:
Divine
Proportions: Rational Trigonometry to Universal Geometry
185:
5641:. The rationals are characterized topologically as the unique
4351:{\displaystyle {\frac {m_{1}}{n_{1}}}={\frac {m_{2}}{n_{2}}}}
1842:{\displaystyle -\left({\frac {a}{b}}\right)={\frac {-a}{b}}.}
1027:
Every rational number may be expressed in a unique way as an
1003:
982:
878:
414:
5633:
All three topologies coincide and turn the rationals into a
7040:"Does rational come from ratio or ratio come from rational"
6553:
6292:
is equivalent to either the usual real absolute value or a
7209:
6776:(illustrated ed.). Courier Corporation. p. 382.
5489:
5440:
Illustration of the countability of the positive rationals
2711:
is in canonical form, the canonical form of the result is
898:(i.e., a point whose coordinates are rational numbers); a
7322:"Rational Number" From MathWorld – A Wolfram Web Resource
7171:. Springer Science & Business Media. p. A.VII.5.
6561:
3485:
are different ways to represent the same rational value.
3067:
can be represented as a finite continued fraction, whose
1885:
is in canonical form, the same is true for its opposite.
2769:
is even. Otherwise, the canonical form of the result is
2500:
The result is in canonical form if the same is true for
1738:—even if both original fractions are in canonical form.
279:
is a rational number, as is every integer (for example,
6329:
Naive height—height of a rational number in lowest term
2976:
2964:
2950:
2938:
2917:
2905:
2884:
2872:
1018:
Fraction (mathematics) § Arithmetic with fractions
876:
refers to the fact that a rational number represents a
475:
4640:
4440:
4156:
4086:
3380:
3365:
3281:
3268:
3152:
3111:
3044:
2979:
2967:
2953:
2941:
2920:
2908:
2887:
2875:
2779:
2721:
2688:
2510:
2316:
2279:
2246:
2213:
2072:
2039:
2006:
1898:
1862:
1756:
1169:
1102:
1040:
681:
297:
256:
205:
8039:
7962:
7923:
7885:
7857:
7829:
7801:
7722:
7689:
7656:
7628:
7600:
6527:
6489:
6451:
6413:
6375:
6274:
6231:
6180:
6149:
6069:
5967:
5932:
5887:
5814:
5775:
5696:
5672:
5612:
5550:
5351:
5322:
5279:
5238:
5205:
5172:
5142:
5108:
5077:
5043:
5010:
4978:
4755:
4681:
4638:
4472:
4438:
4370:
4296:
4227:
4154:
4066:
4036:
3894:
3737:
3602:
3524:
3460:
3412:
3357:
3314:
3260:
3226:
3190:
3147:
3109:
3042:
2855:
2777:
2719:
2686:
2606:
2545:
2508:
2431:
2350:
2314:
2277:
2244:
2211:
2152:
2139:
2133:
2070:
2037:
2004:
1938:
1896:
1860:
1796:
1754:
1671:
1574:
1480:
1430:
1390:
1353:
1327:
1287:
1249:
1209:
1167:
1100:
1038:
858:
785:
751:
679:
545:
453:
385:. The real numbers that are rational are those whose
358:
286:
254:
203:
154:
125:
96:
67:
38:
2831:
7239:
4463:may be represented by infinitely many pairs, since
8047:
7970:
7931:
7893:
7865:
7837:
7809:
7732:
7697:
7664:
7636:
7608:
6908:. India: Oxford University Press. pp. 75–78.
6539:
6501:
6463:
6425:
6387:
6282:
6249:
6207:
6160:
6126:
6049:
5948:
5916:
5856:
5783:
5739:
5680:
5623:
5596:
5517:of the real numbers, the rationals are neither an
5409:
5334:
5305:
5249:
5216:
5180:
5153:
5116:
5088:
5051:
5018:
4986:
4954:
4735:
4656:
4592:. The canonical representative is the unique pair
4575:
4453:
4419:
4350:
4276:
4172:
4117:
4044:
4013:
3879:
3717:
3564:
3474:
3441:
3393:
3338:
3298:
3241:
3206:
3165:
3124:
3057:
3016:
2819:
2748:
2701:
2667:
2580:{\displaystyle \left({\frac {a}{b}}\right)^{0}=1.}
2579:
2526:
2489:
2413:
2402:
2332:
2292:
2259:
2226:
2192:
2098:
2052:
2019:
1985:
1911:
1875:
1841:
1769:
1723:
1635:
1541:
1451:
1416:
1365:
1339:
1313:
1267:
1235:
1195:which is its canonical form as a rational number.
1185:
1118:
1056:
866:
793:
759:
694:
660:
463:
369:
315:
269:
218:
162:
133:
104:
75:
46:
5528:By virtue of their order, the rationals carry an
3394:{\displaystyle 2+{\tfrac {1}{2}}+{\tfrac {1}{6}}}
3299:{\displaystyle 2+{\tfrac {1}{1+{\tfrac {1}{2}}}}}
990:This unusual history originated in the fact that
427:Repeating decimal § Extension to other bases
8330:
7182:Giese, Martin; Schönegge, Arno (December 1995).
7143:Encyclopedic Dictionary of Mathematics, Volume 1
5460:number divided by itself will always equal one.
5306:{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}
1417:{\displaystyle {\frac {a}{b}}<{\frac {c}{d}}}
1278:If both fractions are in canonical form, then:
19:"Rationals" redirects here. For other uses, see
7216:(2nd, revised ed.). CRC Press. p. 1.
1128:its canonical form may be obtained by dividing
7181:
7011:(2nd ed.). Oxford University Press. 1989.
6984:(2nd ed.). Oxford University Press. 1989.
6961:(2nd ed.). Oxford University Press. 1989.
329:of all rational numbers, also referred to as "
8271:
7558:
7342:
5482:real numbers are irrational, in the sense of
1314:{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}
1236:{\displaystyle {\frac {a}{b}}={\frac {c}{d}}}
7243:A First Course in Discrete Dynamical Systems
6928:
4095:
4089:
3553:
3547:
3308:continued fraction in abbreviated notation:
7146:. London, England: MIT Press. p. 578.
7086:The Nature and Growth of Modern Mathematics
6217:is not complete, and its completion is the
389:either terminates after a finite number of
8278:
8264:
8240:
7565:
7551:
7349:
7335:
6806:. American Mathematical Soc. p. 104.
6531:
6493:
6455:
6417:
6379:
3668:
3664:
2820:{\displaystyle {\tfrac {-b^{n}}{-a^{n}}}.}
611:
607:
8041:
7964:
7925:
7887:
7859:
7831:
7803:
7691:
7658:
7630:
7602:
7088:. Princeton University Press. p. 28.
6747:Discrete Mathematics and its Applications
6533:
6495:
6457:
6419:
6381:
6276:
6234:
6185:
6151:
5777:
5674:
5614:
5240:
5207:
5174:
5144:
5110:
5079:
5045:
5012:
4980:
4586:Each equivalence class contains a unique
4082:
4071:
4038:
3540:
3529:
2150:
2137:
860:
787:
753:
360:
156:
127:
98:
69:
40:
7166:
7139:
7117:"Fraction - Encyclopedia of Mathematics"
6799:
6773:Elements of Pure and Applied Mathematics
5435:
3492:
3092:
1159:can be expressed as the rational number
1150:
494:). Since the set of rational numbers is
25:
7062:
6740:
6738:
5490:Real numbers and topological properties
2749:{\displaystyle {\tfrac {b^{n}}{a^{n}}}}
1022:
8331:
7083:
6922:
4420:{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}
4277:{\displaystyle m_{1}n_{2}=m_{2}n_{1}.}
3488:
977:came from the mathematical meaning of
704:then denotes the equivalence class of
409:). This statement is true not only in
8259:
7546:
7356:
7330:
7269:
7210:Anthony Vazzana; David Garth (2015).
7111:
7109:
7107:
7105:
7103:
7101:
7099:
7097:
7095:
6903:
6899:
6897:
6895:
6853:
6803:The Collected Works of Julia Robinson
6744:
5656:. The rational numbers do not form a
3501:The rational numbers may be built as
890:are rational numbers. For example, a
8285:
7034:
6893:
6891:
6889:
6887:
6885:
6883:
6881:
6879:
6877:
6875:
6769:
6735:
6208:{\displaystyle (\mathbb {Q} ,d_{p})}
6127:{\displaystyle d_{p}(x,y)=|x-y|_{p}}
5604:and this yields a third topology on
3034:are integers. Every rational number
1471:Two fractions are added as follows:
1002:, that is "not to be spoken about" (
7754:Set-theoretically definable numbers
5444:The set of all rational numbers is
4600:in the equivalence class such that
2124:are nonzero, the division rule is
316:{\displaystyle -5={\tfrac {-5}{1}}}
13:
7725:
7572:
7092:
3442:{\displaystyle 2^{3}\times 3^{-1}}
3077:can be determined by applying the
2099:{\displaystyle {\tfrac {-b}{-a}},}
413:, but also in every other integer
405:of digits over and over (example:
14:
8370:
7297:
6872:
6847:
6250:{\displaystyle \mathbb {Q} _{p}.}
5750:
3544:
3207:{\displaystyle 2.{\overline {6}}}
2832:Continued fraction representation
1657:
498:, and the set of real numbers is
440:. Irrational numbers include the
8239:
7526:
4657:{\displaystyle {\tfrac {n}{1}}.}
4589:canonical representative element
4173:{\displaystyle {\tfrac {m}{n}}.}
4138:The equivalence class of a pair
3166:{\displaystyle 2{\tfrac {2}{3}}}
2527:{\displaystyle {\tfrac {a}{b}}.}
2422:is a non-negative integer, then
2333:{\displaystyle {\tfrac {c}{d}}:}
1662:The rule for multiplication is:
1186:{\displaystyle {\tfrac {n}{1}},}
1119:{\displaystyle {\tfrac {a}{b}},}
1092:Starting from a rational number
1057:{\displaystyle {\tfrac {a}{b}},}
836:Construction of the real numbers
7263:
7233:
7203:
7175:
7160:
7133:
7077:
7056:
7028:
7001:
6540:{\displaystyle :\;\mathbb {N} }
6502:{\displaystyle :\;\mathbb {Z} }
6464:{\displaystyle :\;\mathbb {Q} }
6426:{\displaystyle :\;\mathbb {R} }
6388:{\displaystyle :\;\mathbb {C} }
5949:{\displaystyle {\frac {a}{b}},}
5857:{\displaystyle |a|_{p}=p_{-n},}
5431:
4898:
4892:
4850:
4798:
4792:
4454:{\displaystyle {\tfrac {m}{n}}}
4024:This equivalence relation is a
3125:{\displaystyle {\tfrac {8}{3}}}
3058:{\displaystyle {\tfrac {a}{b}}}
2702:{\displaystyle {\tfrac {a}{b}}}
2414:Exponentiation to integer power
2293:{\displaystyle {\tfrac {a}{b}}}
2260:{\displaystyle {\tfrac {c}{d}}}
2227:{\displaystyle {\tfrac {a}{b}}}
2053:{\displaystyle {\tfrac {b}{a}}}
2020:{\displaystyle {\tfrac {a}{b}}}
1912:{\displaystyle {\tfrac {a}{b}}}
1876:{\displaystyle {\tfrac {a}{b}}}
1770:{\displaystyle {\tfrac {a}{b}}}
715:Rational numbers together with
695:{\displaystyle {\tfrac {p}{q}}}
436:that is not rational is called
341:is usually denoted by boldface
270:{\displaystyle {\tfrac {3}{7}}}
219:{\displaystyle {\tfrac {p}{q}}}
7733:{\displaystyle {\mathcal {P}}}
7063:Coolman, Robert (2016-01-29).
6974:
6951:
6823:
6793:
6763:
6202:
6181:
6114:
6099:
6092:
6080:
6031:
6022:
6009:
6000:
5898:
5889:
5825:
5816:
5733:
5719:
5712:
5700:
5587:
5573:
5566:
5554:
5536:. The rational numbers form a
5100:With the order defined above,
4942:
4863:
4842:
4763:
4621:representation in lowest terms
4101:
4098:
4078:
4067:
4058:by this equivalence relation,
4005:
3959:
3953:
3927:
3921:
3895:
3871:
3802:
3796:
3770:
3764:
3738:
3665:
3661:
3635:
3629:
3603:
3559:
3556:
3536:
3525:
3333:
3315:
3254:using traditional typography:
3236:
3230:
1563:
841:
814:, the rational numbers form a
608:
604:
578:
572:
546:
114:, while rationals include the
1:
8088:Plane-based geometric algebra
7213:Introduction to Number Theory
6728:
6161:{\displaystyle \mathbb {Q} .}
5804:and for any non-zero integer
5624:{\displaystyle \mathbb {Q} .}
5597:{\displaystyle d(x,y)=|x-y|,}
5250:{\displaystyle \mathbb {Q} ,}
5217:{\displaystyle \mathbb {Z} .}
5154:{\displaystyle \mathbb {Q} .}
5089:{\displaystyle \mathbb {Q} .}
4965:
2269:is equivalent to multiplying
1011:
506:real numbers are irrational.
370:{\displaystyle \mathbb {Q} .}
188:that can be expressed as the
8048:{\displaystyle \mathbb {S} }
7971:{\displaystyle \mathbb {C} }
7932:{\displaystyle \mathbb {R} }
7894:{\displaystyle \mathbb {O} }
7866:{\displaystyle \mathbb {H} }
7838:{\displaystyle \mathbb {C} }
7810:{\displaystyle \mathbb {R} }
7698:{\displaystyle \mathbb {A} }
7665:{\displaystyle \mathbb {Q} }
7637:{\displaystyle \mathbb {Z} }
7609:{\displaystyle \mathbb {N} }
7240:Richard A. Holmgren (2012).
6283:{\displaystyle \mathbb {Q} }
6262:states that any non-trivial
5784:{\displaystyle \mathbb {Q} }
5740:{\displaystyle d(x,y)=|x-y|}
5681:{\displaystyle \mathbb {Q} }
5181:{\displaystyle \mathbb {Q} }
5117:{\displaystyle \mathbb {Q} }
5052:{\displaystyle \mathbb {Q} }
5019:{\displaystyle \mathbb {Q} }
4987:{\displaystyle \mathbb {Q} }
4045:{\displaystyle \mathbb {Q} }
3199:
932:
867:{\displaystyle \mathbb {Q} }
794:{\displaystyle \mathbb {Q} }
760:{\displaystyle \mathbb {Q} }
163:{\displaystyle \mathbb {N} }
143:, which in turn include the
134:{\displaystyle \mathbb {Z} }
105:{\displaystyle \mathbb {C} }
85:, which are included in the
76:{\displaystyle \mathbb {R} }
47:{\displaystyle \mathbb {Q} }
7:
7437:Quadratic irrational number
7423:Pisot–Vijayaraghavan number
7310:Encyclopedia of Mathematics
6302:
5508:regular continued fractions
5454:Cartesian coordinate system
3593:is defined on this set by
2115:
1466:
1376:
1198:
1085:. This is often called the
464:{\displaystyle {\sqrt {2}}}
397:), or eventually begins to
10:
8375:
7169:Algebra II: Chapters 4 - 7
6935:Elements of Modern Algebra
5917:{\displaystyle |0|_{p}=0.}
5793:into a topological field:
5757:
2835:
1888:A nonzero rational number
1741:
1734:where the result may be a
1015:
1004:
983:
18:
8293:
8230:
8172:
8098:
8078:Algebra of physical space
8000:
7908:
7779:
7581:
7522:
7364:
7009:Oxford English Dictionary
6982:Oxford English Dictionary
6959:Oxford English Dictionary
6904:Biggs, Norman L. (2002).
5428:to the rational numbers.
4630:with the rational number
3404:prime power decomposition
2846:is an expression such as
2844:finite continued fraction
2108:depending on the sign of
1452:{\displaystyle ad<bc.}
957:. On the contrary, it is
894:is a point with rational
886:sometimes means that the
339:field of rational numbers
21:Rational (disambiguation)
8134:Extended complex numbers
8117:Extended natural numbers
7292:
7140:Sūgakkai, Nihon (1993).
6800:Robinson, Julia (1996).
6266:on the rational numbers
5924:For any rational number
5868:is the highest power of
4623:of the rational number.
4430:Every equivalence class
3574:be the set of the pairs
3216:repeating decimal using
1089:of the rational number.
941:are defined in terms of
850:in reference to the set
509:Rational numbers can be
16:Quotient of two integers
8349:Fractions (mathematics)
6835:Encyclopedia Britannica
6745:Rosen, Kenneth (2007).
5342:are positive), we have
1138:greatest common divisor
906:of rational numbers; a
771:algebraic number fields
381:A rational number is a
8339:Elementary mathematics
8190:Transcendental numbers
8049:
8026:Hyperbolic quaternions
7972:
7933:
7895:
7867:
7839:
7811:
7734:
7699:
7666:
7638:
7610:
7533:Mathematics portal
7121:encyclopediaofmath.org
6627:Dyadic (finite binary)
6541:
6503:
6465:
6427:
6389:
6284:
6251:
6209:
6162:
6128:
6051:
5950:
5918:
5858:
5785:
5741:
5682:
5664:are the completion of
5625:
5598:
5441:
5411:
5336:
5307:
5251:
5218:
5182:
5155:
5118:
5090:
5053:
5020:
4988:
4956:
4737:
4658:
4577:
4455:
4421:
4352:
4278:
4174:
4119:
4054:is the defined as the
4046:
4015:
3881:
3719:
3566:
3498:
3476:
3443:
3395:
3340:
3300:
3243:
3208:
3167:
3126:
3059:
3018:
2821:
2750:
2703:
2669:
2581:
2528:
2491:
2404:
2334:
2294:
2261:
2228:
2194:
2100:
2054:
2021:
1987:
1923:multiplicative inverse
1913:
1877:
1843:
1771:
1746:Every rational number
1725:
1637:
1543:
1453:
1418:
1367:
1341:
1315:
1269:
1237:
1187:
1120:
1058:
868:
795:
761:
696:
662:
465:
371:
317:
271:
220:
173:
164:
135:
106:
77:
48:
8318:Superparticular ratio
8122:Extended real numbers
8050:
7973:
7943:Split-complex numbers
7934:
7896:
7868:
7840:
7812:
7735:
7700:
7676:Constructible numbers
7667:
7639:
7611:
7167:Bourbaki, N. (2003).
7084:Kramer, Edna (1983).
6542:
6504:
6466:
6428:
6390:
6285:
6252:
6210:
6163:
6129:
6052:
5951:
5919:
5859:
5786:
5742:
5683:
5658:complete metric space
5626:
5599:
5439:
5412:
5337:
5308:
5252:
5219:
5183:
5156:
5119:
5091:
5054:
5021:
4989:
4957:
4738:
4659:
4578:
4456:
4422:
4353:
4279:
4175:
4120:
4047:
4016:
3882:
3720:
3567:
3496:
3477:
3444:
3396:
3341:
3301:
3244:
3242:{\displaystyle 2.(6)}
3209:
3168:
3127:
3093:Other representations
3060:
3019:
2822:
2751:
2704:
2670:
2582:
2529:
2492:
2405:
2335:
2295:
2262:
2229:
2195:
2101:
2055:
2022:
1988:
1914:
1878:
1844:
1772:
1726:
1638:
1544:
1454:
1419:
1368:
1342:
1316:
1270:
1268:{\displaystyle ad=bc}
1238:
1188:
1151:Embedding of integers
1121:
1059:
961:that is derived from
869:
812:mathematical analysis
796:
762:
697:
663:
517:of pairs of integers
466:
372:
318:
272:
221:
165:
136:
107:
78:
49:
30:The rational numbers
29:
8359:Sets of real numbers
8154:Supernatural numbers
8064:Multicomplex numbers
8037:
8021:Dual-complex numbers
7960:
7921:
7883:
7855:
7827:
7799:
7781:Composition algebras
7749:Arithmetical numbers
7720:
7687:
7654:
7626:
7598:
7379:Constructible number
6906:Discrete Mathematics
6770:Lass, Harry (2009).
6672:Algebraic irrational
6525:
6487:
6449:
6411:
6373:
6272:
6229:
6178:
6147:
6067:
5965:
5930:
5885:
5812:
5773:
5694:
5670:
5654:totally disconnected
5652:. The space is also
5610:
5548:
5494:The rationals are a
5349:
5320:
5277:
5266:The rationals are a
5236:
5203:
5170:
5140:
5106:
5075:
5041:
5008:
4976:
4753:
4679:
4636:
4470:
4436:
4368:
4294:
4225:
4152:
4064:
4034:
3892:
3735:
3600:
3591:equivalence relation
3522:
3516:More precisely, let
3458:
3410:
3355:
3312:
3258:
3224:
3188:
3145:
3107:
3040:
2853:
2775:
2717:
2684:
2604:
2543:
2506:
2429:
2348:
2312:
2275:
2242:
2209:
2131:
2068:
2035:
2002:
1936:
1894:
1858:
1794:
1752:
1669:
1572:
1478:
1428:
1388:
1351:
1325:
1285:
1247:
1207:
1165:
1098:
1036:
1029:irreducible fraction
1023:Irreducible fraction
856:
783:
749:
677:
543:
536:defined as follows:
534:equivalence relation
451:
407:9/44 = 0.20454545...
356:
284:
252:
201:
152:
123:
94:
65:
56:are included in the
36:
8344:Field (mathematics)
8059:Split-biquaternions
7771:Eisenstein integers
7709:Closed-form numbers
7505:Supersilver ratio (
7496:Supergolden ratio (
7270:Weisstein, Eric W.
6854:Weisstein, Eric W.
6819:Extract of page 104
6789:Extract of page 382
6363:
6260:Ostrowski's theorem
5542:absolute difference
5335:{\displaystyle b,d}
4619:. It is called the
4026:congruence relation
3503:equivalence classes
3489:Formal construction
3475:{\displaystyle 3'6}
3079:Euclidean algorithm
2978:
2966:
2952:
2940:
2919:
2907:
2886:
2874:
1783:, often called its
1366:{\displaystyle b=d}
1340:{\displaystyle a=c}
965:: the first use of
912:rational expression
908:rational polynomial
737:characteristic zero
727:which contains the
515:equivalence classes
8217:Profinite integers
8180:Irrational numbers
8045:
7968:
7929:
7891:
7863:
7835:
7807:
7764:Gaussian rationals
7744:Computable numbers
7730:
7695:
7662:
7634:
7606:
7399:Eisenstein integer
7259:Extract of page 26
7191:(Technical report)
6537:
6499:
6461:
6423:
6385:
6359:
6339:Rational data type
6280:
6247:
6222:-adic number field
6205:
6158:
6124:
6047:
5946:
5914:
5854:
5781:
5737:
5678:
5621:
5594:
5442:
5407:
5332:
5303:
5247:
5214:
5192:field of fractions
5178:
5151:
5114:
5086:
5049:
5030:field automorphism
5016:
4984:
4952:
4950:
4733:
4654:
4649:
4573:
4451:
4449:
4417:
4348:
4274:
4170:
4165:
4133:field of fractions
4115:
4042:
4011:
3877:
3715:
3562:
3499:
3472:
3439:
3391:
3389:
3374:
3336:
3296:
3294:
3290:
3252:continued fraction
3239:
3204:
3163:
3161:
3122:
3120:
3055:
3053:
3014:
3007:
3002:
2997:
2992:
2973:
2947:
2914:
2881:
2838:Continued fraction
2817:
2812:
2746:
2744:
2699:
2697:
2665:
2577:
2524:
2519:
2487:
2400:
2330:
2325:
2290:
2288:
2257:
2255:
2224:
2222:
2190:
2161:
2148:
2096:
2091:
2050:
2048:
2017:
2015:
1983:
1925:, also called its
1909:
1907:
1873:
1871:
1839:
1767:
1765:
1736:reducible fraction
1721:
1633:
1539:
1449:
1414:
1363:
1337:
1311:
1265:
1233:
1183:
1178:
1116:
1111:
1054:
1049:
998:, in the sense of
937:Although nowadays
864:
791:
757:
735:, and a field has
692:
690:
658:
461:
367:
335:field of rationals
313:
311:
267:
265:
216:
214:
174:
160:
131:
102:
73:
44:
8326:
8325:
8253:
8252:
8164:Superreal numbers
8144:Levi-Civita field
8139:Hyperreal numbers
8083:Spacetime algebra
8069:Geometric algebra
7982:Bicomplex numbers
7948:Split-quaternions
7789:Division algebras
7759:Gaussian integers
7681:Algebraic numbers
7584:definable numbers
7540:
7539:
7514:Twelfth root of 2
7394:Doubling the cube
7384:Conway's constant
7369:Algebraic integer
7358:Algebraic numbers
7305:"Rational number"
7276:Wolfram MathWorld
7253:978-1-4419-8732-7
7229:Extract of page 1
7223:978-1-4987-1752-6
6929:Gilbert, Jimmie;
6915:978-0-19-871369-2
6860:Wolfram MathWorld
6856:"Rational Number"
6831:"Rational number"
6813:978-0-8218-0575-6
6783:978-0-486-47186-0
6756:978-0-07-288008-3
6726:
6725:
6722:
6721:
6718:
6717:
6714:
6713:
6703:
6702:
6699:
6698:
6695:
6694:
6691:
6690:
6679:Irrational period
6653:
6652:
6649:
6648:
6645:
6644:
6641:
6640:
6634:Repeating decimal
6601:
6600:
6597:
6596:
6592:Negative integers
6586:
6585:
6582:
6581:
6577:Composite numbers
6324:Gaussian rational
6172:The metric space
6042:
5981:
5941:
5690:under the metric
5635:topological field
5534:subspace topology
5469:Stern–Brocot tree
5402:
5389:
5360:
5301:
5288:
5261:algebraic numbers
5228:algebraic closure
4896:
4854:
4796:
4731:
4704:
4648:
4562:
4539:
4526:
4503:
4448:
4346:
4319:
4287:This means that
4164:
4131:and produces its
3582:of integers such
3388:
3373:
3349:Egyptian fraction
3293:
3289:
3202:
3178:repeating decimal
3160:
3119:
3052:
3009:
3004:
2999:
2994:
2977:
2965:
2951:
2939:
2918:
2906:
2885:
2873:
2811:
2743:
2696:
2660:
2620:
2559:
2518:
2482:
2445:
2395:
2382:
2369:
2324:
2287:
2254:
2221:
2185:
2162:
2160:
2147:
2090:
2047:
2014:
1978:
1952:
1906:
1870:
1834:
1812:
1764:
1716:
1693:
1680:
1628:
1596:
1583:
1534:
1502:
1489:
1412:
1399:
1309:
1296:
1231:
1218:
1177:
1110:
1048:
916:rational function
805:algebraic numbers
775:algebraic closure
689:
459:
387:decimal expansion
310:
264:
213:
8366:
8354:Rational numbers
8287:Rational numbers
8280:
8273:
8266:
8257:
8256:
8243:
8242:
8210:
8200:
8112:Cardinal numbers
8073:Clifford algebra
8054:
8052:
8051:
8046:
8044:
8016:Dual quaternions
7977:
7975:
7974:
7969:
7967:
7938:
7936:
7935:
7930:
7928:
7900:
7898:
7897:
7892:
7890:
7872:
7870:
7869:
7864:
7862:
7844:
7842:
7841:
7836:
7834:
7816:
7814:
7813:
7808:
7806:
7739:
7737:
7736:
7731:
7729:
7728:
7704:
7702:
7701:
7696:
7694:
7671:
7669:
7668:
7663:
7661:
7648:Rational numbers
7643:
7641:
7640:
7635:
7633:
7615:
7613:
7612:
7607:
7605:
7567:
7560:
7553:
7544:
7543:
7531:
7530:
7508:
7499:
7491:Square root of 7
7486:Square root of 6
7481:Square root of 5
7476:Square root of 3
7471:Square root of 2
7464:
7460:
7431:
7412:
7404:Gaussian integer
7389:Cyclotomic field
7351:
7344:
7337:
7328:
7327:
7318:
7286:
7285:
7283:
7282:
7267:
7261:
7257:
7237:
7231:
7227:
7207:
7201:
7200:
7198:
7196:
7190:
7179:
7173:
7172:
7164:
7158:
7157:
7137:
7131:
7130:
7128:
7127:
7113:
7090:
7089:
7081:
7075:
7074:
7072:
7071:
7060:
7054:
7053:
7051:
7050:
7032:
7026:
7012:
7005:
6999:
6985:
6978:
6972:
6962:
6955:
6949:
6948:
6926:
6920:
6919:
6901:
6870:
6869:
6867:
6866:
6851:
6845:
6844:
6842:
6841:
6827:
6821:
6817:
6797:
6791:
6787:
6767:
6761:
6760:
6742:
6668:
6667:
6659:
6658:
6616:
6615:
6607:
6606:
6550:
6549:
6546:
6544:
6543:
6538:
6536:
6516:
6515:
6512:
6511:
6508:
6506:
6505:
6500:
6498:
6478:
6477:
6474:
6473:
6470:
6468:
6467:
6462:
6460:
6440:
6439:
6436:
6435:
6432:
6430:
6429:
6424:
6422:
6402:
6401:
6398:
6397:
6394:
6392:
6391:
6386:
6384:
6364:
6358:
6355:
6354:
6351:
6350:
6299:absolute value.
6296:
6291:
6289:
6287:
6286:
6281:
6279:
6258:
6256:
6254:
6253:
6248:
6243:
6242:
6237:
6221:
6216:
6214:
6212:
6211:
6206:
6201:
6200:
6188:
6169:
6167:
6165:
6164:
6159:
6154:
6133:
6131:
6130:
6125:
6123:
6122:
6117:
6102:
6079:
6078:
6056:
6054:
6053:
6048:
6043:
6041:
6040:
6039:
6034:
6025:
6019:
6018:
6017:
6012:
6003:
5997:
5992:
5991:
5986:
5982:
5974:
5957:
5955:
5953:
5952:
5947:
5942:
5934:
5923:
5921:
5920:
5915:
5907:
5906:
5901:
5892:
5881:In addition set
5877:
5871:
5867:
5863:
5861:
5860:
5855:
5850:
5849:
5834:
5833:
5828:
5819:
5807:
5799:
5792:
5790:
5788:
5787:
5782:
5780:
5762:
5746:
5744:
5743:
5738:
5736:
5722:
5689:
5687:
5685:
5684:
5679:
5677:
5646:metrizable space
5632:
5630:
5628:
5627:
5622:
5617:
5603:
5601:
5600:
5595:
5590:
5576:
5484:Lebesgue measure
5465:Calkin–Wilf tree
5426:order isomorphic
5416:
5414:
5413:
5408:
5403:
5395:
5390:
5388:
5377:
5366:
5361:
5353:
5341:
5339:
5338:
5333:
5312:
5310:
5309:
5304:
5302:
5294:
5289:
5281:
5258:
5256:
5254:
5253:
5248:
5243:
5225:
5223:
5221:
5220:
5215:
5210:
5189:
5187:
5185:
5184:
5179:
5177:
5162:
5160:
5158:
5157:
5152:
5147:
5125:
5123:
5121:
5120:
5115:
5113:
5097:
5095:
5093:
5092:
5087:
5082:
5060:
5058:
5056:
5055:
5050:
5048:
5027:
5025:
5023:
5022:
5017:
5015:
4995:
4993:
4991:
4990:
4985:
4983:
4961:
4959:
4958:
4953:
4951:
4941:
4940:
4931:
4930:
4918:
4917:
4908:
4907:
4897:
4894:
4885:
4884:
4875:
4874:
4859:
4855:
4852:
4848:
4841:
4840:
4831:
4830:
4818:
4817:
4808:
4807:
4797:
4794:
4785:
4784:
4775:
4774:
4759:
4742:
4740:
4739:
4734:
4732:
4730:
4729:
4720:
4719:
4710:
4705:
4703:
4702:
4693:
4692:
4683:
4665:
4663:
4661:
4660:
4655:
4650:
4641:
4629:
4618:
4607:
4603:
4599:
4582:
4580:
4579:
4574:
4563:
4561:
4553:
4545:
4540:
4532:
4527:
4525:
4517:
4509:
4504:
4502:
4491:
4480:
4462:
4460:
4458:
4457:
4452:
4450:
4441:
4426:
4424:
4423:
4418:
4413:
4412:
4403:
4402:
4390:
4389:
4380:
4379:
4357:
4355:
4354:
4349:
4347:
4345:
4344:
4335:
4334:
4325:
4320:
4318:
4317:
4308:
4307:
4298:
4283:
4281:
4280:
4275:
4270:
4269:
4260:
4259:
4247:
4246:
4237:
4236:
4217:
4199:
4181:
4179:
4177:
4176:
4171:
4166:
4157:
4145:
4126:
4124:
4122:
4121:
4116:
4108:
4085:
4074:
4053:
4051:
4049:
4048:
4043:
4041:
4020:
4018:
4017:
4012:
4004:
4003:
3994:
3993:
3981:
3980:
3971:
3970:
3952:
3951:
3939:
3938:
3920:
3919:
3907:
3906:
3886:
3884:
3883:
3878:
3870:
3869:
3860:
3859:
3847:
3846:
3837:
3836:
3824:
3823:
3814:
3813:
3795:
3794:
3782:
3781:
3763:
3762:
3750:
3749:
3724:
3722:
3721:
3716:
3711:
3710:
3701:
3700:
3688:
3687:
3678:
3677:
3660:
3659:
3647:
3646:
3628:
3627:
3615:
3614:
3588:
3581:
3573:
3571:
3569:
3568:
3563:
3543:
3532:
3481:
3479:
3478:
3473:
3468:
3448:
3446:
3445:
3440:
3438:
3437:
3422:
3421:
3400:
3398:
3397:
3392:
3390:
3381:
3375:
3366:
3345:
3343:
3342:
3339:{\displaystyle }
3337:
3305:
3303:
3302:
3297:
3295:
3292:
3291:
3282:
3269:
3248:
3246:
3245:
3240:
3213:
3211:
3210:
3205:
3203:
3195:
3174:
3172:
3170:
3169:
3164:
3162:
3153:
3133:
3131:
3129:
3128:
3123:
3121:
3112:
3088:
3076:
3066:
3064:
3062:
3061:
3056:
3054:
3045:
3033:
3023:
3021:
3020:
3015:
3010:
3008:
3006:
3005:
3003:
3001:
3000:
2998:
2996:
2995:
2993:
2991:
2990:
2989:
2974:
2972:
2962:
2948:
2946:
2936:
2931:
2930:
2915:
2913:
2903:
2898:
2897:
2882:
2880:
2870:
2865:
2864:
2828:
2826:
2824:
2823:
2818:
2813:
2810:
2809:
2808:
2795:
2794:
2793:
2780:
2768:
2764:
2757:
2755:
2753:
2752:
2747:
2745:
2742:
2741:
2732:
2731:
2722:
2710:
2708:
2706:
2705:
2700:
2698:
2689:
2674:
2672:
2671:
2666:
2661:
2659:
2658:
2649:
2648:
2639:
2634:
2633:
2625:
2621:
2613:
2596:
2586:
2584:
2583:
2578:
2570:
2569:
2564:
2560:
2552:
2536:In particular,
2535:
2533:
2531:
2530:
2525:
2520:
2511:
2496:
2494:
2493:
2488:
2483:
2481:
2480:
2471:
2470:
2461:
2456:
2455:
2450:
2446:
2438:
2421:
2409:
2407:
2406:
2401:
2396:
2388:
2383:
2375:
2370:
2368:
2360:
2352:
2341:
2339:
2337:
2336:
2331:
2326:
2317:
2301:
2299:
2297:
2296:
2291:
2289:
2280:
2268:
2266:
2264:
2263:
2258:
2256:
2247:
2235:
2233:
2231:
2230:
2225:
2223:
2214:
2199:
2197:
2196:
2191:
2186:
2184:
2176:
2168:
2163:
2153:
2151:
2149:
2140:
2135:
2123:
2111:
2107:
2105:
2103:
2102:
2097:
2092:
2089:
2081:
2073:
2061:
2059:
2057:
2056:
2051:
2049:
2040:
2028:
2026:
2024:
2023:
2018:
2016:
2007:
1992:
1990:
1989:
1984:
1979:
1971:
1966:
1965:
1957:
1953:
1945:
1920:
1918:
1916:
1915:
1910:
1908:
1899:
1884:
1882:
1880:
1879:
1874:
1872:
1863:
1848:
1846:
1845:
1840:
1835:
1830:
1822:
1817:
1813:
1805:
1781:additive inverse
1778:
1776:
1774:
1773:
1768:
1766:
1757:
1730:
1728:
1727:
1722:
1717:
1715:
1707:
1699:
1694:
1686:
1681:
1673:
1652:coprime integers
1649:
1642:
1640:
1639:
1634:
1629:
1627:
1619:
1602:
1597:
1589:
1584:
1576:
1558:coprime integers
1555:
1548:
1546:
1545:
1540:
1535:
1533:
1525:
1508:
1503:
1495:
1490:
1482:
1458:
1456:
1455:
1450:
1423:
1421:
1420:
1415:
1413:
1405:
1400:
1392:
1372:
1370:
1369:
1364:
1346:
1344:
1343:
1338:
1320:
1318:
1317:
1312:
1310:
1302:
1297:
1289:
1274:
1272:
1271:
1266:
1242:
1240:
1239:
1234:
1232:
1224:
1219:
1211:
1194:
1192:
1190:
1189:
1184:
1179:
1170:
1158:
1146:
1135:
1131:
1127:
1125:
1123:
1122:
1117:
1112:
1103:
1084:
1076:coprime integers
1073:
1069:
1065:
1063:
1061:
1060:
1055:
1050:
1041:
1007:
1006:
986:
985:
939:rational numbers
875:
873:
871:
870:
865:
863:
824:Cauchy sequences
803:is the field of
802:
800:
798:
797:
792:
790:
768:
766:
764:
763:
758:
756:
711:
703:
701:
699:
698:
693:
691:
682:
667:
665:
664:
659:
654:
653:
644:
643:
631:
630:
621:
620:
603:
602:
590:
589:
571:
570:
558:
557:
531:
524:
493:
485:
478:
474:
472:
470:
468:
467:
462:
460:
455:
442:square root of 2
408:
401:the same finite
396:
378:
376:
374:
373:
368:
363:
346:
324:
322:
320:
319:
314:
312:
306:
298:
278:
276:
274:
273:
268:
266:
257:
245:
238:
227:
225:
223:
222:
217:
215:
206:
171:
169:
167:
166:
161:
159:
142:
140:
138:
137:
132:
130:
113:
111:
109:
108:
103:
101:
84:
82:
80:
79:
74:
72:
55:
53:
51:
50:
45:
43:
8374:
8373:
8369:
8368:
8367:
8365:
8364:
8363:
8329:
8328:
8327:
8322:
8308:Dyadic rational
8289:
8284:
8254:
8249:
8226:
8205:
8195:
8168:
8159:Surreal numbers
8149:Ordinal numbers
8094:
8040:
8038:
8035:
8034:
7996:
7963:
7961:
7958:
7957:
7955:
7953:Split-octonions
7924:
7922:
7919:
7918:
7910:
7904:
7886:
7884:
7881:
7880:
7858:
7856:
7853:
7852:
7830:
7828:
7825:
7824:
7821:Complex numbers
7802:
7800:
7797:
7796:
7775:
7724:
7723:
7721:
7718:
7717:
7690:
7688:
7685:
7684:
7657:
7655:
7652:
7651:
7629:
7627:
7624:
7623:
7601:
7599:
7596:
7595:
7592:Natural numbers
7577:
7571:
7541:
7536:
7525:
7518:
7506:
7497:
7465:
7462:
7458:
7442:Rational number
7429:
7428:Plastic ratio (
7410:
7374:Chebyshev nodes
7360:
7355:
7303:
7300:
7295:
7290:
7289:
7280:
7278:
7272:"p-adic Number"
7268:
7264:
7254:
7238:
7234:
7224:
7208:
7204:
7194:
7192:
7188:
7180:
7176:
7165:
7161:
7154:
7138:
7134:
7125:
7123:
7115:
7114:
7093:
7082:
7078:
7069:
7067:
7061:
7057:
7048:
7046:
7033:
7029:
7007:
7006:
7002:
6980:
6979:
6975:
6957:
6956:
6952:
6945:
6927:
6923:
6916:
6902:
6873:
6864:
6862:
6852:
6848:
6839:
6837:
6829:
6828:
6824:
6814:
6798:
6794:
6784:
6768:
6764:
6757:
6743:
6736:
6731:
6532:
6526:
6523:
6522:
6494:
6488:
6485:
6484:
6456:
6450:
6447:
6446:
6418:
6412:
6409:
6408:
6380:
6374:
6371:
6370:
6334:Niven's theorem
6309:Dyadic rational
6305:
6294:
6275:
6273:
6270:
6269:
6267:
6238:
6233:
6232:
6230:
6227:
6226:
6224:
6219:
6196:
6192:
6184:
6179:
6176:
6175:
6173:
6150:
6148:
6145:
6144:
6142:
6118:
6113:
6112:
6098:
6074:
6070:
6068:
6065:
6064:
6035:
6030:
6029:
6021:
6020:
6013:
6008:
6007:
5999:
5998:
5996:
5987:
5973:
5969:
5968:
5966:
5963:
5962:
5933:
5931:
5928:
5927:
5925:
5902:
5897:
5896:
5888:
5886:
5883:
5882:
5875:
5869:
5865:
5842:
5838:
5829:
5824:
5823:
5815:
5813:
5810:
5809:
5805:
5797:
5776:
5774:
5771:
5770:
5768:
5765:
5760:
5756:
5732:
5718:
5695:
5692:
5691:
5673:
5671:
5668:
5667:
5665:
5650:isolated points
5639:locally compact
5613:
5611:
5608:
5607:
5605:
5586:
5572:
5549:
5546:
5545:
5492:
5434:
5422:totally ordered
5394:
5378:
5367:
5365:
5352:
5350:
5347:
5346:
5321:
5318:
5317:
5293:
5280:
5278:
5275:
5274:
5268:densely ordered
5239:
5237:
5234:
5233:
5231:
5206:
5204:
5201:
5200:
5198:
5173:
5171:
5168:
5167:
5165:
5143:
5141:
5138:
5137:
5135:
5109:
5107:
5104:
5103:
5101:
5078:
5076:
5073:
5072:
5070:
5044:
5042:
5039:
5038:
5036:
5011:
5009:
5006:
5005:
5003:
4979:
4977:
4974:
4973:
4971:
4968:
4949:
4948:
4936:
4932:
4926:
4922:
4913:
4909:
4903:
4899:
4893:
4880:
4876:
4870:
4866:
4857:
4856:
4851:
4846:
4845:
4836:
4832:
4826:
4822:
4813:
4809:
4803:
4799:
4793:
4780:
4776:
4770:
4766:
4756:
4754:
4751:
4750:
4725:
4721:
4715:
4711:
4709:
4698:
4694:
4688:
4684:
4682:
4680:
4677:
4676:
4639:
4637:
4634:
4633:
4631:
4627:
4613:
4605:
4601:
4593:
4554:
4546:
4544:
4531:
4518:
4510:
4508:
4492:
4481:
4479:
4471:
4468:
4467:
4439:
4437:
4434:
4433:
4431:
4408:
4404:
4398:
4394:
4385:
4381:
4375:
4371:
4369:
4366:
4365:
4361:if and only if
4340:
4336:
4330:
4326:
4324:
4313:
4309:
4303:
4299:
4297:
4295:
4292:
4291:
4265:
4261:
4255:
4251:
4242:
4238:
4232:
4228:
4226:
4223:
4222:
4215:
4208:
4201:
4197:
4190:
4183:
4155:
4153:
4150:
4149:
4147:
4139:
4129:integral domain
4104:
4081:
4070:
4065:
4062:
4061:
4059:
4037:
4035:
4032:
4031:
4029:
3999:
3995:
3989:
3985:
3976:
3972:
3966:
3962:
3947:
3943:
3934:
3930:
3915:
3911:
3902:
3898:
3893:
3890:
3889:
3865:
3861:
3855:
3851:
3842:
3838:
3832:
3828:
3819:
3815:
3809:
3805:
3790:
3786:
3777:
3773:
3758:
3754:
3745:
3741:
3736:
3733:
3732:
3706:
3702:
3696:
3692:
3683:
3679:
3673:
3669:
3655:
3651:
3642:
3638:
3623:
3619:
3610:
3606:
3601:
3598:
3597:
3583:
3575:
3539:
3528:
3523:
3520:
3519:
3517:
3491:
3461:
3459:
3456:
3455:
3430:
3426:
3417:
3413:
3411:
3408:
3407:
3379:
3364:
3356:
3353:
3352:
3313:
3310:
3309:
3280:
3273:
3267:
3259:
3256:
3255:
3225:
3222:
3221:
3194:
3189:
3186:
3185:
3151:
3146:
3143:
3142:
3140:
3110:
3108:
3105:
3104:
3102:
3099:common fraction
3095:
3082:
3075:
3071:
3043:
3041:
3038:
3037:
3035:
3032:
3028:
2985:
2981:
2980:
2975:
2968:
2963:
2961:
2954:
2949:
2942:
2937:
2935:
2926:
2922:
2921:
2916:
2909:
2904:
2902:
2893:
2889:
2888:
2883:
2876:
2871:
2869:
2860:
2856:
2854:
2851:
2850:
2840:
2834:
2804:
2800:
2796:
2789:
2785:
2781:
2778:
2776:
2773:
2772:
2770:
2766:
2759:
2737:
2733:
2727:
2723:
2720:
2718:
2715:
2714:
2712:
2687:
2685:
2682:
2681:
2679:
2654:
2650:
2644:
2640:
2638:
2626:
2612:
2608:
2607:
2605:
2602:
2601:
2591:
2565:
2551:
2547:
2546:
2544:
2541:
2540:
2509:
2507:
2504:
2503:
2501:
2476:
2472:
2466:
2462:
2460:
2451:
2437:
2433:
2432:
2430:
2427:
2426:
2419:
2416:
2387:
2374:
2361:
2353:
2351:
2349:
2346:
2345:
2315:
2313:
2310:
2309:
2307:
2278:
2276:
2273:
2272:
2270:
2245:
2243:
2240:
2239:
2237:
2212:
2210:
2207:
2206:
2204:
2203:Thus, dividing
2177:
2169:
2167:
2138:
2136:
2134:
2132:
2129:
2128:
2121:
2118:
2109:
2082:
2074:
2071:
2069:
2066:
2065:
2063:
2038:
2036:
2033:
2032:
2030:
2005:
2003:
2000:
1999:
1997:
1970:
1958:
1944:
1940:
1939:
1937:
1934:
1933:
1897:
1895:
1892:
1891:
1889:
1861:
1859:
1856:
1855:
1853:
1823:
1821:
1804:
1800:
1795:
1792:
1791:
1755:
1753:
1750:
1749:
1747:
1744:
1708:
1700:
1698:
1685:
1672:
1670:
1667:
1666:
1660:
1647:
1620:
1603:
1601:
1588:
1575:
1573:
1570:
1569:
1566:
1553:
1526:
1509:
1507:
1494:
1481:
1479:
1476:
1475:
1469:
1429:
1426:
1425:
1424:if and only if
1404:
1391:
1389:
1386:
1385:
1379:
1352:
1349:
1348:
1326:
1323:
1322:
1321:if and only if
1301:
1288:
1286:
1283:
1282:
1248:
1245:
1244:
1243:if and only if
1223:
1210:
1208:
1205:
1204:
1201:
1168:
1166:
1163:
1162:
1160:
1156:
1153:
1141:
1133:
1129:
1101:
1099:
1096:
1095:
1093:
1079:
1071:
1067:
1039:
1037:
1034:
1033:
1031:
1025:
1020:
1014:
935:
900:rational matrix
859:
857:
854:
853:
851:
844:
786:
784:
781:
780:
778:
752:
750:
747:
746:
744:
705:
680:
678:
675:
674:
672:
649:
645:
639:
635:
626:
622:
616:
612:
598:
594:
585:
581:
566:
562:
553:
549:
544:
541:
540:
526:
518:
491:
481:
476:
454:
452:
449:
448:
446:
444:
406:
394:
359:
357:
354:
353:
351:
349:blackboard bold
342:
299:
296:
285:
282:
281:
280:
255:
253:
250:
249:
247:
246:. For example,
243:
239:and a non-zero
236:
204:
202:
199:
198:
196:
182:rational number
155:
153:
150:
149:
147:
145:natural numbers
126:
124:
121:
120:
118:
97:
95:
92:
91:
89:
87:complex numbers
68:
66:
63:
62:
60:
39:
37:
34:
33:
31:
24:
17:
12:
11:
5:
8372:
8362:
8361:
8356:
8351:
8346:
8341:
8324:
8323:
8321:
8320:
8315:
8310:
8305:
8300:
8294:
8291:
8290:
8283:
8282:
8275:
8268:
8260:
8251:
8250:
8248:
8247:
8237:
8235:Classification
8231:
8228:
8227:
8225:
8224:
8222:Normal numbers
8219:
8214:
8192:
8187:
8182:
8176:
8174:
8170:
8169:
8167:
8166:
8161:
8156:
8151:
8146:
8141:
8136:
8131:
8130:
8129:
8119:
8114:
8108:
8106:
8104:infinitesimals
8096:
8095:
8093:
8092:
8091:
8090:
8085:
8080:
8066:
8061:
8056:
8043:
8028:
8023:
8018:
8013:
8007:
8005:
7998:
7997:
7995:
7994:
7989:
7984:
7979:
7966:
7950:
7945:
7940:
7927:
7914:
7912:
7906:
7905:
7903:
7902:
7889:
7874:
7861:
7846:
7833:
7818:
7805:
7785:
7783:
7777:
7776:
7774:
7773:
7768:
7767:
7766:
7756:
7751:
7746:
7741:
7727:
7711:
7706:
7693:
7678:
7673:
7660:
7645:
7632:
7617:
7604:
7588:
7586:
7579:
7578:
7570:
7569:
7562:
7555:
7547:
7538:
7537:
7523:
7520:
7519:
7517:
7516:
7511:
7502:
7493:
7488:
7483:
7478:
7473:
7468:
7461:
7457:Silver ratio (
7454:
7449:
7444:
7439:
7434:
7425:
7420:
7415:
7409:Golden ratio (
7406:
7401:
7396:
7391:
7386:
7381:
7376:
7371:
7365:
7362:
7361:
7354:
7353:
7346:
7339:
7331:
7325:
7324:
7319:
7299:
7298:External links
7296:
7294:
7291:
7288:
7287:
7262:
7252:
7232:
7222:
7202:
7174:
7159:
7152:
7132:
7091:
7076:
7055:
7044:Stack Exchange
7038:(2017-05-09).
7027:
7000:
6973:
6950:
6943:
6931:Linda, Gilbert
6921:
6914:
6871:
6846:
6822:
6812:
6792:
6782:
6762:
6755:
6733:
6732:
6730:
6727:
6724:
6723:
6720:
6719:
6716:
6715:
6712:
6711:
6705:
6704:
6701:
6700:
6697:
6696:
6693:
6692:
6689:
6688:
6686:Transcendental
6682:
6681:
6675:
6674:
6665:
6655:
6654:
6651:
6650:
6647:
6646:
6643:
6642:
6639:
6638:
6636:
6630:
6629:
6623:
6622:
6620:Finite decimal
6613:
6603:
6602:
6599:
6598:
6595:
6594:
6588:
6587:
6584:
6583:
6580:
6579:
6573:
6572:
6566:
6565:
6558:
6557:
6547:
6535:
6530:
6509:
6497:
6492:
6471:
6459:
6454:
6433:
6421:
6416:
6395:
6383:
6378:
6361:Number systems
6349:
6348:
6341:
6336:
6331:
6326:
6321:
6316:
6314:Floating point
6311:
6304:
6301:
6278:
6264:absolute value
6246:
6241:
6236:
6204:
6199:
6195:
6191:
6187:
6183:
6157:
6153:
6135:
6134:
6121:
6116:
6111:
6108:
6105:
6101:
6097:
6094:
6091:
6088:
6085:
6082:
6077:
6073:
6058:
6057:
6046:
6038:
6033:
6028:
6024:
6016:
6011:
6006:
6002:
5995:
5990:
5985:
5980:
5977:
5972:
5945:
5940:
5937:
5913:
5910:
5905:
5900:
5895:
5891:
5853:
5848:
5845:
5841:
5837:
5832:
5827:
5822:
5818:
5779:
5758:Main article:
5755:
5749:
5735:
5731:
5728:
5725:
5721:
5717:
5714:
5711:
5708:
5705:
5702:
5699:
5676:
5620:
5616:
5593:
5589:
5585:
5582:
5579:
5575:
5571:
5568:
5565:
5562:
5559:
5556:
5553:
5530:order topology
5506:expansions as
5491:
5488:
5450:square lattice
5433:
5430:
5418:
5417:
5406:
5401:
5398:
5393:
5387:
5384:
5381:
5376:
5373:
5370:
5364:
5359:
5356:
5331:
5328:
5325:
5314:
5313:
5300:
5297:
5292:
5287:
5284:
5246:
5242:
5213:
5209:
5176:
5150:
5146:
5112:
5085:
5081:
5067:characteristic
5047:
5014:
4982:
4967:
4964:
4963:
4962:
4947:
4944:
4939:
4935:
4929:
4925:
4921:
4916:
4912:
4906:
4902:
4891:
4888:
4883:
4879:
4873:
4869:
4865:
4862:
4860:
4858:
4849:
4847:
4844:
4839:
4835:
4829:
4825:
4821:
4816:
4812:
4806:
4802:
4791:
4788:
4783:
4779:
4773:
4769:
4765:
4762:
4760:
4758:
4744:
4743:
4728:
4724:
4718:
4714:
4708:
4701:
4697:
4691:
4687:
4653:
4647:
4644:
4584:
4583:
4572:
4569:
4566:
4560:
4557:
4552:
4549:
4543:
4538:
4535:
4530:
4524:
4521:
4516:
4513:
4507:
4501:
4498:
4495:
4490:
4487:
4484:
4478:
4475:
4447:
4444:
4428:
4427:
4416:
4411:
4407:
4401:
4397:
4393:
4388:
4384:
4378:
4374:
4359:
4358:
4343:
4339:
4333:
4329:
4323:
4316:
4312:
4306:
4302:
4285:
4284:
4273:
4268:
4264:
4258:
4254:
4250:
4245:
4241:
4235:
4231:
4213:
4206:
4195:
4188:
4169:
4163:
4160:
4114:
4111:
4107:
4103:
4100:
4097:
4094:
4091:
4088:
4084:
4080:
4077:
4073:
4069:
4040:
4022:
4021:
4010:
4007:
4002:
3998:
3992:
3988:
3984:
3979:
3975:
3969:
3965:
3961:
3958:
3955:
3950:
3946:
3942:
3937:
3933:
3929:
3926:
3923:
3918:
3914:
3910:
3905:
3901:
3897:
3887:
3876:
3873:
3868:
3864:
3858:
3854:
3850:
3845:
3841:
3835:
3831:
3827:
3822:
3818:
3812:
3808:
3804:
3801:
3798:
3793:
3789:
3785:
3780:
3776:
3772:
3769:
3766:
3761:
3757:
3753:
3748:
3744:
3740:
3726:
3725:
3714:
3709:
3705:
3699:
3695:
3691:
3686:
3682:
3676:
3672:
3667:
3663:
3658:
3654:
3650:
3645:
3641:
3637:
3634:
3631:
3626:
3622:
3618:
3613:
3609:
3605:
3561:
3558:
3555:
3552:
3549:
3546:
3542:
3538:
3535:
3531:
3527:
3490:
3487:
3483:
3482:
3471:
3467:
3464:
3452:quote notation
3449:
3436:
3433:
3429:
3425:
3420:
3416:
3401:
3387:
3384:
3378:
3372:
3369:
3363:
3360:
3346:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3306:
3288:
3285:
3279:
3276:
3272:
3266:
3263:
3249:
3238:
3235:
3232:
3229:
3214:
3201:
3198:
3193:
3175:
3159:
3156:
3150:
3134:
3118:
3115:
3094:
3091:
3073:
3051:
3048:
3030:
3025:
3024:
3013:
2988:
2984:
2971:
2960:
2957:
2945:
2934:
2929:
2925:
2912:
2901:
2896:
2892:
2879:
2868:
2863:
2859:
2836:Main article:
2833:
2830:
2816:
2807:
2803:
2799:
2792:
2788:
2784:
2740:
2736:
2730:
2726:
2695:
2692:
2676:
2675:
2664:
2657:
2653:
2647:
2643:
2637:
2632:
2629:
2624:
2619:
2616:
2611:
2588:
2587:
2576:
2573:
2568:
2563:
2558:
2555:
2550:
2523:
2517:
2514:
2498:
2497:
2486:
2479:
2475:
2469:
2465:
2459:
2454:
2449:
2444:
2441:
2436:
2415:
2412:
2411:
2410:
2399:
2394:
2391:
2386:
2381:
2378:
2373:
2367:
2364:
2359:
2356:
2329:
2323:
2320:
2286:
2283:
2253:
2250:
2220:
2217:
2201:
2200:
2189:
2183:
2180:
2175:
2172:
2166:
2159:
2156:
2146:
2143:
2117:
2114:
2095:
2088:
2085:
2080:
2077:
2046:
2043:
2013:
2010:
1994:
1993:
1982:
1977:
1974:
1969:
1964:
1961:
1956:
1951:
1948:
1943:
1905:
1902:
1869:
1866:
1850:
1849:
1838:
1833:
1829:
1826:
1820:
1816:
1811:
1808:
1803:
1799:
1763:
1760:
1743:
1740:
1732:
1731:
1720:
1714:
1711:
1706:
1703:
1697:
1692:
1689:
1684:
1679:
1676:
1659:
1658:Multiplication
1656:
1644:
1643:
1632:
1626:
1623:
1618:
1615:
1612:
1609:
1606:
1600:
1595:
1592:
1587:
1582:
1579:
1565:
1562:
1550:
1549:
1538:
1532:
1529:
1524:
1521:
1518:
1515:
1512:
1506:
1501:
1498:
1493:
1488:
1485:
1468:
1465:
1460:
1459:
1448:
1445:
1442:
1439:
1436:
1433:
1411:
1408:
1403:
1398:
1395:
1378:
1375:
1374:
1373:
1362:
1359:
1356:
1336:
1333:
1330:
1308:
1305:
1300:
1295:
1292:
1276:
1275:
1264:
1261:
1258:
1255:
1252:
1230:
1227:
1222:
1217:
1214:
1200:
1197:
1182:
1176:
1173:
1152:
1149:
1115:
1109:
1106:
1087:canonical form
1053:
1047:
1044:
1024:
1021:
1013:
1010:
992:ancient Greeks
934:
931:
924:rational curve
892:rational point
862:
843:
840:
830:, or infinite
789:
755:
721:multiplication
688:
685:
669:
668:
657:
652:
648:
642:
638:
634:
629:
625:
619:
615:
610:
606:
601:
597:
593:
588:
584:
580:
577:
574:
569:
565:
561:
556:
552:
548:
458:
417:, such as the
366:
362:
309:
305:
302:
295:
292:
289:
263:
260:
212:
209:
158:
129:
100:
71:
42:
15:
9:
6:
4:
3:
2:
8371:
8360:
8357:
8355:
8352:
8350:
8347:
8345:
8342:
8340:
8337:
8336:
8334:
8319:
8316:
8314:
8311:
8309:
8306:
8304:
8301:
8299:
8296:
8295:
8292:
8288:
8281:
8276:
8274:
8269:
8267:
8262:
8261:
8258:
8246:
8238:
8236:
8233:
8232:
8229:
8223:
8220:
8218:
8215:
8212:
8208:
8202:
8198:
8193:
8191:
8188:
8186:
8185:Fuzzy numbers
8183:
8181:
8178:
8177:
8175:
8171:
8165:
8162:
8160:
8157:
8155:
8152:
8150:
8147:
8145:
8142:
8140:
8137:
8135:
8132:
8128:
8125:
8124:
8123:
8120:
8118:
8115:
8113:
8110:
8109:
8107:
8105:
8101:
8097:
8089:
8086:
8084:
8081:
8079:
8076:
8075:
8074:
8070:
8067:
8065:
8062:
8060:
8057:
8032:
8029:
8027:
8024:
8022:
8019:
8017:
8014:
8012:
8009:
8008:
8006:
8004:
7999:
7993:
7990:
7988:
7987:Biquaternions
7985:
7983:
7980:
7954:
7951:
7949:
7946:
7944:
7941:
7916:
7915:
7913:
7907:
7878:
7875:
7850:
7847:
7822:
7819:
7794:
7790:
7787:
7786:
7784:
7782:
7778:
7772:
7769:
7765:
7762:
7761:
7760:
7757:
7755:
7752:
7750:
7747:
7745:
7742:
7715:
7712:
7710:
7707:
7682:
7679:
7677:
7674:
7649:
7646:
7621:
7618:
7593:
7590:
7589:
7587:
7585:
7580:
7575:
7568:
7563:
7561:
7556:
7554:
7549:
7548:
7545:
7535:
7534:
7529:
7521:
7515:
7512:
7510:
7503:
7501:
7494:
7492:
7489:
7487:
7484:
7482:
7479:
7477:
7474:
7472:
7469:
7467:
7455:
7453:
7450:
7448:
7447:Root of unity
7445:
7443:
7440:
7438:
7435:
7433:
7426:
7424:
7421:
7419:
7418:Perron number
7416:
7414:
7407:
7405:
7402:
7400:
7397:
7395:
7392:
7390:
7387:
7385:
7382:
7380:
7377:
7375:
7372:
7370:
7367:
7366:
7363:
7359:
7352:
7347:
7345:
7340:
7338:
7333:
7332:
7329:
7323:
7320:
7316:
7312:
7311:
7306:
7302:
7301:
7277:
7273:
7266:
7260:
7255:
7249:
7245:
7244:
7236:
7230:
7225:
7219:
7215:
7214:
7206:
7187:
7186:
7178:
7170:
7163:
7155:
7153:0-2625-9020-4
7149:
7145:
7144:
7136:
7122:
7118:
7112:
7110:
7108:
7106:
7104:
7102:
7100:
7098:
7096:
7087:
7080:
7066:
7059:
7045:
7041:
7037:
7031:
7024:
7020:
7016:
7010:
7004:
6997:
6993:
6989:
6983:
6977:
6970:
6966:
6960:
6954:
6946:
6944:0-534-40264-X
6940:
6936:
6932:
6925:
6917:
6911:
6907:
6900:
6898:
6896:
6894:
6892:
6890:
6888:
6886:
6884:
6882:
6880:
6878:
6876:
6861:
6857:
6850:
6836:
6832:
6826:
6820:
6815:
6809:
6805:
6804:
6796:
6790:
6785:
6779:
6775:
6774:
6766:
6758:
6752:
6748:
6741:
6739:
6734:
6710:
6707:
6706:
6687:
6684:
6683:
6680:
6677:
6676:
6673:
6670:
6669:
6666:
6664:
6661:
6660:
6657:
6656:
6637:
6635:
6632:
6631:
6628:
6625:
6624:
6621:
6618:
6617:
6614:
6612:
6609:
6608:
6605:
6604:
6593:
6590:
6589:
6578:
6575:
6574:
6571:
6570:Prime numbers
6568:
6567:
6563:
6560:
6559:
6555:
6552:
6551:
6548:
6528:
6521:
6518:
6517:
6514:
6513:
6510:
6490:
6483:
6480:
6479:
6476:
6475:
6472:
6452:
6445:
6442:
6441:
6438:
6437:
6434:
6414:
6407:
6404:
6403:
6400:
6399:
6396:
6376:
6369:
6366:
6365:
6362:
6357:
6356:
6353:
6352:
6347:
6346:
6342:
6340:
6337:
6335:
6332:
6330:
6327:
6325:
6322:
6320:
6317:
6315:
6312:
6310:
6307:
6306:
6300:
6298:
6265:
6261:
6244:
6239:
6223:
6197:
6193:
6189:
6170:
6155:
6140:
6119:
6109:
6106:
6103:
6095:
6089:
6086:
6083:
6075:
6071:
6063:
6062:
6061:
6044:
6036:
6026:
6014:
6004:
5993:
5988:
5983:
5978:
5975:
5970:
5961:
5960:
5959:
5943:
5938:
5935:
5911:
5908:
5903:
5893:
5879:
5874:
5851:
5846:
5843:
5839:
5835:
5830:
5820:
5803:
5794:
5764:
5754:-adic numbers
5753:
5748:
5729:
5726:
5723:
5715:
5709:
5706:
5703:
5697:
5663:
5659:
5655:
5651:
5647:
5644:
5640:
5636:
5618:
5591:
5583:
5580:
5577:
5569:
5563:
5560:
5557:
5551:
5543:
5540:by using the
5539:
5535:
5531:
5526:
5524:
5520:
5516:
5513:In the usual
5511:
5509:
5505:
5501:
5497:
5487:
5485:
5481:
5477:
5472:
5470:
5466:
5461:
5459:
5455:
5451:
5447:
5438:
5429:
5427:
5423:
5404:
5399:
5396:
5391:
5385:
5382:
5379:
5374:
5371:
5368:
5362:
5357:
5354:
5345:
5344:
5343:
5329:
5326:
5323:
5298:
5295:
5290:
5285:
5282:
5273:
5272:
5271:
5269:
5264:
5262:
5244:
5229:
5211:
5197:
5193:
5163:
5148:
5133:
5129:
5128:ordered field
5098:
5083:
5068:
5064:
5034:
5031:
5001:
4999:
4945:
4937:
4933:
4927:
4923:
4919:
4914:
4910:
4904:
4900:
4889:
4886:
4881:
4877:
4871:
4867:
4861:
4837:
4833:
4827:
4823:
4819:
4814:
4810:
4804:
4800:
4789:
4786:
4781:
4777:
4771:
4767:
4761:
4749:
4748:
4747:
4726:
4722:
4716:
4712:
4706:
4699:
4695:
4689:
4685:
4675:
4674:
4673:
4671:
4666:
4651:
4645:
4642:
4624:
4622:
4616:
4611:
4597:
4591:
4590:
4570:
4567:
4564:
4558:
4555:
4550:
4547:
4541:
4536:
4533:
4528:
4522:
4519:
4514:
4511:
4505:
4499:
4496:
4493:
4488:
4485:
4482:
4476:
4473:
4466:
4465:
4464:
4445:
4442:
4414:
4409:
4405:
4399:
4395:
4391:
4386:
4382:
4376:
4372:
4364:
4363:
4362:
4341:
4337:
4331:
4327:
4321:
4314:
4310:
4304:
4300:
4290:
4289:
4288:
4271:
4266:
4262:
4256:
4252:
4248:
4243:
4239:
4233:
4229:
4221:
4220:
4219:
4212:
4205:
4194:
4187:
4167:
4161:
4158:
4143:
4136:
4134:
4130:
4112:
4109:
4105:
4092:
4075:
4057:
4027:
4008:
4000:
3996:
3990:
3986:
3982:
3977:
3973:
3967:
3963:
3956:
3948:
3944:
3940:
3935:
3931:
3924:
3916:
3912:
3908:
3903:
3899:
3888:
3874:
3866:
3862:
3856:
3852:
3848:
3843:
3839:
3833:
3829:
3825:
3820:
3816:
3810:
3806:
3799:
3791:
3787:
3783:
3778:
3774:
3767:
3759:
3755:
3751:
3746:
3742:
3731:
3730:
3729:
3712:
3707:
3703:
3697:
3693:
3689:
3684:
3680:
3674:
3670:
3656:
3652:
3648:
3643:
3639:
3632:
3624:
3620:
3616:
3611:
3607:
3596:
3595:
3594:
3592:
3586:
3579:
3550:
3533:
3514:
3512:
3508:
3507:ordered pairs
3504:
3495:
3486:
3469:
3465:
3462:
3453:
3450:
3434:
3431:
3427:
3423:
3418:
3414:
3405:
3402:
3385:
3382:
3376:
3370:
3367:
3361:
3358:
3350:
3347:
3330:
3327:
3324:
3321:
3318:
3307:
3286:
3283:
3277:
3274:
3270:
3264:
3261:
3253:
3250:
3233:
3227:
3219:
3215:
3196:
3191:
3183:
3179:
3176:
3157:
3154:
3148:
3138:
3137:mixed numeral
3135:
3116:
3113:
3100:
3097:
3096:
3090:
3086:
3080:
3070:
3049:
3046:
3011:
2986:
2982:
2969:
2958:
2955:
2943:
2932:
2927:
2923:
2910:
2899:
2894:
2890:
2877:
2866:
2861:
2857:
2849:
2848:
2847:
2845:
2839:
2829:
2814:
2805:
2801:
2797:
2790:
2786:
2782:
2762:
2738:
2734:
2728:
2724:
2693:
2690:
2662:
2655:
2651:
2645:
2641:
2635:
2630:
2627:
2622:
2617:
2614:
2609:
2600:
2599:
2598:
2594:
2574:
2571:
2566:
2561:
2556:
2553:
2548:
2539:
2538:
2537:
2521:
2515:
2512:
2484:
2477:
2473:
2467:
2463:
2457:
2452:
2447:
2442:
2439:
2434:
2425:
2424:
2423:
2397:
2392:
2389:
2384:
2379:
2376:
2371:
2365:
2362:
2357:
2354:
2344:
2343:
2342:
2327:
2321:
2318:
2305:
2284:
2281:
2251:
2248:
2218:
2215:
2187:
2181:
2178:
2173:
2170:
2164:
2157:
2154:
2144:
2141:
2127:
2126:
2125:
2113:
2093:
2086:
2083:
2078:
2075:
2044:
2041:
2011:
2008:
1980:
1975:
1972:
1967:
1962:
1959:
1954:
1949:
1946:
1941:
1932:
1931:
1930:
1928:
1924:
1903:
1900:
1886:
1867:
1864:
1836:
1831:
1827:
1824:
1818:
1814:
1809:
1806:
1801:
1797:
1790:
1789:
1788:
1786:
1782:
1761:
1758:
1739:
1737:
1718:
1712:
1709:
1704:
1701:
1695:
1690:
1687:
1682:
1677:
1674:
1665:
1664:
1663:
1655:
1653:
1630:
1624:
1621:
1616:
1613:
1610:
1607:
1604:
1598:
1593:
1590:
1585:
1580:
1577:
1568:
1567:
1561:
1559:
1536:
1530:
1527:
1522:
1519:
1516:
1513:
1510:
1504:
1499:
1496:
1491:
1486:
1483:
1474:
1473:
1472:
1464:
1446:
1443:
1440:
1437:
1434:
1431:
1409:
1406:
1401:
1396:
1393:
1384:
1383:
1382:
1360:
1357:
1354:
1334:
1331:
1328:
1306:
1303:
1298:
1293:
1290:
1281:
1280:
1279:
1262:
1259:
1256:
1253:
1250:
1228:
1225:
1220:
1215:
1212:
1203:
1202:
1196:
1180:
1174:
1171:
1148:
1144:
1139:
1113:
1107:
1104:
1090:
1088:
1082:
1077:
1051:
1045:
1042:
1030:
1019:
1009:
1001:
997:
993:
988:
980:
976:
972:
968:
964:
960:
956:
952:
948:
944:
940:
930:
928:
925:
921:
917:
913:
909:
905:
901:
897:
893:
889:
885:
881:
880:
849:
839:
837:
833:
829:
828:Dedekind cuts
825:
821:
817:
813:
808:
806:
776:
772:
742:
738:
734:
730:
726:
722:
718:
713:
709:
686:
683:
671:The fraction
655:
650:
646:
640:
636:
632:
627:
623:
617:
613:
599:
595:
591:
586:
582:
575:
567:
563:
559:
554:
550:
539:
538:
537:
535:
529:
522:
516:
512:
507:
505:
501:
497:
489:
484:
479:
456:
443:
439:
435:
430:
428:
424:
420:
416:
412:
404:
400:
392:
388:
384:
379:
364:
350:
345:
340:
336:
332:
331:the rationals
328:
307:
303:
300:
293:
290:
287:
261:
258:
242:
235:
231:
210:
207:
195:
191:
187:
183:
179:
146:
117:
88:
59:
28:
22:
8313:Half-integer
8303:Dedekind cut
8286:
8206:
8196:
8011:Dual numbers
8003:hypercomplex
7793:Real numbers
7647:
7524:
7452:Salem number
7441:
7308:
7279:. Retrieved
7275:
7265:
7242:
7235:
7212:
7205:
7193:. Retrieved
7184:
7177:
7168:
7162:
7142:
7135:
7124:. Retrieved
7120:
7085:
7079:
7068:. Retrieved
7058:
7047:. Retrieved
7043:
7030:
7022:
7018:
7014:
7008:
7003:
6998:, sense 5.a.
6995:
6991:
6987:
6981:
6976:
6971:, sense 2.a.
6968:
6964:
6958:
6953:
6934:
6924:
6905:
6863:. Retrieved
6859:
6849:
6838:. Retrieved
6834:
6825:
6802:
6795:
6772:
6765:
6746:
6443:
6343:
6319:Ford circles
6171:
6136:
6059:
5880:
5802:prime number
5795:
5766:
5763:-adic number
5751:
5662:real numbers
5538:metric space
5527:
5512:
5500:real numbers
5496:dense subset
5493:
5473:
5462:
5443:
5432:Countability
5419:
5315:
5265:
5164:
5099:
5035:
5002:
4969:
4745:
4667:
4625:
4614:
4595:
4587:
4585:
4429:
4360:
4286:
4210:
4203:
4192:
4185:
4141:
4137:
4056:quotient set
4023:
3727:
3584:
3577:
3515:
3500:
3484:
3084:
3069:coefficients
3026:
2843:
2841:
2760:
2677:
2592:
2589:
2499:
2417:
2202:
2119:
1995:
1926:
1887:
1851:
1784:
1745:
1733:
1661:
1645:
1551:
1470:
1461:
1380:
1277:
1155:Any integer
1154:
1142:
1091:
1080:
1026:
999:
995:
989:
978:
974:
970:
966:
962:
958:
954:
946:
942:
938:
936:
926:
907:
899:
888:coefficients
883:
877:
847:
845:
816:dense subset
809:
714:
707:
670:
532:, using the
527:
520:
508:
488:golden ratio
431:
380:
343:
338:
334:
330:
181:
175:
58:real numbers
8173:Other types
7992:Bioctonions
7849:Quaternions
7036:Shor, Peter
5478:, that is,
5063:prime field
4670:total order
4146:is denoted
3218:parentheses
1564:Subtraction
1008:in Greek).
945:, the term
896:coordinates
842:Terminology
769:are called
733:prime field
513:defined as
500:uncountable
434:real number
423:hexadecimal
383:real number
241:denominator
178:mathematics
8333:Categories
8127:Projective
8100:Infinities
7281:2021-08-17
7126:2021-08-17
7070:2021-03-20
7049:2021-03-19
7025:, sense 3.
7015:irrational
6865:2020-08-11
6840:2020-08-11
6729:References
6663:Irrational
6137:defines a
5660:, and the
5523:closed set
5480:almost all
5132:isomorphic
4966:Properties
4182:Two pairs
2304:reciprocal
1927:reciprocal
1140:, and, if
1016:See also:
1012:Arithmetic
996:irrational
979:irrational
951:derivation
920:polynomial
820:completion
773:, and the
741:extensions
504:almost all
486:, and the
438:irrational
425:ones (see
395:3/4 = 0.75
393:(example:
8211:solenoids
8031:Sedenions
7877:Octonions
7315:EMS Press
7195:17 August
6992:a. (adv.)
6709:Imaginary
6107:−
5844:−
5727:−
5643:countable
5581:−
5446:countable
4920:≥
4820:≤
4707:≤
4568:⋯
4520:−
4512:−
4494:−
4483:−
4474:⋯
4110:∼
4087:∖
4076:×
3957:≡
3925:×
3800:≡
3666:⟺
3633:∼
3545:∖
3534:×
3432:−
3424:×
3200:¯
2956:⋱
2798:−
2783:−
2628:−
2385:⋅
2084:−
2076:−
1960:−
1825:−
1798:−
1683:⋅
1611:−
1586:−
1136:by their
1000:illogical
949:is not a
933:Etymology
846:The term
609:⟺
576:∼
496:countable
301:−
288:−
234:numerator
7620:Integers
7582:Sets of
6988:rational
6933:(2005).
6611:Fraction
6444:Rational
6303:See also
5958:we set
5873:dividing
5648:without
5519:open set
5515:topology
5476:null set
5458:non-zero
5452:as in a
5196:integers
4970:The set
3511:integers
3466:′
3182:vinculum
3180:using a
2116:Division
1785:opposite
1467:Addition
1377:Ordering
1199:Equality
975:rational
971:rational
963:rational
947:rational
884:rational
848:rational
832:decimals
822:, using
729:integers
717:addition
511:formally
403:sequence
230:integers
194:fraction
190:quotient
116:integers
8298:Integer
8201:numbers
8033: (
7879: (
7851: (
7823: (
7795: (
7716: (
7714:Periods
7683: (
7650: (
7622: (
7594: (
7576:systems
7317:, 2001
6520:Natural
6482:Integer
6368:Complex
6290:
6268:
6257:
6225:
6215:
6174:
6168:
6143:
5956:
5926:
5791:
5769:
5747:above.
5688:
5666:
5631:
5606:
5544:metric
5498:of the
5316:(where
5257:
5232:
5224:
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5194:of the
5190:is the
5188:
5166:
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5124:
5102:
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5071:
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5028:has no
5026:
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4461:
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4060:
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2534:
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2302:by the
2300:
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2267:
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2122:b, c, d
2106:
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1779:has an
1777:
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1193:
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1064:
1032:
914:" and "
874:
852:
801:
779:
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745:
723:form a
702:
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411:base 10
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352:
337:or the
333:", the
277:
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228:of two
226:
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8001:Other
7574:Number
7250:
7220:
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7013:Entry
6986:Entry
6963:Entry
6941:
6912:
6810:
6780:
6753:
6139:metric
6060:Then
5864:where
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5521:nor a
5504:finite
5126:is an
4617:> 0
4612:, and
3027:where
2763:> 0
1921:has a
1145:< 0
1083:> 0
1066:where
1005:ἄλογος
984:ἄλογος
943:ratios
927:is not
904:matrix
419:binary
399:repeat
391:digits
186:number
8209:-adic
8199:-adic
7956:Over
7917:Over
7911:types
7909:Split
7293:Notes
7189:(PDF)
6965:ratio
6297:-adic
5800:be a
5061:is a
4998:field
3589:. An
967:ratio
959:ratio
955:ratio
918:" (a
902:is a
879:ratio
834:(see
725:field
525:with
347:, or
184:is a
8245:List
8102:and
7248:ISBN
7218:ISBN
7197:2021
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7021:and
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6939:ISBN
6910:ISBN
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6564:: 1
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415:base
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2012:b
2009:a
1981:.
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1968:=
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