3391:
40:
306:
8761:
6827:
516:
Mathematicians have noted tendencies in which definition is used, such as algebra texts including 0, number theory and analysis texts excluding 0, logic and set theory texts including 0, dictionaries excluding 0, school books (through high-school level) excluding 0, and upper-division college-level
5071:
2710:
well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite,
588:
for natural numbers, thus stating they were not really natural—but a consequence of definitions. Later, two classes of such formal definitions emerged, using set theory and Peano's axioms respectively. Later still, they were shown to be equivalent in most practical applications.
2658:. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an
838:
6032:
2568:
A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. The numbering of cardinals usually begins at zero, to accommodate the
1232:
440:, for example, defined a unit first and then a number as a multitude of units, thus by his definition, a unit is not a number and there are no unique numbers (e.g., any two units from indefinitely many units is a 2). However, in the definition of
2832:
that can be stated and proved in Peano arithmetic can also be proved in set theory. However, the two definitions are not equivalent, as there are theorems that can be stated in terms of Peano arithmetic and proved in set theory, which are not
1117:
2852:(as it is usually guessed), then Peano arithmetic is consistent. In other words, if a contradiction could be proved in Peano arithmetic, then set theory would be contradictory, and every theorem of set theory would be both true and wrong.
948:
603:. To avoid such paradoxes, the formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.
3876:
A tablet found at Kish ... thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty
1304:
6044:
470:(natural progression) in 1484. The earliest known use of "natural number" as a complete English phrase is in 1763. The 1771 Encyclopaedia Britannica defines natural numbers in the logarithm article.
1373:
599:. He initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set. However, this definition turned out to lead to paradoxes, including
565:
stated that axioms can only be demonstrated in their finite application, and concluded that it is "the power of the mind" which allows conceiving of the indefinite repetition of the same act.
720:
or not, it may be important to know which version is referred to. This is often specified by the context, but may also be done by using a subscript or a superscript in the notation, such as:
3952:
1744:. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that
360:
system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one—its value being determined from context.
371:
in place-value notation (within other numbers) dates back as early as 700 BCE by the
Babylonians, who omitted such a digit when it would have been the last symbol in the number. The
2144:
1684:
727:
2966:: If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.
1875:
329:
for each object is another primitive method. Later, a set of objects could be tested for equality, excess or shortage—by striking out a mark and removing an object from the set.
2970:
These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of
1501:
1123:
3635:
3597:
3559:
3521:
3483:
6324:
705:
6639:
6562:
6523:
6485:
6457:
6429:
6401:
6289:
6256:
6228:
6200:
3978:
2764:
1790:
1764:
1334:
983:
163:
2648:
2590:
1013:
1614:
4294:
987:
they may be referred to as the positive, or the non-negative integers, respectively. To be unambiguous about whether 0 is included or not, sometimes a superscript "
3855:, p. x) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers."
3014:
845:
226:
3330:, the natural numbers may not form a set. Nevertheless, the natural numbers can still be individually defined as above, and they still satisfy the Peano axioms.
2988:
2957:
2937:
2917:
2897:
2662:(more than a bijection) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as
1005:
3997:
The
English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891–1892, 19, quoting from a lecture of Kronecker's of 1886."
5086:
6863:
4242:
3315:
It follows from the definition that each natural number is equal to the set of all natural numbers less than it. This definition, can be extended to the
43:
Natural numbers can be used for counting: one apple; two apples are one apple added to another apple, three apples are one apple added to two apples, ...
92:
refer to the natural numbers in common language, particularly in primary school education, and are similarly ambiguous although typically exclude zero.
618:
proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of
Dedekind's axioms in his book
241:
4612:
4268:
5055:
517:
books including 0. There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include
5014:
3404:
592:
4357:
4006:"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (
1245:
1811:
If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with
628:
5106:
5001:
4908:
4311:
4276:
4250:
3434:
3025:
2805:
is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set
314:
4396:
5564:
3379:
th element of a sequence) is immediate. Unlike von
Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.
2627:: "first", "second", "third", and so forth. The numbering of ordinals usually begins at zero, to accommodate the order type of the
6856:
6150:
2052:
While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of
4728:
5468:
4452:
549:
standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as
436:. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.
6076:
6018:
5997:
5976:
5951:
5930:
5906:
5896:
5878:
5856:
5832:
5808:
5787:
5763:
5737:
5693:
5669:
5648:
5627:
5448:
5395:
5370:
5345:
5065:
4974:
4880:
4818:
4776:
4510:
4431:
4390:
4164:
4766:
4170:
4031:, p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I)
6339:
5438:
4841:
561:
In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers.
4712:
4305:
2225:
The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:
1342:
352:
in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The
30:
This article is about "positive integers" and "non-negative integers". For all the numbers ..., −2, −1, 0, 1, 2, ..., see
7663:
6849:
477:
wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0. In 1889,
6334:
1770:
under subtraction (that is, subtracting one natural from another does not always result in another natural), means that
7658:
6037:
Acta
Litterarum AC Scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum
644:: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is
7673:
5310:"Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions"
5207:
4575:
4333:
17:
7653:
3905:
8366:
7946:
5556:
5488:
5100:
2719:
833:{\displaystyle \{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}}
6294:
2009:: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an
8790:
6712:
4552:
In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example
2678:
2650:. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any
2097:
660:
replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include
474:
80:
as the natural numbers together with zero, excluding zero from the natural numbers, while in other writings, the
6790:
1648:
7668:
5777:
5455:...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
4254:
4104:
8805:
8452:
6673:
6109:
5638:
5560:
2624:
5798:
8118:
7768:
7437:
7230:
6143:
573:
5504:
Fletcher, Peter; Hrbacek, Karel; Kanovei, Vladimir; Katz, Mikhail G.; Lobry, Claude; Sanders, Sam (2017).
5239:
Baratella, Stefano; Ferro, Ruggero (1993). "A theory of sets with the negation of the axiom of infinity".
5223:
4678:
3077:, although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to
1839:
8294:
8153:
7984:
7798:
7788:
7442:
7422:
6299:
6104:
4280:
3768:
1227:{\displaystyle \{0,1,2,\dots \}=\{x\in \mathbb {Z} :x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}}
8123:
1007:" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:
8243:
7866:
7708:
7623:
7432:
7414:
7308:
7298:
7288:
7124:
4538:
2845:
577:
8148:
4695:
4478:
3337:
provided a construction that is nowadays only of historical interest, and is sometimes referred to as
2430:
If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number
1472:
8785:
8371:
7916:
7537:
7323:
7318:
7313:
7303:
7280:
6707:
6663:
3614:
3576:
3538:
3500:
3462:
3051:
2814:
538:
279:
275:
8128:
6305:
6099:
4159:. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer).
336:
to represent numbers. This allowed systems to be developed for recording large numbers. The ancient
7793:
7703:
7356:
6830:
6702:
4349:
2838:
2730:
numbers are an uncountable model that can be constructed from the ordinary natural numbers via the
2556:
Two important generalizations of natural numbers arise from the two uses of counting and ordering:
1112:{\displaystyle \{1,2,3,\dots \}=\{x\in \mathbb {Z} :x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}}
685:
337:
6622:
6545:
6506:
6468:
6440:
6412:
6384:
6272:
6239:
6211:
6183:
3961:
2747:
1773:
1747:
1317:
966:
611:
146:
8482:
8447:
8233:
8143:
8017:
7992:
7901:
7891:
7613:
7503:
7485:
7405:
6136:
5114:
2731:
1943:
1687:
2633:
2575:
8800:
8742:
8012:
7886:
7517:
7293:
7073:
7000:
5968:
5140:
3323:, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."
2774:
There are two standard methods for formally defining natural numbers. The first one, named for
661:
607:
198:
4872:
4424:
The
Development of Mathematics Throughout the Centuries: A brief history in a cultural context
2600:
between them. The set of natural numbers itself, and any bijective image of it, is said to be
8706:
8346:
7997:
7851:
7778:
6933:
6775:
6611:
6064:
5191:
4896:
4419:
4380:
3775:
3152:
2229:
1767:
1599:
1519:
4864:
4595:
1737:. These properties of addition and multiplication make the natural numbers an instance of a
943:{\displaystyle \;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}}
8639:
8533:
8497:
8238:
7961:
7941:
7758:
7427:
7215:
6528:
6261:
5548:
5274:
Kirby, Laurie; Paris, Jeff (1982). "Accessible
Independence Results for Peano Arithmetic".
5260:
5176:
4744:
4643:
4543:
4501:
4483:
3887:
3086:
3059:
600:
585:
542:
530:
341:
300:
7718:
7187:
2848:
of Peano arithmetic inside set theory. An important consequence is that, if set theory is
344:
for 1, 10, and all powers of 10 up to over 1 million. A stone carving from
321:
The most primitive method of representing a natural number is to use one's fingers, as in
255:(and also the sums and products thereof); and so on. This chain of extensions canonically
8:
8361:
8225:
8220:
8188:
7951:
7926:
7921:
7896:
7826:
7822:
7753:
7643:
7475:
7271:
7240:
6739:
6649:
6606:
6588:
6366:
4662:
3316:
2993:
2735:
2722:
satisfying the Peano
Arithmetic (that is, the first-order Peano axioms) was developed by
2214:
2046:
1902:
1523:
1467:
506:
490:
357:
237:
203:
8764:
8518:
8513:
8427:
8401:
8299:
8278:
8050:
7931:
7881:
7803:
7773:
7713:
7480:
7460:
7391:
7104:
6644:
6356:
5683:
5517:
5487:
Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962).
5164:
4947:
3396:
2973:
2963:
2942:
2922:
2902:
2882:
2057:
1797:
1337:
990:
510:
387:. The use of a numeral 0 in modern times originated with the Indian mathematician
7648:
6117:
2610:
8795:
8760:
8658:
8603:
8457:
8432:
8406:
7861:
7856:
7783:
7763:
7748:
7470:
7452:
7371:
7361:
7346:
7109:
6802:
6765:
6729:
6668:
6654:
6349:
6329:
6072:
6014:
5993:
5972:
5947:
5926:
5916:
5902:
5888:
5874:
5852:
5828:
5804:
5783:
5759:
5733:
5689:
5665:
5644:
5623:
5444:
5391:
5366:
5341:
5291:
5203:
5196:
5061:
4876:
4865:
4814:
4772:
4571:
4516:
4506:
4427:
4386:
4329:
4193:
4160:
4100:
3985:
3752:
3723:
3390:
3148:
3047:
3035:
2798:
2659:
673:
657:
581:
566:
526:
396:
376:
176:
129:
76:
Some authors acknowledge both definitions whenever convenient. Some texts define the
8183:
5532:
5505:
5412:
4951:
4196:
2828:
The sets used to define natural numbers satisfy Peano axioms. It follows that every
562:
317:) is believed to have been used 20,000 years ago for natural number arithmetic.
8694:
8487:
8073:
8045:
8035:
8027:
7911:
7876:
7871:
7838:
7532:
7495:
7386:
7381:
7376:
7366:
7338:
7225:
7172:
7129:
7068:
6820:
6749:
6724:
6658:
6567:
6533:
6374:
6344:
6266:
6169:
6060:
6028:
5892:
5725:
5712:. Translated by Beman, Wooster Woodruff. Chicago, IL: Open Court Publishing Company
5703:
5679:
5527:
5506:"Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others"
5283:
5248:
5156:
4939:
4448:
4126:
4051:
gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1:
3798:
3761:
3681:
3666:
3074:
2783:
2779:
2398:
1511:
633:
615:
518:
498:
494:
180:
7177:
4966:
489:. Historically, most definitions have excluded 0, but many mathematicians such as
481:
used N for the positive integers and started at 1, but he later changed to using N
8670:
8559:
8492:
8418:
8341:
8315:
8133:
7846:
7638:
7608:
7598:
7593:
7259:
7167:
7114:
6958:
6898:
6697:
6601:
6233:
6008:
5987:
5962:
5941:
5920:
5846:
5822:
5818:
5753:
5659:
5617:
5256:
5172:
4904:
4591:
4152:
4065:
3716:
3533:
3428:
2557:
2446:. However, the "existence of additive identity element" property is not satisfied
2015:
463:
368:
322:
194:
139:
122:
101:
4871:(1. ed., 1. print ed.). Boca Raton, Fla. : Chapman & Hall/CRC. p.
4835:
4420:"Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations"
125:
in sports)—which do not have the properties of numbers in a mathematical sense.
8675:
8543:
8528:
8392:
8356:
8331:
8207:
8178:
8163:
8040:
7936:
7906:
7633:
7588:
7465:
7063:
7058:
7053:
7025:
7010:
6923:
6908:
6886:
6744:
6734:
6719:
6538:
6406:
3709:
3457:
3422:
3320:
3082:
2775:
2593:
2561:
2451:
2010:
1703:
1593:
645:
641:
637:
623:
569:
summarized his belief as "God made the integers, all else is the work of man".
478:
441:
425:
400:
333:
249:
245:
117:
111:
59:
0, 1, 2, 3, etc., possibly excluding 0. Some define the natural numbers as the
5029:
2592:. This concept of "size" relies on maps between sets, such that two sets have
8779:
8598:
8582:
8523:
8477:
8173:
8158:
8068:
7351:
7220:
7182:
7139:
7020:
7005:
6995:
6953:
6943:
6918:
6841:
6807:
6780:
6689:
5842:
5309:
5295:
5252:
4069:
3450:
3410:
3372:
3368:
3334:
3301:
2602:
2343:
2267:
1376:
596:
550:
283:
271:
5287:
5144:
4730:
Advanced
Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166
4520:
1690:
with identity element 1; a generator set for this monoid is the set of
39:
8634:
8623:
8538:
8376:
8351:
8268:
8168:
8138:
8113:
8097:
8002:
7969:
7692:
7603:
7542:
7119:
7015:
6948:
6928:
6903:
6770:
6572:
4272:
4246:
3659:
3234:
3078:
2861:
2787:
2741:
2727:
2655:
2507:
2210:
1691:
444:
which comes shortly afterward, Euclid treats 1 as a number like any other.
310:
267:
263:
5008:
4943:
8593:
8468:
8273:
7737:
7628:
7583:
7578:
7328:
7235:
7134:
6963:
6938:
6913:
6596:
6378:
5866:
5773:
5749:
5707:
5688:. Translated by Beman, Wooster Woodruff (reprint ed.). Dover Books.
5335:
5051:
4217:
3902:, p. 15) include zero in the natural numbers: 'Intuitively, the set
3495:
3305:
2844:
The definition of the integers as sets satisfying Peano axioms provide a
2607:
1909:
1738:
1515:
1504:
502:
452:
421:
388:
384:
367:
can be considered as a number, with its own numeral. The use of a 0
305:
233:
48:
3170:
It follows that the natural numbers are defined iteratively as follows:
2220:
1299:{\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}}
447:
Independent studies on numbers also occurred at around the same time in
8730:
8711:
8007:
7618:
6577:
6434:
5168:
3309:
3055:
2849:
2794:
2707:
2666:; this is also the ordinal number of the set of natural numbers itself.
2651:
2006:
649:
433:
429:
364:
326:
168:
4022:, pp. 117 ff) calls them "Peano's Postulates" and begins with "1.
1702:
Addition and multiplication are compatible, which is expressed in the
8336:
8263:
8255:
8060:
7974:
7092:
4765:
Křížek, Michal; Somer, Lawrence; Šolcová, Alena (21 September 2021).
4201:
3255:
3101:
2628:
2597:
2570:
2169:
546:
522:
391:
in 628 CE. However, 0 had been used as a number in the medieval
353:
256:
6007:
Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008).
5160:
4549:
A perfect number is that which is equal to the sum of its own parts.
4449:"Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius"
2873:
Every natural number has a successor which is also a natural number.
417:, the Latin word for "none", was employed to denote a 0 value.
171:
are built by successively extending the set of natural numbers: the
8437:
6685:
6616:
6462:
6069:
From Frege to Gödel: A source book in mathematical logic, 1879–1931
5522:
3700:
3416:
3327:
3159:. The intersection of all inductive sets is still an inductive set.
2744:
used to claim provocatively that "The naïve integers don't fill up
2712:
2157:
1801:
1741:
1697:
534:
392:
2209:. This Euclidean division is key to the several other properties (
473:
Starting at 0 or 1 has long been a matter of definition. In 1727,
8442:
8101:
8095:
6205:
6128:
5943:
Bridge to
Abstract Mathematics: Mathematical proof and structures
4840:(Winter 2014 ed.). The Stanford Encyclopedia of Philosophy.
3571:
3375:
is not directly accessible; only the ordinal property (being the
2829:
1527:
958:
172:
85:
31:
5961:
Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013).
6159:
5363:
Discrete and Combinatorial Mathematics: An applied introduction
5099:]. pp. 2:5–23. (The quote is on p. 19). Archived from
4534:
4474:
4299:
3291:
2723:
1507:
954:
437:
363:
A much later advance was the development of the idea that
349:
345:
95:
The natural numbers can be used for counting (as in "there are
56:
7157:
5194:. In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.).
3867:, p. 1): "Numbers make up the foundation of mathematics."
3333:
There are other set theoretical constructions. In particular,
3058:, as such an equivalence class would not be a set (because of
2005:
An important property of the natural numbers is that they are
4644:"Earliest Known Uses of Some of the Words of Mathematics (N)"
448:
399:
in 525 CE, without being denoted by a numeral. Standard
372:
115:. Natural numbers are sometimes used as labels—also known as
6071:(3rd ed.). Harvard University Press. pp. 346–354.
5964:
Mathematics for Elementary Teachers: A contemporary approach
5074:
from the original on 29 March 2017 – via Google Books.
3062:). The standard solution is to define a particular set with
1375:
sending each natural number to the next one, one can define
545:. Including 0 began to rise in popularity in the 1960s. The
3643:
2232:
under addition and multiplication: for all natural numbers
1526:. The smallest group containing the natural numbers is the
109:
largest city in the country"), in which case they serve as
6035:[On the Introduction of the Transfinite Numbers].
5503:
5057:
Plato's Ghost: The modernist transformation of mathematics
2060:
is available as a substitute: for any two natural numbers
953:
Alternatively, since the natural numbers naturally form a
232:(and also the product of these inverses by integers); the
5486:
4733:. United States Armed Forces Institute. 1958. p. 12.
3651:
3407: – Representation of a number as a product of primes
653:
105:. They may also be used for ordering (as in "this is the
4191:
2454:
of multiplication over addition for all natural numbers
1518:
on one generator. This commutative monoid satisfies the
395:(the calculation of the date of Easter), beginning with
6006:
5579:
4796:
4768:
From Great Discoveries in Number Theory to Applications
4185:
3371:. So, the property of the natural numbers to represent
3233:
It can be checked that the natural numbers satisfy the
3155:
under the successor function. Such sets are said to be
348:, dating back from around 1500 BCE and now at the
5097:
Annual report of the German Mathematicians Association
3893:
3419: – Function of the natural numbers in another set
3367:
With this definition each nonzero natural number is a
620:
The principles of arithmetic presented by a new method
340:
developed a powerful system of numerals with distinct
332:
The first major advance in abstraction was the use of
7821:
6625:
6548:
6509:
6471:
6443:
6415:
6387:
6308:
6275:
6242:
6214:
6186:
5390:(5th ed.). Boston: Addison-Wesley. p. 133.
4813:. Princeton: Princeton university press. p. 17.
4505:. Mineola, New York: Dover Publications. p. 58.
4500:
Philosophy of mathematics and deductive structure in
4382:
1491: New Revelations of the Americas before Columbus
4053:
An Axiomatization for the System of Positive Integers
3964:
3908:
3617:
3579:
3541:
3503:
3465:
3254:
elements" can be formally defined as "there exists a
2996:
2976:
2945:
2925:
2905:
2885:
2750:
2636:
2578:
2221:
Algebraic properties satisfied by the natural numbers
2100:
1842:
1776:
1750:
1651:
1602:
1592:
Analogously, given that addition has been defined, a
1475:
1345:
1320:
1248:
1126:
1016:
993:
969:
848:
730:
688:
274:. Problems concerning counting and ordering, such as
206:
149:
8206:
5960:
5109:
Jahresbericht der Deutschen Mathematiker-Vereinigung
5092:
Jahresbericht der Deutschen Mathematiker-Vereinigung
4295:"The Ishango Bone, Democratic Republic of the Congo"
3386:
1368:{\displaystyle S\colon \mathbb {N} \to \mathbb {N} }
576:
saw a need to improve upon the logical rigor in the
5985:
5040:]. Translated by Greenstreet, William John. VI.
4930:Brown, Jim (1978). "In defense of index origin 0".
3984:; ...'. They follow that with their version of the
1922:if and only if there exists another natural number
6633:
6556:
6517:
6479:
6451:
6423:
6395:
6318:
6283:
6250:
6222:
6194:
5437:Fletcher, Harold; Howell, Arnold A. (9 May 2014).
5388:A review of discrete and combinatorial mathematics
5195:
4764:
4491:
3972:
3946:
3629:
3591:
3553:
3515:
3477:
3425: – Generalization of "n-th" to infinite cases
3008:
2982:
2951:
2931:
2911:
2891:
2758:
2642:
2584:
2138:
1869:
1784:
1758:
1678:
1608:
1495:
1367:
1328:
1298:
1226:
1111:
999:
977:
942:
832:
699:
220:
157:
99:coins on the table"), in which case they serve as
7205:
5986:Szczepanski, Amy F.; Kositsky, Andrew P. (2008).
4700:(in French). Paris, Gauthier-Villars. p. 39.
4568:Mathematical Thought from Ancient to Modern Times
4072:exists and Russel's paradox cannot be formulated.
3085:, the sets considered below are sometimes called
379:used 0 as a separate number as early as the
259:the natural numbers in the other number systems.
8777:
5887:
5803:(Revised ed.). Cambridge University Press.
5657:
3899:
3073:The following definition was first published by
3066:elements that will be called the natural number
2692:but many well-ordered sets with cardinal number
1698:Relationship between addition and multiplication
7091:
5901:(3rd ed.). American Mathematical Society.
5658:Clapham, Christopher; Nicholson, James (2014).
3413: – Mathematical set that can be enumerated
2837:inside Peano arithmetic. A probable example is
6885:
6871:
6013:(Second ed.). ClassicalRealAnalysis.com.
5922:Number Systems and the Foundations of Analysis
5436:
5238:
5198:Studies in the Logic of Charles Sanders Peirce
5015:International Organization for Standardization
4867:Classic Set Theory: A guided independent study
4467:
3947:{\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}
3890:, see D. Joyce's web edition of Book VII.
3405:Canonical representation of a positive integer
2029:In this section, juxtaposed variables such as
2013:; for the natural numbers, this is denoted as
1885:In this section, juxtaposed variables such as
593:Set-theoretical definitions of natural numbers
6857:
6144:
5755:An Introduction to the History of Mathematics
5622:(Second ed.). McGraw-Hill Professional.
4151:Ganssle, Jack G. & Barr, Michael (2003).
3240:With this definition, given a natural number
2876:0 is not the successor of any natural number.
2797:. It defines the natural numbers as specific
1912:on the natural numbers is defined by letting
676:of all natural numbers is standardly denoted
629:Arithmetices principia, nova methodo exposita
8693:
7043:
6118:"Axioms and construction of natural numbers"
6065:"On the introduction of transfinite numbers"
5661:The Concise Oxford Dictionary of Mathematics
5202:. Indiana University Press. pp. 43–52.
4683:(in Latin). Fratres Bocca. 1889. p. 12.
3941:
3917:
1293:
1287:
1183:
1157:
1151:
1127:
1073:
1047:
1041:
1017:
937:
931:
880:
850:
812:
806:
755:
731:
6084:
6059:
6027:
5664:(Fifth ed.). Oxford University Press.
5585:
5475:Addition of natural numbers is associative.
5276:Bulletin of the London Mathematical Society
5060:. Princeton University Press. p. 153.
4996:
4994:
4992:
4638:
4636:
4634:
4632:
4630:
4628:
4626:
4624:
4312:Royal Belgian Institute of Natural Sciences
4277:Royal Belgian Institute of Natural Sciences
4261:
4251:Royal Belgian Institute of Natural Sciences
4235:
4150:
3843:Any Cauchy sequence in the Reals converges,
3435:Set-theoretic definition of natural numbers
3151:, there exist sets which contain 0 and are
3026:Set-theoretic definition of natural numbers
3019:
2139:{\displaystyle a=bq+r{\text{ and }}r<b.}
412:
315:Royal Belgian Institute of Natural Sciences
262:Properties of the natural numbers, such as
7158:Possessing a specific set of other numbers
6981:
6864:
6850:
6826:
6151:
6137:
5817:
5273:
5192:"3. Peirce's Axiomatization of Arithmetic"
5084:
4660:
4090:
4088:
3621:
3583:
3545:
3507:
3469:
1679:{\displaystyle (\mathbb {N} ^{*},\times )}
1379:of natural numbers recursively by setting
849:
403:do not have a symbol for 0; instead,
8621:
7568:
6627:
6550:
6511:
6473:
6445:
6417:
6389:
6277:
6244:
6216:
6188:
5989:The Complete Idiot's Guide to Pre-algebra
5915:
5782:. Springer Science & Business Media.
5636:
5531:
5521:
5030:"On the nature of mathematical reasoning"
4792:
4760:
4758:
3966:
3910:
3886:This convention is used, for example, in
3852:
3623:
3585:
3547:
3509:
3471:
3042:elements. So, it seems natural to define
2866:The five Peano axioms are the following:
2752:
2734:. Other generalizations are discussed in
1852:
1848:
1778:
1752:
1657:
1480:
1361:
1353:
1322:
1274:
1259:
1250:
1211:
1191:
1167:
1096:
1081:
1057:
971:
918:
903:
888:
820:
793:
778:
763:
690:
420:The first systematic study of numbers as
151:
6033:"Zur Einführung der transfiniten Zahlen"
5946:(Second ed.). Mcgraw-Hill College.
5796:
5724:
5702:
5678:
5493:. Vol. 8. Laidlaw Bros. p. 25.
5466:
5385:
5365:(5th ed.). Pearson Addison Wesley.
5360:
5354:
5027:
4989:
4621:
4099:. New York: Academic Press. p. 66.
4094:
4019:
3832:
3308:on the natural numbers. This order is a
304:
38:
5595:
5593:
5189:
4862:
4833:
4610:
4590:
4497:
4446:
4085:
3431: – Size of a possibly infinite set
2736:Number § Extensions of the concept
707:Older texts have occasionally employed
614:of natural-number arithmetic. In 1888,
383:, but this usage did not spread beyond
14:
8778:
8729:
5939:
5851:(Third ed.). Chelsea Publishing.
5841:
5827:(Fifth ed.). Chapman & Hall.
5772:
5615:
5139:
4811:The Princeton companion to mathematics
4808:
4797:Thomson, Bruckner & Bruckner (2008
4755:
4302:'s Portal to the Heritage of Astronomy
4287:
4048:
4028:
3864:
3222:= {{ }, {{ }}, ..., {{ }, {{ }}, ...}}
2801:. More precisely, each natural number
556:
458:
8728:
8692:
8656:
8620:
8580:
8205:
8094:
7820:
7735:
7690:
7567:
7257:
7204:
7156:
7090:
7042:
6980:
6884:
6845:
6132:
5873:. Springer-Verlag Berlin Heidelberg.
5410:
5340:. New York: McGraw-Hill. p. 25.
5333:
4929:
4911:from the original on 13 December 2019
4693:
4597:Le Triparty en la science des nombres
4565:
4417:
4323:
4192:
4124:
3328:does not accept the axiom of infinity
2769:
248:, by adjoining to the real numbers a
7258:
5865:
5748:
5599:
5590:
5567:from the original on 13 October 2014
5228:(in German). F. Vieweg. 1893. 71-73.
5127:
5050:
5002:"Standard number sets and intervals"
4977:from the original on 20 October 2015
4710:
4680:Arithmetices principia: nova methodo
4664:Eléments de la géométrie de l'infini
4455:from the original on 15 January 2019
4378:
4360:from the original on 19 January 2013
4120:
4118:
4116:
4007:
3162:This intersection is the set of the
3092:The definition proceeds as follows:
2813:elements" means that there exists a
2699:have an ordinal number greater than
1942:. This order is compatible with the
1870:{\displaystyle (\mathbb {N^{*}} ,+)}
8657:
6340:Set-theoretically definable numbers
5337:Principles of Mathematical Analysis
5225:Was sind und was sollen die Zahlen?
4447:Deckers, Michael (25 August 2003).
4046:is the set of all natural numbers).
3266:. This formalizes the operation of
3189:2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}}
3050:under the relation "can be made in
88:(including negative integers). The
24:
8581:
6311:
6158:
5467:Davisson, Schuyler Colfax (1910).
4844:from the original on 14 March 2015
4533:
4473:
4173:from the original on 29 March 2017
3317:von Neumann definition of ordinals
2793:The second definition is based on
2637:
2579:
2551:
716:Since natural numbers may contain
66:, while others define them as the
25:
8817:
6092:
6067:. In van Heijenoort, Jean (ed.).
5546:
5473:. Macmillian Company. p. 2.
4222:Brilliant Math & Science Wiki
4113:
2670:The least ordinal of cardinality
2623:Natural numbers are also used as
1587:
1242:This section uses the convention
8759:
8367:Perfect digit-to-digit invariant
7736:
6825:
5085:Weber, Heinrich L. (1891–1892).
4967:"Is index origin 0 a hindrance?"
4399:from the original on 14 May 2015
4326:The Universal History of Numbers
3389:
3030:Intuitively, the natural number
2720:non-standard model of arithmetic
1496:{\displaystyle (\mathbb {N} ,+)}
294:
189:for each nonzero natural number
132:, commonly symbolized as a bold
5730:Essays on the Theory of Numbers
5709:Essays on the Theory of Numbers
5685:Essays on the Theory of Numbers
5608:
5540:
5533:10.14321/realanalexch.42.2.0193
5497:
5480:
5460:
5430:
5404:
5379:
5327:
5302:
5267:
5232:
5216:
5183:
5149:American Journal of Mathematics
5133:
5121:
5078:
5044:
5021:
4964:
4958:
4923:
4889:
4856:
4827:
4802:
4785:
4737:
4721:
4704:
4687:
4671:
4661:Fontenelle, Bernard de (1727).
4654:
4604:
4584:
4559:
4527:
4479:"Book VII, definitions 1 and 2"
4440:
4411:
4372:
4354:MacTutor History of Mathematics
4342:
4317:
4058:
4013:
4000:
3991:
3880:
3870:
3630:{\displaystyle :\;\mathbb {N} }
3592:{\displaystyle :\;\mathbb {Z} }
3554:{\displaystyle :\;\mathbb {Q} }
3516:{\displaystyle :\;\mathbb {R} }
3478:{\displaystyle :\;\mathbb {C} }
2855:
2654:countably infinite set without
2217:), and ideas in number theory.
632:). This approach is now called
475:Bernard Le Bovier de Fontenelle
6319:{\displaystyle {\mathcal {P}}}
5720:– via Project Gutenberg.
5643:. Cambridge University Press.
5440:Mathematics with Understanding
4771:. Springer Nature. p. 6.
4310:, on permanent display at the
4210:
4144:
3858:
3846:
3837:
3825:
3437: – Axiom(s) of Set Theory
3034:is the common property of all
2522:are natural numbers such that
1864:
1843:
1673:
1652:
1490:
1476:
1357:
13:
1:
7206:Expressible via specific sums
6674:Plane-based geometric algebra
5732:. Kessinger Publishing, LLC.
5561:European Mathematical Society
4711:Fine, Henry Burchard (1904).
4095:Enderton, Herbert B. (1977).
4079:
3980:contains an "initial" number
3958:may be described as follows:
3900:Mac Lane & Birkhoff (1999
3054:". This does not work in all
2786:, based on few axioms called
1522:, so it can be embedded in a
1237:
700:{\displaystyle \mathbb {N} .}
648:with several weak systems of
513:have preferred to include 0.
6634:{\displaystyle \mathbb {S} }
6557:{\displaystyle \mathbb {C} }
6518:{\displaystyle \mathbb {R} }
6480:{\displaystyle \mathbb {O} }
6452:{\displaystyle \mathbb {H} }
6424:{\displaystyle \mathbb {C} }
6396:{\displaystyle \mathbb {R} }
6284:{\displaystyle \mathbb {A} }
6251:{\displaystyle \mathbb {Q} }
6223:{\displaystyle \mathbb {Z} }
6195:{\displaystyle \mathbb {N} }
5241:Mathematical Logic Quarterly
4863:Goldrei, Derek (1998). "3".
4697:Formulaire des mathematiques
4064:In some set theories, e.g.,
3973:{\displaystyle \mathbb {N} }
3198:= {{ }, {{ }}, {{ }, {{ }}}}
2778:, consists of an autonomous
2759:{\displaystyle \mathbb {N} }
2596:, exactly if there exists a
1785:{\displaystyle \mathbb {N} }
1759:{\displaystyle \mathbb {N} }
1329:{\displaystyle \mathbb {N} }
978:{\displaystyle \mathbb {Z} }
713:as the symbol for this set.
533:when enumerating items like
158:{\displaystyle \mathbb {N} }
7:
8295:Multiplicative digital root
6105:Encyclopedia of Mathematics
5553:Encyclopedia of Mathematics
5386:Grimaldi, Ralph P. (2003).
5361:Grimaldi, Ralph P. (2004).
4570:. Oxford University Press.
4157:Embedded Systems Dictionary
3382:
2401:: for every natural number
2213:), algorithms (such as the
2197:are uniquely determined by
2024:
1946:in the following sense: if
1578:is simply the successor of
1336:of natural numbers and the
1309:
667:
424:is usually credited to the
128:The natural numbers form a
10:
8822:
7691:
5940:Morash, Ronald P. (1991).
5559:, in cooperation with the
5490:Laidlaw mathematics series
3833:§ Emergence as a term
3345:. It consists in defining
3023:
2859:
2643:{\displaystyle \emptyset }
2625:linguistic ordinal numbers
2585:{\displaystyle \emptyset }
2346:: for all natural numbers
2270:: for all natural numbers
2079:there are natural numbers
578:foundations of mathematics
298:
289:
29:
8755:
8738:
8724:
8702:
8688:
8666:
8652:
8630:
8616:
8589:
8576:
8552:
8506:
8466:
8417:
8391:
8372:Perfect digital invariant
8324:
8308:
8287:
8254:
8219:
8215:
8201:
8109:
8090:
8059:
8026:
7983:
7960:
7947:Superior highly composite
7837:
7833:
7816:
7744:
7731:
7699:
7686:
7574:
7563:
7525:
7516:
7494:
7451:
7413:
7404:
7337:
7279:
7270:
7266:
7253:
7211:
7200:
7163:
7152:
7100:
7086:
7049:
7038:
6991:
6976:
6894:
6880:
6816:
6758:
6684:
6664:Algebra of physical space
6586:
6494:
6365:
6167:
6083:– English translation of
6043:: 199–208. Archived from
6023:– via Google Books.
6002:– via Google Books.
5981:– via Google Books.
5956:– via Google Books.
5935:– via Google Books.
5911:– via Google Books.
5861:– via Google Books.
5837:– via Google Books.
5813:– via Google Books.
5792:– via Google Books.
5768:– via Google Books.
5758:(6th ed.). Thomson.
5674:– via Google Books.
5653:– via Google Books.
5632:– via Google Books.
5443:. Elsevier. p. 116.
5034:La Science et l'hypothèse
5028:Poincaré, Henri (1905) .
5017:. 19 May 2020. p. 4.
4932:ACM SIGAPL APL Quote Quad
4611:Emerson, William (1763).
4539:"Book VII, definition 22"
4436:– via Google Books.
4426:. John Wiley & Sons.
4407:– via Google Books.
4379:Mann, Charles C. (2005).
4181:– via Google Books.
3183:1 = 0 ∪ {0} = {0} = {{ }}
3052:one to one correspondence
2815:one to one correspondence
1877:has no identity element.
228:for each nonzero integer
179:0 (if not yet in) and an
7985:Euler's totient function
7769:Euler–Jacobi pseudoprime
7044:Other polynomial numbers
6720:Extended complex numbers
6703:Extended natural numbers
6010:Elementary Real Analysis
5800:Logic for Mathematicians
5698:– via Archive.org.
5637:Carothers, N.L. (2000).
5253:10.1002/malq.19930390138
5145:"On the Logic of Number"
4809:Gowers, Timothy (2008).
4694:Peano, Giuseppe (1901).
4614:The method of increments
3818:
3020:Set-theoretic definition
2899:equals the successor of
1964:are natural numbers and
1880:
266:and the distribution of
27:Number used for counting
7799:Somer–Lucas pseudoprime
7789:Lucas–Carmichael number
7624:Lazy caterer's sequence
5848:Foundations of Analysis
5821:; James, Glenn (1992).
5797:Hamilton, A.G. (1988).
5619:Pre-Algebra DeMYSTiFieD
5586:von Neumann (1923)
4667:(in French). p. 3.
4566:Kline, Morris (1990) .
4324:Ifrah, Georges (2000).
4026:0 is a natural number."
3195:3 = 2 ∪ {2} = {0, 1, 2}
2732:ultrapower construction
2054:division with remainder
1944:arithmetical operations
1688:free commutative monoid
1609:{\displaystyle \times }
724:Naturals without zero:
8791:Elementary mathematics
7674:Wedderburn–Etherington
7074:Lucky numbers of Euler
6776:Transcendental numbers
6635:
6612:Hyperbolic quaternions
6558:
6519:
6481:
6453:
6425:
6397:
6320:
6285:
6252:
6224:
6196:
6061:von Neumann, John
5969:Wiley Global Education
5925:. Dover Publications.
5824:Mathematics Dictionary
5616:Bluman, Allan (2010).
5510:Real Analysis Exchange
5190:Shields, Paul (1997).
5038:Science and Hypothesis
4834:Bagaria, Joan (2017).
4541:. In Joyce, D. (ed.).
4481:. In Joyce, D. (ed.).
4097:Elements of set theory
3974:
3948:
3717:Dyadic (finite binary)
3631:
3593:
3555:
3517:
3479:
3349:as the empty set, and
3300:. In other words, the
3246:, the sentence "a set
3010:
2984:
2953:
2933:
2913:
2893:
2870:0 is a natural number.
2760:
2644:
2586:
2140:
1871:
1786:
1760:
1680:
1610:
1497:
1369:
1330:
1300:
1228:
1113:
1001:
979:
944:
834:
701:
627:
608:Charles Sanders Peirce
413:
407:(or the genitive form
318:
313:(on exhibition at the
222:
199:multiplicative inverse
159:
44:
7962:Prime omega functions
7779:Frobenius pseudoprime
7569:Combinatorial numbers
7438:Centered dodecahedral
7231:Primary pseudoperfect
6708:Extended real numbers
6636:
6559:
6529:Split-complex numbers
6520:
6482:
6454:
6426:
6398:
6321:
6286:
6262:Constructible numbers
6253:
6225:
6197:
6085:von Neumann 1923
5417:mathworld.wolfram.com
5314:functions.wolfram.com
5288:10.1112/blms/14.4.285
5282:(4). Wiley: 285–293.
4944:10.1145/586050.586053
4498:Mueller, Ian (2006).
4418:Evans, Brian (2014).
4385:. Knopf. p. 19.
4275:. Brussels, Belgium:
4249:. Brussels, Belgium:
4131:mathworld.wolfram.com
3975:
3949:
3632:
3594:
3556:
3518:
3480:
3011:
2985:
2954:
2934:
2914:
2894:
2839:Fermat's Last Theorem
2817:between the two sets
2761:
2645:
2587:
2141:
2035:indicate the product
1891:indicate the product
1872:
1787:
1761:
1681:
1611:
1520:cancellation property
1498:
1370:
1331:
1301:
1229:
1114:
1002:
980:
945:
835:
702:
652:. One such system is
640:of the properties of
468:progression naturelle
308:
299:Further information:
223:
160:
61:non-negative integers
42:
8806:Sets of real numbers
8421:-composition related
8221:Arithmetic functions
7823:Arithmetic functions
7759:Elliptic pseudoprime
7443:Centered icosahedral
7423:Centered tetrahedral
6740:Supernatural numbers
6650:Multicomplex numbers
6623:
6607:Dual-complex numbers
6546:
6507:
6469:
6441:
6413:
6385:
6367:Composition algebras
6335:Arithmetical numbers
6306:
6273:
6240:
6212:
6184:
4556:is a perfect number.
4547:. Clark University.
4451:. Hbar.phys.msu.ru.
4314:, Brussels, Belgium.
4308:on 10 November 2014.
4269:"Flash presentation"
3962:
3906:
3762:Algebraic irrational
3615:
3577:
3539:
3501:
3463:
3087:von Neumann ordinals
2994:
2974:
2943:
2923:
2903:
2883:
2879:If the successor of
2748:
2634:
2576:
2264:are natural numbers.
2098:
1840:
1774:
1748:
1649:
1600:
1473:
1343:
1318:
1246:
1124:
1014:
991:
967:
846:
842:Naturals with zero:
728:
686:
636:. It is based on an
586:recursive definition
521:and the size of the
301:Prehistoric counting
204:
147:
84:refer to all of the
8347:Kaprekar's constant
7867:Colossally abundant
7754:Catalan pseudoprime
7654:Schröder–Hipparchus
7433:Centered octahedral
7309:Centered heptagonal
7299:Centered pentagonal
7289:Centered triangular
6889:and related numbers
6645:Split-biquaternions
6357:Eisenstein integers
6295:Closed-form numbers
6047:on 18 December 2014
5547:Mints, G.E. (ed.).
5411:Weisstein, Eric W.
4901:Merriam-Webster.com
4749:archive.lib.msu.edu
4487:. Clark University.
4350:"A history of Zero"
4125:Weisstein, Eric W.
4038:(where, of course,
3453:
3081:as a definition of
3009:{\displaystyle x+1}
2215:Euclidean algorithm
2173:of the division of
2047:order of operations
2045:, and the standard
1903:order of operations
1901:, and the standard
1616:can be defined via
1533:If 1 is defined as
1468:algebraic structure
1205:
662:Goodstein's theorem
610:provided the first
557:Formal construction
507:Stephen Cole Kleene
491:George A. Wentworth
459:Emergence as a term
221:{\displaystyle 1/n}
8765:Mathematics portal
8707:Aronson's sequence
8453:Smarandache–Wellin
8210:-dependent numbers
7917:Primitive abundant
7804:Strong pseudoprime
7794:Perrin pseudoprime
7774:Fermat pseudoprime
7714:Wolstenholme prime
7538:Squared triangular
7324:Centered decagonal
7319:Centered nonagonal
7314:Centered octagonal
7304:Centered hexagonal
6803:Profinite integers
6766:Irrational numbers
6631:
6554:
6515:
6477:
6449:
6421:
6393:
6350:Gaussian rationals
6330:Computable numbers
6316:
6281:
6248:
6220:
6192:
5917:Mendelson, Elliott
5889:Mac Lane, Saunders
5334:Rudin, W. (1976).
5141:Peirce, C. S.
5117:on 20 August 2017.
4791:See, for example,
4717:. Ginn. p. 6.
4194:Weisstein, Eric W.
3970:
3944:
3627:
3589:
3551:
3513:
3475:
3449:
3397:Mathematics portal
3304:defines the usual
3215:−1} = {0, 1, ...,
3006:
2980:
2964:axiom of induction
2949:
2929:
2909:
2889:
2770:Formal definitions
2756:
2640:
2603:countably infinite
2582:
2136:
2058:Euclidean division
1867:
1800:; instead it is a
1782:
1756:
1676:
1606:
1493:
1365:
1338:successor function
1326:
1296:
1224:
1189:
1109:
997:
975:
940:
830:
697:
595:were initiated by
527:Computer languages
511:John Horton Conway
377:Maya civilizations
319:
244:of rationals; the
218:
175:, by including an
155:
45:
8773:
8772:
8751:
8750:
8720:
8719:
8684:
8683:
8648:
8647:
8612:
8611:
8572:
8571:
8568:
8567:
8387:
8386:
8197:
8196:
8086:
8085:
8082:
8081:
8028:Aliquot sequences
7839:Divisor functions
7812:
7811:
7784:Lucas pseudoprime
7764:Euler pseudoprime
7749:Carmichael number
7727:
7726:
7682:
7681:
7559:
7558:
7555:
7554:
7551:
7550:
7512:
7511:
7400:
7399:
7357:Square triangular
7249:
7248:
7196:
7195:
7148:
7147:
7082:
7081:
7034:
7033:
6972:
6971:
6839:
6838:
6750:Superreal numbers
6730:Levi-Civita field
6725:Hyperreal numbers
6669:Spacetime algebra
6655:Geometric algebra
6568:Bicomplex numbers
6534:Split-quaternions
6375:Division algebras
6345:Gaussian integers
6267:Algebraic numbers
6170:definable numbers
6078:978-0-674-32449-7
6063:(January 2002) .
6029:von Neumann, John
6020:978-1-4348-4367-8
5999:978-1-59257-772-9
5992:. Penguin Group.
5978:978-1-118-45744-3
5967:(10th ed.).
5953:978-0-07-043043-3
5932:978-0-486-45792-5
5908:978-0-8218-1646-2
5893:Birkhoff, Garrett
5880:978-3-662-02310-5
5858:978-0-8218-2693-5
5834:978-0-412-99041-0
5810:978-0-521-36865-0
5789:978-0-387-90092-6
5765:978-0-03-029558-4
5739:978-0-548-08985-9
5726:Dedekind, Richard
5704:Dedekind, Richard
5695:978-0-486-21010-0
5680:Dedekind, Richard
5671:978-0-19-967959-1
5650:978-0-521-49756-5
5629:978-0-07-174251-1
5450:978-1-4832-8079-0
5397:978-0-201-72634-3
5372:978-0-201-72634-3
5347:978-0-07-054235-8
5103:on 9 August 2018;
5067:978-1-4008-2904-0
4882:978-0-412-60610-6
4820:978-0-691-11880-2
4778:978-3-030-83899-7
4714:A College Algebra
4512:978-0-486-45300-2
4502:Euclid's Elements
4433:978-1-118-85397-9
4392:978-1-4000-4006-3
4218:"Natural Numbers"
4197:"Counting Number"
4166:978-1-57820-120-4
3888:Euclid's Elements
3816:
3815:
3812:
3811:
3808:
3807:
3804:
3803:
3793:
3792:
3789:
3788:
3785:
3784:
3781:
3780:
3769:Irrational period
3743:
3742:
3739:
3738:
3735:
3734:
3731:
3730:
3724:Repeating decimal
3691:
3690:
3687:
3686:
3682:Negative integers
3676:
3675:
3672:
3671:
3667:Composite numbers
3319:for defining all
3149:axiom of infinity
3060:Russell's paradox
3048:equivalence class
2983:{\displaystyle x}
2952:{\displaystyle y}
2932:{\displaystyle x}
2912:{\displaystyle y}
2892:{\displaystyle x}
2660:order isomorphism
2399:identity elements
2122:
1804:(also known as a
1514: 0. It is a
1466:, and so on. The
1000:{\displaystyle *}
658:axiom of infinity
601:Russell's paradox
582:Hermann Grassmann
567:Leopold Kronecker
397:Dionysius Exiguus
325:. Putting down a
282:, are studied in
270:, are studied in
236:by including the
197:, by including a
177:additive identity
68:positive integers
18:Positive integers
16:(Redirected from
8813:
8786:Cardinal numbers
8763:
8726:
8725:
8695:Natural language
8690:
8689:
8654:
8653:
8622:Generated via a
8618:
8617:
8578:
8577:
8483:Digit-reassembly
8448:Self-descriptive
8252:
8251:
8217:
8216:
8203:
8202:
8154:Lucas–Carmichael
8144:Harmonic divisor
8092:
8091:
8018:Sparsely totient
7993:Highly cototient
7902:Multiply perfect
7892:Highly composite
7835:
7834:
7818:
7817:
7733:
7732:
7688:
7687:
7669:Telephone number
7565:
7564:
7523:
7522:
7504:Square pyramidal
7486:Stella octangula
7411:
7410:
7277:
7276:
7268:
7267:
7260:Figurate numbers
7255:
7254:
7202:
7201:
7154:
7153:
7088:
7087:
7040:
7039:
6978:
6977:
6882:
6881:
6866:
6859:
6852:
6843:
6842:
6829:
6828:
6796:
6786:
6698:Cardinal numbers
6659:Clifford algebra
6640:
6638:
6637:
6632:
6630:
6602:Dual quaternions
6563:
6561:
6560:
6555:
6553:
6524:
6522:
6521:
6516:
6514:
6486:
6484:
6483:
6478:
6476:
6458:
6456:
6455:
6450:
6448:
6430:
6428:
6427:
6422:
6420:
6402:
6400:
6399:
6394:
6392:
6325:
6323:
6322:
6317:
6315:
6314:
6290:
6288:
6287:
6282:
6280:
6257:
6255:
6254:
6249:
6247:
6234:Rational numbers
6229:
6227:
6226:
6221:
6219:
6201:
6199:
6198:
6193:
6191:
6153:
6146:
6139:
6130:
6129:
6125:
6113:
6100:"Natural number"
6082:
6056:
6054:
6052:
6024:
6003:
5982:
5957:
5936:
5912:
5884:
5871:Basic Set Theory
5862:
5838:
5819:James, Robert C.
5814:
5793:
5779:Naive Set Theory
5769:
5743:
5721:
5719:
5717:
5699:
5675:
5654:
5633:
5603:
5597:
5588:
5583:
5577:
5576:
5574:
5572:
5544:
5538:
5537:
5535:
5525:
5501:
5495:
5494:
5484:
5478:
5477:
5464:
5458:
5457:
5434:
5428:
5427:
5425:
5423:
5413:"Multiplication"
5408:
5402:
5401:
5383:
5377:
5376:
5358:
5352:
5351:
5331:
5325:
5324:
5322:
5320:
5306:
5300:
5299:
5271:
5265:
5264:
5236:
5230:
5229:
5220:
5214:
5213:
5201:
5187:
5181:
5180:
5137:
5131:
5125:
5119:
5118:
5113:. Archived from
5104:
5082:
5076:
5075:
5048:
5042:
5041:
5025:
5019:
5018:
5010:ISO 80000-2:2019
5006:
4998:
4987:
4986:
4984:
4982:
4962:
4956:
4955:
4927:
4921:
4920:
4918:
4916:
4897:"natural number"
4893:
4887:
4886:
4870:
4860:
4854:
4853:
4851:
4849:
4831:
4825:
4824:
4806:
4800:
4795:, p. 3) or
4789:
4783:
4782:
4762:
4753:
4752:
4745:"Natural Number"
4741:
4735:
4734:
4725:
4719:
4718:
4708:
4702:
4701:
4691:
4685:
4684:
4675:
4669:
4668:
4658:
4652:
4651:
4640:
4619:
4618:
4608:
4602:
4601:
4592:Chuquet, Nicolas
4588:
4582:
4581:
4563:
4557:
4555:
4551:
4531:
4525:
4524:
4495:
4489:
4488:
4471:
4465:
4464:
4462:
4460:
4444:
4438:
4437:
4415:
4409:
4408:
4406:
4404:
4376:
4370:
4369:
4367:
4365:
4346:
4340:
4339:
4321:
4315:
4309:
4304:. Archived from
4291:
4285:
4284:
4279:. Archived from
4265:
4259:
4258:
4257:on 4 March 2016.
4253:. Archived from
4239:
4233:
4232:
4230:
4228:
4214:
4208:
4207:
4206:
4189:
4183:
4182:
4180:
4178:
4148:
4142:
4141:
4139:
4137:
4127:"Natural Number"
4122:
4111:
4110:
4092:
4073:
4062:
4056:
4045:
4041:
4037:
4034:
4025:
4017:
4011:
4004:
3998:
3995:
3989:
3983:
3979:
3977:
3976:
3971:
3969:
3953:
3951:
3950:
3945:
3913:
3897:
3891:
3884:
3878:
3874:
3868:
3862:
3856:
3850:
3844:
3841:
3835:
3829:
3758:
3757:
3749:
3748:
3706:
3705:
3697:
3696:
3640:
3639:
3636:
3634:
3633:
3628:
3626:
3606:
3605:
3602:
3601:
3598:
3596:
3595:
3590:
3588:
3568:
3567:
3564:
3563:
3560:
3558:
3557:
3552:
3550:
3530:
3529:
3526:
3525:
3522:
3520:
3519:
3514:
3512:
3492:
3491:
3488:
3487:
3484:
3482:
3481:
3476:
3474:
3454:
3448:
3445:
3444:
3441:
3440:
3399:
3394:
3393:
3378:
3363:
3348:
3343:
3342:
3341:Zermelo ordinals
3299:
3289:
3283:
3273:
3270:the elements of
3265:
3261:
3253:
3249:
3245:
3223:
3220:
3199:
3196:
3190:
3184:
3178:
3143:
3124:
3120:
3099:
3075:John von Neumann
3069:
3065:
3045:
3041:
3033:
3015:
3013:
3012:
3007:
2989:
2987:
2986:
2981:
2958:
2956:
2955:
2950:
2938:
2936:
2935:
2930:
2918:
2916:
2915:
2910:
2898:
2896:
2895:
2890:
2824:
2820:
2812:
2808:
2804:
2784:Peano arithmetic
2780:axiomatic theory
2765:
2763:
2762:
2757:
2755:
2702:
2698:
2691:
2687:
2676:
2665:
2649:
2647:
2646:
2641:
2619:
2591:
2589:
2588:
2583:
2558:cardinal numbers
2546:
2539:
2532:
2521:
2515:
2502:
2471:
2465:
2459:
2445:
2435:
2426:
2416:
2406:
2393:
2375:
2357:
2351:
2339:
2313:
2287:
2281:
2275:
2263:
2253:
2243:
2237:
2208:
2202:
2196:
2190:
2184:
2178:
2166:
2154:
2145:
2143:
2142:
2137:
2123:
2120:
2090:
2084:
2078:
2071:
2065:
2044:
2034:
2020:
2001:
1991:
1973:
1963:
1957:
1951:
1941:
1927:
1921:
1900:
1890:
1876:
1874:
1873:
1868:
1857:
1856:
1855:
1835:
1825:
1791:
1789:
1788:
1783:
1781:
1765:
1763:
1762:
1757:
1755:
1736:
1704:distribution law
1685:
1683:
1682:
1677:
1666:
1665:
1660:
1644:
1622:
1615:
1613:
1612:
1607:
1583:
1577:
1570:
1539:
1512:identity element
1502:
1500:
1499:
1494:
1483:
1465:
1446:
1427:
1421:
1415:
1388:
1374:
1372:
1371:
1366:
1364:
1356:
1335:
1333:
1332:
1327:
1325:
1305:
1303:
1302:
1297:
1283:
1282:
1277:
1268:
1267:
1262:
1253:
1233:
1231:
1230:
1225:
1223:
1222:
1214:
1204:
1199:
1194:
1170:
1118:
1116:
1115:
1110:
1108:
1107:
1099:
1090:
1089:
1084:
1060:
1006:
1004:
1003:
998:
986:
984:
982:
981:
976:
974:
949:
947:
946:
941:
927:
926:
921:
912:
911:
906:
897:
896:
891:
839:
837:
836:
831:
829:
828:
823:
802:
801:
796:
787:
786:
781:
772:
771:
766:
719:
712:
706:
704:
703:
698:
693:
681:
634:Peano arithmetic
616:Richard Dedekind
580:. In the 1860s,
519:division by zero
499:Nicolas Bourbaki
495:Bertrand Russell
416:
382:
253:
242:Cauchy sequences
231:
227:
225:
224:
219:
214:
195:rational numbers
192:
188:
181:additive inverse
166:
164:
162:
161:
156:
154:
137:
102:cardinal numbers
90:counting numbers
75:
73:
65:
21:
8821:
8820:
8816:
8815:
8814:
8812:
8811:
8810:
8776:
8775:
8774:
8769:
8747:
8743:Strobogrammatic
8734:
8716:
8698:
8680:
8662:
8644:
8626:
8608:
8585:
8564:
8548:
8507:Divisor-related
8502:
8462:
8413:
8383:
8320:
8304:
8283:
8250:
8223:
8211:
8193:
8105:
8104:related numbers
8078:
8055:
8022:
8013:Perfect totient
7979:
7956:
7887:Highly abundant
7829:
7808:
7740:
7723:
7695:
7678:
7664:Stirling second
7570:
7547:
7508:
7490:
7447:
7396:
7333:
7294:Centered square
7262:
7245:
7207:
7192:
7159:
7144:
7096:
7095:defined numbers
7078:
7045:
7030:
7001:Double Mersenne
6987:
6968:
6890:
6876:
6874:natural numbers
6870:
6840:
6835:
6812:
6791:
6781:
6754:
6745:Surreal numbers
6735:Ordinal numbers
6680:
6626:
6624:
6621:
6620:
6582:
6549:
6547:
6544:
6543:
6541:
6539:Split-octonions
6510:
6508:
6505:
6504:
6496:
6490:
6472:
6470:
6467:
6466:
6444:
6442:
6439:
6438:
6416:
6414:
6411:
6410:
6407:Complex numbers
6388:
6386:
6383:
6382:
6361:
6310:
6309:
6307:
6304:
6303:
6276:
6274:
6271:
6270:
6243:
6241:
6238:
6237:
6215:
6213:
6210:
6209:
6187:
6185:
6182:
6181:
6178:Natural numbers
6163:
6157:
6116:
6098:
6095:
6090:
6079:
6050:
6048:
6021:
6000:
5979:
5954:
5933:
5909:
5881:
5859:
5835:
5811:
5790:
5766:
5740:
5715:
5713:
5696:
5672:
5651:
5630:
5611:
5606:
5598:
5591:
5584:
5580:
5570:
5568:
5545:
5541:
5502:
5498:
5485:
5481:
5470:College Algebra
5465:
5461:
5451:
5435:
5431:
5421:
5419:
5409:
5405:
5398:
5384:
5380:
5373:
5359:
5355:
5348:
5332:
5328:
5318:
5316:
5308:
5307:
5303:
5272:
5268:
5237:
5233:
5222:
5221:
5217:
5210:
5188:
5184:
5161:10.2307/2369151
5138:
5134:
5126:
5122:
5105:
5083:
5079:
5068:
5049:
5045:
5026:
5022:
5004:
5000:
4999:
4990:
4980:
4978:
4963:
4959:
4928:
4924:
4914:
4912:
4905:Merriam-Webster
4895:
4894:
4890:
4883:
4861:
4857:
4847:
4845:
4832:
4828:
4821:
4807:
4803:
4793:Carothers (2000
4790:
4786:
4779:
4763:
4756:
4743:
4742:
4738:
4727:
4726:
4722:
4709:
4705:
4692:
4688:
4677:
4676:
4672:
4659:
4655:
4642:
4641:
4622:
4609:
4605:
4589:
4585:
4578:
4564:
4560:
4553:
4532:
4528:
4513:
4496:
4492:
4472:
4468:
4458:
4456:
4445:
4441:
4434:
4416:
4412:
4402:
4400:
4393:
4377:
4373:
4363:
4361:
4348:
4347:
4343:
4336:
4322:
4318:
4293:
4292:
4288:
4283:on 27 May 2016.
4267:
4266:
4262:
4241:
4240:
4236:
4226:
4224:
4216:
4215:
4211:
4190:
4186:
4176:
4174:
4167:
4149:
4145:
4135:
4133:
4123:
4114:
4107:
4093:
4086:
4082:
4077:
4076:
4066:New Foundations
4063:
4059:
4047:
4043:
4039:
4035:
4032:
4027:
4023:
4018:
4014:
4010:, p. 606)
4005:
4001:
3996:
3992:
3981:
3965:
3963:
3960:
3959:
3956:natural numbers
3909:
3907:
3904:
3903:
3898:
3894:
3885:
3881:
3875:
3871:
3863:
3859:
3853:Mendelson (2008
3851:
3847:
3842:
3838:
3830:
3826:
3821:
3622:
3616:
3613:
3612:
3584:
3578:
3575:
3574:
3546:
3540:
3537:
3536:
3508:
3502:
3499:
3498:
3470:
3464:
3461:
3460:
3429:Cardinal number
3395:
3388:
3385:
3376:
3350:
3346:
3340:
3339:
3321:ordinal numbers
3295:
3285:
3284:if and only if
3275:
3271:
3263:
3259:
3251:
3247:
3241:
3221:
3203:
3197:
3194:
3188:
3182:
3176:
3164:natural numbers
3126:
3122:
3111:
3097:
3067:
3063:
3043:
3039:
3031:
3028:
3022:
2995:
2992:
2991:
2975:
2972:
2971:
2944:
2941:
2940:
2924:
2921:
2920:
2904:
2901:
2900:
2884:
2881:
2880:
2864:
2858:
2822:
2818:
2810:
2806:
2802:
2772:
2751:
2749:
2746:
2745:
2700:
2697:
2693:
2689:
2686:
2682:
2679:initial ordinal
2675:
2671:
2663:
2635:
2632:
2631:
2618:
2614:
2577:
2574:
2573:
2562:ordinal numbers
2554:
2552:Generalizations
2541:
2534:
2523:
2517:
2511:
2473:
2467:
2461:
2455:
2437:
2431:
2418:
2408:
2402:
2377:
2359:
2353:
2347:
2315:
2289:
2283:
2277:
2271:
2255:
2245:
2239:
2233:
2223:
2204:
2198:
2192:
2186:
2180:
2174:
2162:
2150:
2121: and
2119:
2099:
2096:
2095:
2086:
2080:
2073:
2067:
2061:
2036:
2030:
2027:
2014:
1993:
1975:
1965:
1959:
1953:
1947:
1929:
1923:
1913:
1892:
1886:
1883:
1851:
1847:
1846:
1841:
1838:
1837:
1836:. Furthermore,
1827:
1812:
1777:
1775:
1772:
1771:
1751:
1749:
1746:
1745:
1707:
1700:
1661:
1656:
1655:
1650:
1647:
1646:
1624:
1617:
1601:
1598:
1597:
1590:
1579:
1572:
1541:
1534:
1479:
1474:
1471:
1470:
1448:
1429:
1423:
1417:
1390:
1380:
1360:
1352:
1344:
1341:
1340:
1321:
1319:
1316:
1315:
1312:
1278:
1273:
1272:
1263:
1258:
1257:
1249:
1247:
1244:
1243:
1240:
1215:
1210:
1209:
1200:
1195:
1190:
1166:
1125:
1122:
1121:
1100:
1095:
1094:
1085:
1080:
1079:
1056:
1015:
1012:
1011:
992:
989:
988:
970:
968:
965:
964:
962:
922:
917:
916:
907:
902:
901:
892:
887:
886:
847:
844:
843:
824:
819:
818:
797:
792:
791:
782:
777:
776:
767:
762:
761:
729:
726:
725:
717:
708:
689:
687:
684:
683:
677:
670:
642:ordinal numbers
574:constructivists
559:
531:start from zero
488:
484:
464:Nicolas Chuquet
461:
381:1st century BCE
380:
323:finger counting
303:
297:
292:
251:
250:square root of
246:complex numbers
229:
210:
205:
202:
201:
190:
183:
150:
148:
145:
144:
142:
140:blackboard bold
133:
118:nominal numbers
112:ordinal numbers
71:
70:
64:0, 1, 2, 3, ...
63:
53:natural numbers
35:
28:
23:
22:
15:
12:
11:
5:
8819:
8809:
8808:
8803:
8798:
8793:
8788:
8771:
8770:
8768:
8767:
8756:
8753:
8752:
8749:
8748:
8746:
8745:
8739:
8736:
8735:
8722:
8721:
8718:
8717:
8715:
8714:
8709:
8703:
8700:
8699:
8686:
8685:
8682:
8681:
8679:
8678:
8676:Sorting number
8673:
8671:Pancake number
8667:
8664:
8663:
8650:
8649:
8646:
8645:
8643:
8642:
8637:
8631:
8628:
8627:
8614:
8613:
8610:
8609:
8607:
8606:
8601:
8596:
8590:
8587:
8586:
8583:Binary numbers
8574:
8573:
8570:
8569:
8566:
8565:
8563:
8562:
8556:
8554:
8550:
8549:
8547:
8546:
8541:
8536:
8531:
8526:
8521:
8516:
8510:
8508:
8504:
8503:
8501:
8500:
8495:
8490:
8485:
8480:
8474:
8472:
8464:
8463:
8461:
8460:
8455:
8450:
8445:
8440:
8435:
8430:
8424:
8422:
8415:
8414:
8412:
8411:
8410:
8409:
8398:
8396:
8393:P-adic numbers
8389:
8388:
8385:
8384:
8382:
8381:
8380:
8379:
8369:
8364:
8359:
8354:
8349:
8344:
8339:
8334:
8328:
8326:
8322:
8321:
8319:
8318:
8312:
8310:
8309:Coding-related
8306:
8305:
8303:
8302:
8297:
8291:
8289:
8285:
8284:
8282:
8281:
8276:
8271:
8266:
8260:
8258:
8249:
8248:
8247:
8246:
8244:Multiplicative
8241:
8230:
8228:
8213:
8212:
8208:Numeral system
8199:
8198:
8195:
8194:
8192:
8191:
8186:
8181:
8176:
8171:
8166:
8161:
8156:
8151:
8146:
8141:
8136:
8131:
8126:
8121:
8116:
8110:
8107:
8106:
8088:
8087:
8084:
8083:
8080:
8079:
8077:
8076:
8071:
8065:
8063:
8057:
8056:
8054:
8053:
8048:
8043:
8038:
8032:
8030:
8024:
8023:
8021:
8020:
8015:
8010:
8005:
8000:
7998:Highly totient
7995:
7989:
7987:
7981:
7980:
7978:
7977:
7972:
7966:
7964:
7958:
7957:
7955:
7954:
7949:
7944:
7939:
7934:
7929:
7924:
7919:
7914:
7909:
7904:
7899:
7894:
7889:
7884:
7879:
7874:
7869:
7864:
7859:
7854:
7852:Almost perfect
7849:
7843:
7841:
7831:
7830:
7814:
7813:
7810:
7809:
7807:
7806:
7801:
7796:
7791:
7786:
7781:
7776:
7771:
7766:
7761:
7756:
7751:
7745:
7742:
7741:
7729:
7728:
7725:
7724:
7722:
7721:
7716:
7711:
7706:
7700:
7697:
7696:
7684:
7683:
7680:
7679:
7677:
7676:
7671:
7666:
7661:
7659:Stirling first
7656:
7651:
7646:
7641:
7636:
7631:
7626:
7621:
7616:
7611:
7606:
7601:
7596:
7591:
7586:
7581:
7575:
7572:
7571:
7561:
7560:
7557:
7556:
7553:
7552:
7549:
7548:
7546:
7545:
7540:
7535:
7529:
7527:
7520:
7514:
7513:
7510:
7509:
7507:
7506:
7500:
7498:
7492:
7491:
7489:
7488:
7483:
7478:
7473:
7468:
7463:
7457:
7455:
7449:
7448:
7446:
7445:
7440:
7435:
7430:
7425:
7419:
7417:
7408:
7402:
7401:
7398:
7397:
7395:
7394:
7389:
7384:
7379:
7374:
7369:
7364:
7359:
7354:
7349:
7343:
7341:
7335:
7334:
7332:
7331:
7326:
7321:
7316:
7311:
7306:
7301:
7296:
7291:
7285:
7283:
7274:
7264:
7263:
7251:
7250:
7247:
7246:
7244:
7243:
7238:
7233:
7228:
7223:
7218:
7212:
7209:
7208:
7198:
7197:
7194:
7193:
7191:
7190:
7185:
7180:
7175:
7170:
7164:
7161:
7160:
7150:
7149:
7146:
7145:
7143:
7142:
7137:
7132:
7127:
7122:
7117:
7112:
7107:
7101:
7098:
7097:
7084:
7083:
7080:
7079:
7077:
7076:
7071:
7066:
7061:
7056:
7050:
7047:
7046:
7036:
7035:
7032:
7031:
7029:
7028:
7023:
7018:
7013:
7008:
7003:
6998:
6992:
6989:
6988:
6974:
6973:
6970:
6969:
6967:
6966:
6961:
6956:
6951:
6946:
6941:
6936:
6931:
6926:
6921:
6916:
6911:
6906:
6901:
6895:
6892:
6891:
6878:
6877:
6869:
6868:
6861:
6854:
6846:
6837:
6836:
6834:
6833:
6823:
6821:Classification
6817:
6814:
6813:
6811:
6810:
6808:Normal numbers
6805:
6800:
6778:
6773:
6768:
6762:
6760:
6756:
6755:
6753:
6752:
6747:
6742:
6737:
6732:
6727:
6722:
6717:
6716:
6715:
6705:
6700:
6694:
6692:
6690:infinitesimals
6682:
6681:
6679:
6678:
6677:
6676:
6671:
6666:
6652:
6647:
6642:
6629:
6614:
6609:
6604:
6599:
6593:
6591:
6584:
6583:
6581:
6580:
6575:
6570:
6565:
6552:
6536:
6531:
6526:
6513:
6500:
6498:
6492:
6491:
6489:
6488:
6475:
6460:
6447:
6432:
6419:
6404:
6391:
6371:
6369:
6363:
6362:
6360:
6359:
6354:
6353:
6352:
6342:
6337:
6332:
6327:
6313:
6297:
6292:
6279:
6264:
6259:
6246:
6231:
6218:
6203:
6190:
6174:
6172:
6165:
6164:
6156:
6155:
6148:
6141:
6133:
6127:
6126:
6114:
6094:
6093:External links
6091:
6089:
6088:
6077:
6057:
6025:
6019:
6004:
5998:
5983:
5977:
5958:
5952:
5937:
5931:
5913:
5907:
5885:
5879:
5863:
5857:
5843:Landau, Edmund
5839:
5833:
5815:
5809:
5794:
5788:
5770:
5764:
5746:
5745:
5744:
5738:
5722:
5694:
5676:
5670:
5655:
5649:
5634:
5628:
5612:
5610:
5607:
5605:
5604:
5589:
5578:
5549:"Peano axioms"
5539:
5516:(2): 193–253.
5496:
5479:
5459:
5449:
5429:
5403:
5396:
5378:
5371:
5353:
5346:
5326:
5301:
5266:
5247:(3): 338–352.
5231:
5215:
5208:
5182:
5132:
5120:
5077:
5066:
5043:
5020:
4988:
4957:
4922:
4888:
4881:
4855:
4826:
4819:
4801:
4784:
4777:
4754:
4736:
4720:
4703:
4686:
4670:
4653:
4620:
4617:. p. 113.
4603:
4583:
4576:
4558:
4526:
4511:
4490:
4466:
4439:
4432:
4410:
4391:
4371:
4341:
4334:
4316:
4286:
4260:
4243:"Introduction"
4234:
4209:
4184:
4165:
4143:
4112:
4105:
4083:
4081:
4078:
4075:
4074:
4057:
4020:Hamilton (1988
4012:
3999:
3990:
3986:Peano's axioms
3968:
3943:
3940:
3937:
3934:
3931:
3928:
3925:
3922:
3919:
3916:
3912:
3892:
3879:
3869:
3857:
3845:
3836:
3823:
3822:
3820:
3817:
3814:
3813:
3810:
3809:
3806:
3805:
3802:
3801:
3795:
3794:
3791:
3790:
3787:
3786:
3783:
3782:
3779:
3778:
3776:Transcendental
3772:
3771:
3765:
3764:
3755:
3745:
3744:
3741:
3740:
3737:
3736:
3733:
3732:
3729:
3728:
3726:
3720:
3719:
3713:
3712:
3710:Finite decimal
3703:
3693:
3692:
3689:
3688:
3685:
3684:
3678:
3677:
3674:
3673:
3670:
3669:
3663:
3662:
3656:
3655:
3648:
3647:
3637:
3625:
3620:
3599:
3587:
3582:
3561:
3549:
3544:
3523:
3511:
3506:
3485:
3473:
3468:
3451:Number systems
3439:
3438:
3432:
3426:
3423:Ordinal number
3420:
3414:
3408:
3401:
3400:
3384:
3381:
3231:
3230:
3229:
3228:
3225:
3201:
3192:
3186:
3180:
3168:
3167:
3160:
3145:
3105:
3083:ordinal number
3024:Main article:
3021:
3018:
3005:
3002:
2999:
2979:
2968:
2967:
2960:
2948:
2928:
2908:
2888:
2877:
2874:
2871:
2860:Main article:
2857:
2854:
2776:Giuseppe Peano
2771:
2768:
2754:
2695:
2684:
2677:(that is, the
2673:
2668:
2667:
2639:
2621:
2616:
2581:
2553:
2550:
2549:
2548:
2504:
2452:Distributivity
2449:
2448:
2447:
2395:
2341:
2265:
2222:
2219:
2185:. The numbers
2167:is called the
2155:is called the
2147:
2146:
2135:
2132:
2129:
2126:
2118:
2115:
2112:
2109:
2106:
2103:
2026:
2023:
2011:ordinal number
1882:
1879:
1866:
1863:
1860:
1854:
1850:
1845:
1780:
1754:
1699:
1696:
1675:
1672:
1669:
1664:
1659:
1654:
1605:
1594:multiplication
1589:
1588:Multiplication
1586:
1492:
1489:
1486:
1482:
1478:
1363:
1359:
1355:
1351:
1348:
1324:
1314:Given the set
1311:
1308:
1295:
1292:
1289:
1286:
1281:
1276:
1271:
1266:
1261:
1256:
1252:
1239:
1236:
1235:
1234:
1221:
1218:
1213:
1208:
1203:
1198:
1193:
1188:
1185:
1182:
1179:
1176:
1173:
1169:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1119:
1106:
1103:
1098:
1093:
1088:
1083:
1078:
1075:
1072:
1069:
1066:
1063:
1059:
1055:
1052:
1049:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
996:
973:
951:
950:
939:
936:
933:
930:
925:
920:
915:
910:
905:
900:
895:
890:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
852:
840:
827:
822:
817:
814:
811:
808:
805:
800:
795:
790:
785:
780:
775:
770:
765:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
696:
692:
669:
666:
646:equiconsistent
638:axiomatization
612:axiomatization
563:Henri Poincaré
558:
555:
543:array-elements
486:
482:
479:Giuseppe Peano
466:used the term
460:
457:
442:perfect number
401:Roman numerals
296:
293:
291:
288:
217:
213:
209:
153:
123:jersey numbers
26:
9:
6:
4:
3:
2:
8818:
8807:
8804:
8802:
8801:Number theory
8799:
8797:
8794:
8792:
8789:
8787:
8784:
8783:
8781:
8766:
8762:
8758:
8757:
8754:
8744:
8741:
8740:
8737:
8732:
8727:
8723:
8713:
8710:
8708:
8705:
8704:
8701:
8696:
8691:
8687:
8677:
8674:
8672:
8669:
8668:
8665:
8660:
8655:
8651:
8641:
8638:
8636:
8633:
8632:
8629:
8625:
8619:
8615:
8605:
8602:
8600:
8597:
8595:
8592:
8591:
8588:
8584:
8579:
8575:
8561:
8558:
8557:
8555:
8551:
8545:
8542:
8540:
8537:
8535:
8534:Polydivisible
8532:
8530:
8527:
8525:
8522:
8520:
8517:
8515:
8512:
8511:
8509:
8505:
8499:
8496:
8494:
8491:
8489:
8486:
8484:
8481:
8479:
8476:
8475:
8473:
8470:
8465:
8459:
8456:
8454:
8451:
8449:
8446:
8444:
8441:
8439:
8436:
8434:
8431:
8429:
8426:
8425:
8423:
8420:
8416:
8408:
8405:
8404:
8403:
8400:
8399:
8397:
8394:
8390:
8378:
8375:
8374:
8373:
8370:
8368:
8365:
8363:
8360:
8358:
8355:
8353:
8350:
8348:
8345:
8343:
8340:
8338:
8335:
8333:
8330:
8329:
8327:
8323:
8317:
8314:
8313:
8311:
8307:
8301:
8298:
8296:
8293:
8292:
8290:
8288:Digit product
8286:
8280:
8277:
8275:
8272:
8270:
8267:
8265:
8262:
8261:
8259:
8257:
8253:
8245:
8242:
8240:
8237:
8236:
8235:
8232:
8231:
8229:
8227:
8222:
8218:
8214:
8209:
8204:
8200:
8190:
8187:
8185:
8182:
8180:
8177:
8175:
8172:
8170:
8167:
8165:
8162:
8160:
8157:
8155:
8152:
8150:
8147:
8145:
8142:
8140:
8137:
8135:
8132:
8130:
8127:
8125:
8124:Erdős–Nicolas
8122:
8120:
8117:
8115:
8112:
8111:
8108:
8103:
8099:
8093:
8089:
8075:
8072:
8070:
8067:
8066:
8064:
8062:
8058:
8052:
8049:
8047:
8044:
8042:
8039:
8037:
8034:
8033:
8031:
8029:
8025:
8019:
8016:
8014:
8011:
8009:
8006:
8004:
8001:
7999:
7996:
7994:
7991:
7990:
7988:
7986:
7982:
7976:
7973:
7971:
7968:
7967:
7965:
7963:
7959:
7953:
7950:
7948:
7945:
7943:
7942:Superabundant
7940:
7938:
7935:
7933:
7930:
7928:
7925:
7923:
7920:
7918:
7915:
7913:
7910:
7908:
7905:
7903:
7900:
7898:
7895:
7893:
7890:
7888:
7885:
7883:
7880:
7878:
7875:
7873:
7870:
7868:
7865:
7863:
7860:
7858:
7855:
7853:
7850:
7848:
7845:
7844:
7842:
7840:
7836:
7832:
7828:
7824:
7819:
7815:
7805:
7802:
7800:
7797:
7795:
7792:
7790:
7787:
7785:
7782:
7780:
7777:
7775:
7772:
7770:
7767:
7765:
7762:
7760:
7757:
7755:
7752:
7750:
7747:
7746:
7743:
7739:
7734:
7730:
7720:
7717:
7715:
7712:
7710:
7707:
7705:
7702:
7701:
7698:
7694:
7689:
7685:
7675:
7672:
7670:
7667:
7665:
7662:
7660:
7657:
7655:
7652:
7650:
7647:
7645:
7642:
7640:
7637:
7635:
7632:
7630:
7627:
7625:
7622:
7620:
7617:
7615:
7612:
7610:
7607:
7605:
7602:
7600:
7597:
7595:
7592:
7590:
7587:
7585:
7582:
7580:
7577:
7576:
7573:
7566:
7562:
7544:
7541:
7539:
7536:
7534:
7531:
7530:
7528:
7524:
7521:
7519:
7518:4-dimensional
7515:
7505:
7502:
7501:
7499:
7497:
7493:
7487:
7484:
7482:
7479:
7477:
7474:
7472:
7469:
7467:
7464:
7462:
7459:
7458:
7456:
7454:
7450:
7444:
7441:
7439:
7436:
7434:
7431:
7429:
7428:Centered cube
7426:
7424:
7421:
7420:
7418:
7416:
7412:
7409:
7407:
7406:3-dimensional
7403:
7393:
7390:
7388:
7385:
7383:
7380:
7378:
7375:
7373:
7370:
7368:
7365:
7363:
7360:
7358:
7355:
7353:
7350:
7348:
7345:
7344:
7342:
7340:
7336:
7330:
7327:
7325:
7322:
7320:
7317:
7315:
7312:
7310:
7307:
7305:
7302:
7300:
7297:
7295:
7292:
7290:
7287:
7286:
7284:
7282:
7278:
7275:
7273:
7272:2-dimensional
7269:
7265:
7261:
7256:
7252:
7242:
7239:
7237:
7234:
7232:
7229:
7227:
7224:
7222:
7219:
7217:
7216:Nonhypotenuse
7214:
7213:
7210:
7203:
7199:
7189:
7186:
7184:
7181:
7179:
7176:
7174:
7171:
7169:
7166:
7165:
7162:
7155:
7151:
7141:
7138:
7136:
7133:
7131:
7128:
7126:
7123:
7121:
7118:
7116:
7113:
7111:
7108:
7106:
7103:
7102:
7099:
7094:
7089:
7085:
7075:
7072:
7070:
7067:
7065:
7062:
7060:
7057:
7055:
7052:
7051:
7048:
7041:
7037:
7027:
7024:
7022:
7019:
7017:
7014:
7012:
7009:
7007:
7004:
7002:
6999:
6997:
6994:
6993:
6990:
6985:
6979:
6975:
6965:
6962:
6960:
6957:
6955:
6954:Perfect power
6952:
6950:
6947:
6945:
6944:Seventh power
6942:
6940:
6937:
6935:
6932:
6930:
6927:
6925:
6922:
6920:
6917:
6915:
6912:
6910:
6907:
6905:
6902:
6900:
6897:
6896:
6893:
6888:
6883:
6879:
6875:
6867:
6862:
6860:
6855:
6853:
6848:
6847:
6844:
6832:
6824:
6822:
6819:
6818:
6815:
6809:
6806:
6804:
6801:
6798:
6794:
6788:
6784:
6779:
6777:
6774:
6772:
6771:Fuzzy numbers
6769:
6767:
6764:
6763:
6761:
6757:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6726:
6723:
6721:
6718:
6714:
6711:
6710:
6709:
6706:
6704:
6701:
6699:
6696:
6695:
6693:
6691:
6687:
6683:
6675:
6672:
6670:
6667:
6665:
6662:
6661:
6660:
6656:
6653:
6651:
6648:
6646:
6643:
6618:
6615:
6613:
6610:
6608:
6605:
6603:
6600:
6598:
6595:
6594:
6592:
6590:
6585:
6579:
6576:
6574:
6573:Biquaternions
6571:
6569:
6566:
6540:
6537:
6535:
6532:
6530:
6527:
6502:
6501:
6499:
6493:
6464:
6461:
6436:
6433:
6408:
6405:
6380:
6376:
6373:
6372:
6370:
6368:
6364:
6358:
6355:
6351:
6348:
6347:
6346:
6343:
6341:
6338:
6336:
6333:
6331:
6328:
6301:
6298:
6296:
6293:
6268:
6265:
6263:
6260:
6235:
6232:
6207:
6204:
6179:
6176:
6175:
6173:
6171:
6166:
6161:
6154:
6149:
6147:
6142:
6140:
6135:
6134:
6131:
6123:
6119:
6115:
6111:
6107:
6106:
6101:
6097:
6096:
6086:
6080:
6074:
6070:
6066:
6062:
6058:
6046:
6042:
6038:
6034:
6030:
6026:
6022:
6016:
6012:
6011:
6005:
6001:
5995:
5991:
5990:
5984:
5980:
5974:
5970:
5966:
5965:
5959:
5955:
5949:
5945:
5944:
5938:
5934:
5928:
5924:
5923:
5918:
5914:
5910:
5904:
5900:
5899:
5894:
5890:
5886:
5882:
5876:
5872:
5868:
5864:
5860:
5854:
5850:
5849:
5844:
5840:
5836:
5830:
5826:
5825:
5820:
5816:
5812:
5806:
5802:
5801:
5795:
5791:
5785:
5781:
5780:
5775:
5771:
5767:
5761:
5757:
5756:
5751:
5747:
5741:
5735:
5731:
5727:
5723:
5711:
5710:
5705:
5701:
5700:
5697:
5691:
5687:
5686:
5681:
5677:
5673:
5667:
5663:
5662:
5656:
5652:
5646:
5642:
5641:
5640:Real Analysis
5635:
5631:
5625:
5621:
5620:
5614:
5613:
5601:
5596:
5594:
5587:
5582:
5566:
5562:
5558:
5554:
5550:
5543:
5534:
5529:
5524:
5519:
5515:
5511:
5507:
5500:
5492:
5491:
5483:
5476:
5472:
5471:
5463:
5456:
5452:
5446:
5442:
5441:
5433:
5418:
5414:
5407:
5399:
5393:
5389:
5382:
5374:
5368:
5364:
5357:
5349:
5343:
5339:
5338:
5330:
5315:
5311:
5305:
5297:
5293:
5289:
5285:
5281:
5277:
5270:
5262:
5258:
5254:
5250:
5246:
5242:
5235:
5227:
5226:
5219:
5211:
5209:0-253-33020-3
5205:
5200:
5199:
5193:
5186:
5178:
5174:
5170:
5166:
5162:
5158:
5154:
5150:
5146:
5142:
5136:
5129:
5124:
5116:
5112:
5110:
5102:
5098:
5094:
5091:
5088:
5081:
5073:
5069:
5063:
5059:
5058:
5053:
5047:
5039:
5035:
5031:
5024:
5016:
5012:
5011:
5003:
4997:
4995:
4993:
4976:
4972:
4971:jsoftware.com
4968:
4961:
4953:
4949:
4945:
4941:
4937:
4933:
4926:
4910:
4906:
4902:
4898:
4892:
4884:
4878:
4874:
4869:
4868:
4859:
4843:
4839:
4838:
4830:
4822:
4816:
4812:
4805:
4798:
4794:
4788:
4780:
4774:
4770:
4769:
4761:
4759:
4750:
4746:
4740:
4732:
4731:
4724:
4716:
4715:
4707:
4699:
4698:
4690:
4682:
4681:
4674:
4666:
4665:
4657:
4649:
4648:Maths History
4645:
4639:
4637:
4635:
4633:
4631:
4629:
4627:
4625:
4616:
4615:
4607:
4599:
4598:
4593:
4587:
4579:
4577:0-19-506135-7
4573:
4569:
4562:
4554:6 = 1 + 2 + 3
4550:
4546:
4545:
4540:
4536:
4530:
4522:
4518:
4514:
4508:
4504:
4503:
4494:
4486:
4485:
4480:
4476:
4470:
4454:
4450:
4443:
4435:
4429:
4425:
4421:
4414:
4398:
4394:
4388:
4384:
4383:
4375:
4359:
4355:
4351:
4345:
4337:
4335:0-471-37568-3
4331:
4327:
4320:
4313:
4307:
4303:
4301:
4296:
4290:
4282:
4278:
4274:
4270:
4264:
4256:
4252:
4248:
4244:
4238:
4223:
4219:
4213:
4204:
4203:
4198:
4195:
4188:
4172:
4168:
4162:
4158:
4154:
4147:
4132:
4128:
4121:
4119:
4117:
4108:
4102:
4098:
4091:
4089:
4084:
4071:
4070:universal set
4067:
4061:
4054:
4050:
4049:Morash (1991)
4030:
4021:
4016:
4009:
4003:
3994:
3987:
3957:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3914:
3901:
3896:
3889:
3883:
3873:
3866:
3861:
3854:
3849:
3840:
3834:
3828:
3824:
3800:
3797:
3796:
3777:
3774:
3773:
3770:
3767:
3766:
3763:
3760:
3759:
3756:
3754:
3751:
3750:
3747:
3746:
3727:
3725:
3722:
3721:
3718:
3715:
3714:
3711:
3708:
3707:
3704:
3702:
3699:
3698:
3695:
3694:
3683:
3680:
3679:
3668:
3665:
3664:
3661:
3660:Prime numbers
3658:
3657:
3653:
3650:
3649:
3645:
3642:
3641:
3638:
3618:
3611:
3608:
3607:
3604:
3603:
3600:
3580:
3573:
3570:
3569:
3566:
3565:
3562:
3542:
3535:
3532:
3531:
3528:
3527:
3524:
3504:
3497:
3494:
3493:
3490:
3489:
3486:
3466:
3459:
3456:
3455:
3452:
3447:
3446:
3443:
3442:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3411:Countable set
3409:
3406:
3403:
3402:
3398:
3392:
3387:
3380:
3374:
3373:cardinalities
3370:
3369:singleton set
3365:
3361:
3357:
3353:
3344:
3336:
3335:Ernst Zermelo
3331:
3329:
3324:
3322:
3318:
3313:
3311:
3307:
3303:
3302:set inclusion
3298:
3293:
3288:
3282:
3278:
3269:
3257:
3244:
3238:
3236:
3226:
3218:
3214:
3210:
3206:
3202:
3193:
3187:
3181:
3175:
3174:
3173:
3172:
3171:
3165:
3161:
3158:
3154:
3150:
3146:
3141:
3137:
3133:
3129:
3118:
3114:
3110:
3106:
3103:
3095:
3094:
3093:
3090:
3088:
3084:
3080:
3076:
3071:
3061:
3057:
3053:
3049:
3037:
3027:
3017:
3003:
3000:
2997:
2977:
2965:
2961:
2946:
2926:
2906:
2886:
2878:
2875:
2872:
2869:
2868:
2867:
2863:
2853:
2851:
2847:
2842:
2840:
2836:
2831:
2826:
2816:
2800:
2796:
2791:
2789:
2785:
2781:
2777:
2767:
2743:
2739:
2737:
2733:
2729:
2726:in 1933. The
2725:
2721:
2716:
2714:
2709:
2704:
2680:
2661:
2657:
2653:
2630:
2626:
2622:
2612:
2609:
2605:
2604:
2599:
2595:
2594:the same size
2572:
2567:
2566:
2565:
2563:
2559:
2544:
2537:
2530:
2526:
2520:
2514:
2509:
2508:zero divisors
2505:
2500:
2496:
2492:
2488:
2484:
2480:
2476:
2470:
2464:
2458:
2453:
2450:
2444:
2440:
2434:
2429:
2428:
2425:
2421:
2415:
2411:
2405:
2400:
2397:Existence of
2396:
2392:
2388:
2384:
2380:
2374:
2370:
2366:
2362:
2356:
2350:
2345:
2344:Commutativity
2342:
2338:
2334:
2330:
2326:
2322:
2318:
2312:
2308:
2304:
2300:
2296:
2292:
2286:
2280:
2274:
2269:
2268:Associativity
2266:
2262:
2258:
2252:
2248:
2242:
2236:
2231:
2228:
2227:
2226:
2218:
2216:
2212:
2207:
2201:
2195:
2189:
2183:
2177:
2172:
2171:
2165:
2160:
2159:
2153:
2133:
2130:
2127:
2124:
2116:
2113:
2110:
2107:
2104:
2101:
2094:
2093:
2092:
2089:
2083:
2076:
2070:
2064:
2059:
2055:
2050:
2048:
2043:
2039:
2033:
2022:
2019:
2018:
2012:
2008:
2003:
2000:
1996:
1990:
1986:
1982:
1978:
1972:
1968:
1962:
1956:
1950:
1945:
1940:
1936:
1932:
1926:
1920:
1916:
1911:
1906:
1904:
1899:
1895:
1889:
1878:
1861:
1858:
1834:
1830:
1823:
1819:
1815:
1809:
1807:
1803:
1799:
1795:
1769:
1743:
1740:
1734:
1730:
1726:
1722:
1718:
1714:
1710:
1705:
1695:
1693:
1692:prime numbers
1689:
1670:
1667:
1662:
1645:. This turns
1643:
1639:
1635:
1631:
1627:
1620:
1603:
1595:
1585:
1582:
1575:
1568:
1564:
1560:
1556:
1552:
1548:
1544:
1537:
1531:
1529:
1525:
1521:
1517:
1513:
1509:
1506:
1487:
1484:
1469:
1463:
1459:
1455:
1451:
1444:
1440:
1436:
1432:
1426:
1420:
1413:
1409:
1405:
1401:
1397:
1393:
1387:
1383:
1378:
1349:
1346:
1339:
1307:
1290:
1284:
1279:
1269:
1264:
1254:
1219:
1216:
1206:
1201:
1196:
1186:
1180:
1177:
1174:
1171:
1163:
1160:
1154:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1120:
1104:
1101:
1091:
1086:
1076:
1070:
1067:
1064:
1061:
1053:
1050:
1044:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1010:
1009:
1008:
994:
960:
956:
934:
928:
923:
913:
908:
898:
893:
883:
877:
874:
871:
868:
865:
862:
859:
856:
853:
841:
825:
815:
809:
803:
798:
788:
783:
773:
768:
758:
752:
749:
746:
743:
740:
737:
734:
723:
722:
721:
714:
711:
694:
680:
675:
665:
663:
659:
655:
651:
647:
643:
639:
635:
631:
630:
625:
621:
617:
613:
609:
604:
602:
598:
594:
590:
587:
583:
579:
575:
570:
568:
564:
554:
552:
548:
544:
540:
536:
535:loop counters
532:
528:
524:
520:
514:
512:
508:
504:
500:
496:
492:
480:
476:
471:
469:
465:
456:
454:
451:, China, and
450:
445:
443:
439:
435:
431:
428:philosophers
427:
423:
418:
415:
410:
406:
402:
398:
394:
390:
386:
378:
374:
370:
366:
361:
359:
355:
351:
347:
343:
339:
335:
330:
328:
324:
316:
312:
307:
302:
295:Ancient roots
287:
285:
284:combinatorics
281:
277:
273:
272:number theory
269:
268:prime numbers
265:
260:
258:
254:
247:
243:
239:
235:
215:
211:
207:
200:
196:
187:
182:
178:
174:
170:
167:. Many other
141:
136:
131:
126:
124:
120:
119:
114:
113:
108:
104:
103:
98:
93:
91:
87:
83:
82:whole numbers
79:
78:whole numbers
69:
62:
58:
54:
50:
41:
37:
33:
19:
8498:Transposable
8362:Narcissistic
8269:Digital root
8189:Super-Poulet
8149:Jordan–Pólya
8098:prime factor
8003:Noncototient
7970:Almost prime
7952:Superperfect
7927:Refactorable
7922:Quasiperfect
7897:Hyperperfect
7738:Pseudoprimes
7709:Wall–Sun–Sun
7644:Ordered Bell
7614:Fuss–Catalan
7526:non-centered
7476:Dodecahedral
7453:non-centered
7339:non-centered
7241:Wolstenholme
6986:× 2 ± 1
6983:
6982:Of the form
6949:Eighth power
6929:Fourth power
6873:
6792:
6782:
6597:Dual numbers
6589:hypercomplex
6379:Real numbers
6177:
6121:
6103:
6068:
6051:15 September
6049:. Retrieved
6045:the original
6040:
6036:
6009:
5988:
5963:
5942:
5921:
5897:
5870:
5867:Levy, Azriel
5847:
5823:
5799:
5778:
5774:Halmos, Paul
5754:
5750:Eves, Howard
5729:
5714:. Retrieved
5708:
5684:
5660:
5639:
5618:
5609:Bibliography
5602:, p. 52
5581:
5569:. Retrieved
5552:
5542:
5513:
5509:
5499:
5489:
5482:
5474:
5469:
5462:
5454:
5439:
5432:
5420:. Retrieved
5416:
5406:
5387:
5381:
5362:
5356:
5336:
5329:
5317:. Retrieved
5313:
5304:
5279:
5275:
5269:
5244:
5240:
5234:
5224:
5218:
5197:
5185:
5155:(1): 85–95.
5152:
5148:
5135:
5130:, Chapter 15
5123:
5115:the original
5108:
5101:the original
5096:
5093:
5090:
5080:
5056:
5052:Gray, Jeremy
5046:
5037:
5033:
5023:
5009:
4979:. Retrieved
4970:
4965:Hui, Roger.
4960:
4935:
4931:
4925:
4913:. Retrieved
4900:
4891:
4866:
4858:
4846:. Retrieved
4836:
4829:
4810:
4804:
4799:, p. 2)
4787:
4767:
4748:
4739:
4729:
4723:
4713:
4706:
4696:
4689:
4679:
4673:
4663:
4656:
4647:
4613:
4606:
4600:(in French).
4596:
4586:
4567:
4561:
4548:
4542:
4529:
4499:
4493:
4482:
4469:
4457:. Retrieved
4442:
4423:
4413:
4401:. Retrieved
4381:
4374:
4362:. Retrieved
4353:
4344:
4325:
4319:
4306:the original
4298:
4289:
4281:the original
4273:Ishango bone
4263:
4255:the original
4247:Ishango bone
4237:
4225:. Retrieved
4221:
4212:
4200:
4187:
4175:. Retrieved
4156:
4146:
4134:. Retrieved
4130:
4096:
4060:
4052:
4033:
4029:Halmos (1960
4024:
4015:
4002:
3993:
3955:
3895:
3882:
3872:
3865:Bluman (2010
3860:
3848:
3839:
3827:
3609:
3366:
3359:
3355:
3351:
3338:
3332:
3325:
3314:
3296:
3286:
3280:
3276:
3267:
3242:
3239:
3235:Peano axioms
3232:
3216:
3212:
3208:
3204:
3169:
3163:
3156:
3139:
3135:
3131:
3127:
3116:
3112:
3108:
3091:
3079:infinite set
3072:
3056:set theories
3029:
2969:
2865:
2862:Peano axioms
2856:Peano axioms
2843:
2834:
2827:
2792:
2788:Peano axioms
2773:
2742:Georges Reeb
2740:
2728:hypernatural
2718:A countable
2717:
2705:
2669:
2656:limit points
2652:well-ordered
2606:and to have
2601:
2555:
2542:
2535:
2528:
2524:
2518:
2512:
2498:
2494:
2490:
2486:
2482:
2478:
2474:
2468:
2462:
2456:
2442:
2438:
2432:
2423:
2419:
2413:
2409:
2403:
2390:
2386:
2382:
2378:
2372:
2368:
2364:
2360:
2354:
2348:
2336:
2332:
2328:
2324:
2320:
2316:
2310:
2306:
2302:
2298:
2294:
2290:
2284:
2278:
2272:
2260:
2256:
2250:
2246:
2240:
2234:
2224:
2211:divisibility
2205:
2199:
2193:
2187:
2181:
2175:
2168:
2163:
2156:
2151:
2148:
2087:
2081:
2074:
2068:
2062:
2053:
2051:
2049:is assumed.
2041:
2037:
2031:
2028:
2016:
2007:well-ordered
2004:
1998:
1994:
1988:
1984:
1980:
1976:
1970:
1966:
1960:
1954:
1948:
1938:
1934:
1930:
1924:
1918:
1914:
1907:
1905:is assumed.
1897:
1893:
1887:
1884:
1832:
1828:
1821:
1817:
1813:
1810:
1805:
1793:
1732:
1728:
1724:
1720:
1716:
1712:
1708:
1701:
1641:
1637:
1633:
1629:
1625:
1618:
1591:
1580:
1573:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1535:
1532:
1461:
1457:
1453:
1449:
1442:
1438:
1434:
1430:
1424:
1418:
1411:
1407:
1403:
1399:
1395:
1391:
1385:
1381:
1313:
1241:
952:
715:
709:
678:
671:
619:
605:
591:
584:suggested a
571:
560:
515:
472:
467:
462:
446:
422:abstractions
419:
408:
404:
362:
331:
320:
311:Ishango bone
280:enumerations
276:partitioning
264:divisibility
261:
234:real numbers
185:
134:
127:
116:
110:
106:
100:
96:
94:
89:
81:
77:
72:1, 2, 3, ...
67:
60:
52:
46:
36:
8519:Extravagant
8514:Equidigital
8469:permutation
8428:Palindromic
8402:Automorphic
8300:Sum-product
8279:Sum-product
8234:Persistence
8129:Erdős–Woods
8051:Untouchable
7932:Semiperfect
7882:Hemiperfect
7543:Tesseractic
7481:Icosahedral
7461:Tetrahedral
7392:Dodecagonal
7093:Recursively
6964:Prime power
6939:Sixth power
6934:Fifth power
6914:Power of 10
6872:Classes of
6759:Other types
6578:Bioctonions
6435:Quaternions
6122:apronus.com
5600:Levy (1979)
5107:"access to
5087:"Kronecker"
4848:13 February
4459:13 February
3306:total order
3121:of any set
3107:Define the
2608:cardinality
2506:No nonzero
2149:The number
1910:total order
1739:commutative
1571:. That is,
1516:free monoid
1505:commutative
1456:+ S(1) = S(
1437:+ S(0) = S(
551:ISO 80000-2
503:Paul Halmos
453:Mesoamerica
389:Brahmagupta
385:Mesoamerica
358:place-value
354:Babylonians
342:hieroglyphs
169:number sets
49:mathematics
8780:Categories
8731:Graphemics
8604:Pernicious
8458:Undulating
8433:Pandigital
8407:Trimorphic
8008:Nontotient
7857:Arithmetic
7471:Octahedral
7372:Heptagonal
7362:Pentagonal
7347:Triangular
7188:Sierpiński
7110:Jacobsthal
6909:Power of 3
6904:Power of 2
6713:Projective
6686:Infinities
5523:1703.00425
4981:19 January
4837:Set Theory
4403:3 February
4364:23 January
4106:0122384407
4080:References
3753:Irrational
3310:well-order
3038:that have
2850:consistent
2795:set theory
2611:aleph-null
2547:(or both).
2091:such that
1460:+1) = S(S(
1238:Properties
650:set theory
434:Archimedes
430:Pythagoras
327:tally mark
8488:Parasitic
8337:Factorion
8264:Digit sum
8256:Digit sum
8074:Fortunate
8061:Primorial
7975:Semiprime
7912:Practical
7877:Descartes
7872:Deficient
7862:Betrothed
7704:Wieferich
7533:Pentatope
7496:pyramidal
7387:Decagonal
7382:Nonagonal
7377:Octagonal
7367:Hexagonal
7226:Practical
7173:Congruent
7105:Fibonacci
7069:Loeschian
6797:solenoids
6617:Sedenions
6463:Octonions
6110:EMS Press
5919:(2008) .
5728:(2007) .
5716:13 August
5682:(1963) .
5571:8 October
5296:0024-6093
5128:Eves 1990
4915:4 October
4594:(1881) .
4328:. Wiley.
4227:11 August
4202:MathWorld
4153:"integer"
4136:11 August
4008:Eves 1990
3939:…
3799:Imaginary
3256:bijection
3157:inductive
3109:successor
3102:empty set
2638:∅
2629:empty set
2598:bijection
2580:∅
2571:empty set
2203:and
2170:remainder
2021:(omega).
1853:∗
1671:×
1663:∗
1604:×
1596:operator
1358:→
1350::
1285:∪
1280:∗
1217:≥
1178:≥
1164:∈
1149:…
1054:∈
1039:…
995:∗
929:∪
924:∗
804:∖
769:∗
656:with the
606:In 1881,
547:ISO 31-11
523:empty set
338:Egyptians
8796:Integers
8560:Friedman
8493:Primeval
8438:Repdigit
8395:-related
8342:Kaprekar
8316:Meertens
8239:Additive
8226:dynamics
8134:Friendly
8046:Sociable
8036:Amicable
7847:Abundant
7827:dynamics
7649:Schröder
7639:Narayana
7609:Eulerian
7599:Delannoy
7594:Dedekind
7415:centered
7281:centered
7168:Amenable
7125:Narayana
7115:Leonardo
7011:Mersenne
6959:Powerful
6899:Achilles
6206:Integers
6168:Sets of
6031:(1923).
5895:(1999).
5869:(1979).
5845:(1966).
5776:(1960).
5752:(1990).
5706:(1901).
5565:Archived
5557:Springer
5143:(1881).
5072:Archived
5054:(2008).
4975:Archived
4952:40187000
4938:(2): 7.
4909:Archived
4842:Archived
4544:Elements
4521:69792712
4484:Elements
4453:Archived
4397:Archived
4358:Archived
4177:28 March
4171:Archived
3701:Fraction
3534:Rational
3417:Sequence
3383:See also
3274:. Also,
3268:counting
2835:provable
2713:sequence
2179:by
2158:quotient
2025:Division
1802:semiring
1742:semiring
1528:integers
1441:+0) = S(
1428:. Thus,
1416:for all
1377:addition
1310:Addition
963:denoted
959:integers
668:Notation
393:computus
334:numerals
173:integers
121:, (e.g.
86:integers
55:are the
8733:related
8697:related
8661:related
8659:Sorting
8544:Vampire
8529:Harshad
8471:related
8443:Repunit
8357:Lychrel
8332:Dudeney
8184:Størmer
8179:Sphenic
8164:Regular
8102:divisor
8041:Perfect
7937:Sublime
7907:Perfect
7634:Motzkin
7589:Catalan
7130:Padovan
7064:Leyland
7059:Idoneal
7054:Hilbert
7026:Woodall
6787:numbers
6619: (
6465: (
6437: (
6409: (
6381: (
6302: (
6300:Periods
6269: (
6236: (
6208: (
6180: (
6162:systems
6112:, 2001
5898:Algebra
5422:27 July
5319:27 July
5261:1270381
5177:1507856
5169:2369151
3954:of all
3610:Natural
3572:Integer
3458:Complex
3326:If one
3177:0 = { }
3147:By the
3098:0 = { }
2939:equals
2919:, then
2830:theorem
2782:called
2533:, then
2244:, both
2230:Closure
1974:, then
1766:is not
1686:into a
1621:× 0 = 0
1561:+ 0) =
1540:, then
961:(often
957:of the
539:string-
411:) from
290:History
165:
143:
57:numbers
32:Integer
8599:Odious
8524:Frugal
8478:Cyclic
8467:Digit-
8174:Smooth
8159:Pronic
8119:Cyclic
8096:Other
8069:Euclid
7719:Wilson
7693:Primes
7352:Square
7221:Polite
7183:Riesel
7178:Knödel
7140:Perrin
7021:Thabit
7006:Fermat
6996:Cullen
6919:Square
6887:Powers
6587:Other
6160:Number
6075:
6017:
5996:
5975:
5950:
5929:
5905:
5877:
5855:
5831:
5807:
5786:
5762:
5736:
5692:
5668:
5647:
5626:
5447:
5394:
5369:
5344:
5294:
5259:
5206:
5175:
5167:
5064:
4950:
4879:
4817:
4775:
4574:
4535:Euclid
4519:
4509:
4475:Euclid
4430:
4389:
4332:
4300:UNESCO
4163:
4103:
3877:place.
3292:subset
3211:−1 ∪ {
3153:closed
3100:, the
3046:as an
2724:Skolem
2708:finite
2466:, and
2441:× 1 =
2422:× 1 =
2412:+ 0 =
2282:, and
1928:where
1831:× 1 =
1816:+ 1 =
1768:closed
1553:(0) =
1545:+ 1 =
1508:monoid
1452:+ 2 =
1433:+ 1 =
1384:+ 0 =
955:subset
529:often
509:, and
438:Euclid
414:nullus
409:nullae
356:had a
350:Louvre
346:Karnak
257:embeds
238:limits
193:; the
51:, the
8640:Prime
8635:Lucky
8624:sieve
8553:Other
8539:Smith
8419:Digit
8377:Happy
8352:Keith
8325:Other
8169:Rough
8139:Giuga
7604:Euler
7466:Cubic
7120:Lucas
7016:Proth
6795:-adic
6785:-adic
6542:Over
6503:Over
6497:types
6495:Split
5518:arXiv
5165:JSTOR
5095:[
5036:[
5005:(PDF)
4948:S2CID
4040:0 = ∅
4036:0 ∈ ω
3819:Notes
3358:) = {
3290:is a
3258:from
3096:Call
2846:model
2688:) is
2510:: if
2493:) + (
2485:) = (
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