3391:
40:
306:
8729:
6795:
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
5064:
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:
6025:
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.
3869:
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:
6037:
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:
3945:
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.
6831:
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:
705:
6607:
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6453:
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6397:
6369:
6282:
6249:
6221:
6193:
3971:
2764:
1790:
1764:
1334:
983:
163:
2648:
2590:
1013:
1614:
4287:
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 "
3848:, 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."
5007:
3404:
3014:
845:
592:
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:
628:
3990:
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."
5079:
4304:
4269:
4243:
3434:
3025:
314:
4235:
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
4605:
6824:
6143:
4261:
5048:
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
4350:
3999:"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (
1245:
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6011:
5990:
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5944:
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5441:
5388:
5363:
5338:
5058:
4873:
4811:
4769:
4503:
4424:
4383:
4157:
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
5099:
4994:
4901:
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
7631:
6817:
4389:
7626:
5557:
7641:
7621:
5200:
4568:
4326:
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
5549:
8334:
7914:
2719:
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
241:
8758:
6680:
4721:
2678:
474:
5461:
4445:
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.
7636:
5889:
4967:
4097:
8420:
4759:
4163:
4024:, p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I)
5431:
4834:
561:
In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers.
8773:
6641:
6102:
5553:
4705:
4298:
2624:
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
7736:
477:
wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0. In 1889,
8086:
7405:
7198:
6136:
1770:
under subtraction (that is, subtracting one natural from another does not always result in another natural), means that
573:
8121:
8091:
7766:
7756:
8262:
7676:
7410:
7390:
6292:
6097:
6030:
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
7952:
5303:"Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions"
8116:
3898:
8211:
7834:
7591:
7400:
7382:
7276:
7266:
7256:
5481:
5093:
2845:
833:{\displaystyle \{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}}
577:
8096:
2009:: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an
8753:
8339:
7884:
7505:
7291:
7286:
7281:
7271:
7248:
6675:
6631:
4545:
In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example
3051:
2814:
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
538:
279:
275:
80:
as the natural numbers together with zero, excluding zero from the natural numbers, while in other writings, the
1648:
7324:
6798:
6670:
5770:
5448:...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
4247:
337:
7581:
2838:
5631:
8450:
8415:
8201:
8111:
7985:
7960:
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7859:
7471:
7453:
7373:
6129:
5791:
2731:
1943:
1687:
8768:
8710:
7980:
7854:
7485:
7261:
7041:
6968:
6758:
6287:
5961:
5497:
Fletcher, Peter; Hrbacek, Karel; Kanovei, Vladimir; Katz, Mikhail G.; Lobry, Claude; Sanders, Sam (2017).
5232:
Baratella, Stefano; Ferro, Ruggero (1993). "A theory of sets with the negation of the axiom of infinity".
5216:
5133:
4671:
3077:, although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to
1839:
661:
607:
198:
4273:
1227:{\displaystyle \{0,1,2,\dots \}=\{x\in \mathbb {Z} :x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}}
7965:
7819:
7746:
6901:
6743:
6579:
6307:
6302:
3768:
3152:
2229:
1767:
1519:
1007:" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:
8674:
8314:
4531:
3390:
4688:
4471:
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:
8607:
8501:
8465:
8206:
7929:
7909:
7726:
7395:
7183:
7155:
6496:
6254:
4536:
4494:
4476:
3880:
3614:
3576:
3538:
3500:
3462:
3086:
3059:
600:
585:
542:
530:
341:
300:
6092:
4152:. 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
8329:
8193:
8188:
8156:
7919:
7894:
7889:
7864:
7794:
7790:
7721:
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7443:
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6707:
6617:
6574:
6556:
6334:
4342:
3316:
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:
2214:
2046:
1902:
1523:
1467:
1112:{\displaystyle \{1,2,3,\dots \}=\{x\in \mathbb {Z} :x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}}
685:
506:
490:
357:
237:
8728:
6590:
6513:
6474:
6436:
6408:
6380:
6352:
6265:
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3954:
2747:
1773:
1747:
1317:
966:
146:
8732:
8486:
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8267:
8246:
8018:
7899:
7849:
7771:
7741:
7681:
7448:
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7359:
7072:
6612:
6324:
5510:
5157:
5107:
4940:
3396:
2963:
2057:
1797:
1337:
510:
7616:
2633:
2575:
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
8763:
8626:
8571:
8425:
8400:
8374:
8151:
7829:
7824:
7751:
7731:
7716:
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7339:
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7314:
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6697:
6636:
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6317:
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6065:
6007:
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5965:
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5909:
5895:
5881:
5867:
5845:
5821:
5797:
5776:
5752:
5726:
5682:
5658:
5637:
5616:
5437:
5384:
5359:
5334:
5284:
5196:
5054:
4869:
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4807:
4765:
4564:
4509:
4499:
4420:
4417:
The
Development of Mathematics Throughout the Centuries: A brief history in a cultural context
4379:
4322:
4186:
4153:
4093:
3978:
3752:
3723:
3148:
3047:
3035:
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2659:
2600:
between them. The set of natural numbers itself, and any bijective image of it, is said to be
673:
657:
581:
566:
562:
526:
396:
376:
176:
129:
6057:
5184:
4889:
4412:
4373:
1599:
8662:
8455:
8041:
8013:
8003:
7995:
7879:
7844:
7839:
7806:
7500:
7463:
7354:
7349:
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7334:
7306:
7193:
7145:
7140:
7097:
7036:
6717:
6692:
6626:
6535:
6501:
6342:
6312:
6259:
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6053:
6021:
5885:
5718:
5696:
5672:
5520:
5276:
5241:
5149:
4932:
4857:
4588:
3791:
3761:
3681:
3666:
3074:
2783:
2779:
2398:
1737:. These properties of addition and multiplication make the natural numbers an instance of a
1511:
943:{\displaystyle \;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}}
633:
615:
611:
518:
498:
494:
180:
5541:
5267:
Kirby, Laurie; Paris, Jeff (1982). "Accessible
Independence Results for Peano Arithmetic".
5253:
5169:
4737:
4636:
39:
8638:
8527:
8460:
8386:
8309:
8283:
8101:
7814:
7671:
7606:
7576:
7566:
7561:
7227:
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6926:
6866:
6665:
6569:
6226:
5811:
5249:
5165:
4897:
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4058:
3716:
3533:
3428:
2848:
of Peano arithmetic inside set theory. An important consequence is that, if set theory is
2557:
2015:
463:
368:
344:
for 1, 10, and all powers of 10 up to over 1 million. A stone carving from
322:
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
194:
139:
122:
101:
4655:
2993:
2722:
satisfying the Peano
Arithmetic (that is, the first-order Peano axioms) was developed by
203:
8643:
8511:
8496:
8360:
8324:
8299:
8175:
8146:
8131:
8008:
7904:
7874:
7601:
7556:
7433:
7031:
7026:
7021:
6993:
6978:
6891:
6876:
6854:
6712:
6702:
6687:
6506:
6374:
5676:
5480:
Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962).
3709:
3457:
3422:
3320:
3082:
2973:
2942:
2922:
2902:
2882:
2775:
2593:
2561:
2451:
2010:
1703:
1593:
990:
645:
641:
637:
623:
478:
441:
425:
400:
387:. The use of a numeral 0 in modern times originated with the Indian mathematician
333:
305:
249:
245:
117:
111:
6110:
8747:
8566:
8550:
8491:
8445:
8141:
8126:
8036:
7761:
7319:
7188:
7150:
7107:
6988:
6973:
6963:
6921:
6911:
6886:
6775:
6748:
6657:
5835:
5189:
4858:
4062:
3450:
3410:
3372:
3368:
3334:
3301:
2602:
2343:
2267:
1376:
596:
550:
283:
271:
76:
Some authors acknowledge both definitions whenever convenient. Some texts define the
5525:
5498:
5405:
4944:
4189:
2828:
The sets used to define natural numbers satisfy Peano axioms. It follows that every
317:) is believed to have been used 20,000 years ago for natural number arithmetic.
8602:
8591:
8506:
8344:
8319:
8236:
8136:
8106:
8081:
8065:
7970:
7937:
7686:
7660:
7571:
7510:
7087:
6983:
6916:
6896:
6871:
6738:
6540:
5705:. Translated by Beman, Wooster Woodruff. Chicago, IL: Open Court Publishing Company
5499:"Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others"
4441:
4265:
4239:
4119:
4044:
gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1:
3659:
3234:
3078:
2861:
2787:
2741:
2727:
2655:
2610:
2507:
2210:
1691:
310:
267:
263:
4959:
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
6001:
5980:
5955:
5934:
5913:
5839:
5815:
5746:
5652:
5610:
4145:
2446:. However, the "existence of additive identity element" property is not satisfied
8561:
8436:
8241:
7705:
7596:
7551:
7546:
7296:
7203:
7102:
6931:
6906:
6881:
6564:
6346:
5859:
5766:
5742:
5044:
4864:(1. ed., 1. print ed.). Boca Raton, Fla. : Chapman & Hall/CRC. p.
4828:
4413:"Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations"
3495:
3305:
2607:
1909:
1738:
1515:
1504:
502:
452:
421:
388:
384:
233:
125:
in sports)—which do not have the properties of numbers in a mathematical sense.
48:
569:
summarized his belief as "God made the integers, all else is the work of man".
59:
0, 1, 2, 3, etc., possibly excluding 0. Some define the natural numbers as the
17:
8698:
8679:
7975:
7586:
6545:
6402:
5022:
3386:
3309:
3055:
2849:
2794:
2707:
2651:
2592:. This concept of "size" relies on maps between sets, such that two sets have
2006:
649:
433:
429:
364:
326:
168:
6809:
5302:
5288:
5245:
8304:
8231:
8223:
8028:
7942:
7060:
5280:
5137:
4723:
Advanced
Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166
4513:
4194:
3255:
3101:
2628:
2597:
2570:
2169:
1690:
with identity element 1; a generator set for this monoid is the set of
546:
522:
353:
256:
444:
which comes shortly afterward, Euclid treats 1 as a number like any other.
5001:
4936:
8405:
6653:
6584:
6430:
5700:
5681:. Translated by Beman, Wooster Woodruff (reprint ed.). Dover Books.
5328:
4210:
3895:, p. 15) include zero in the natural numbers: 'Intuitively, the set
3700:
3416:
3327:
2844:
The definition of the integers as sets satisfying Peano axioms provide a
2712:
2157:
1801:
1741:
534:
392:
367:
can be considered as a number, with its own numeral. The use of a 0
3170:
It follows that the natural numbers are defined iteratively as follows:
1299:{\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}}
447:
Independent studies on numbers also occurred at around the same time in
8410:
8069:
6198:
5161:
3571:
2829:
2666:; this is also the ordinal number of the set of natural numbers itself.
1527:
958:
172:
85:
31:
4015:, pp. 117 ff) calls them "Peano's Postulates" and begins with "1.
1702:
Addition and multiplication are compatible, which is expressed in the
6788:
6152:
4758:
Křížek, Michal; Somer, Lawrence; Šolcová, Alena (21 September 2021).
4527:
4467:
4292:
3291:
2735:
2723:
1507:
954:
437:
391:
in 628 CE. However, 0 had been used as a number in the medieval
349:
345:
56:
6000:
Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008).
5153:
4542:
A perfect number is that which is equal to the sum of its own parts.
4442:"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
6062:
From Frege to Gödel: A source book in mathematical logic, 1879–1931
5515:
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
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,
448:
372:
304:
38:
6121:
5936:
Bridge to
Abstract Mathematics: Mathematical proof and structures
4833:(Winter 2014 ed.). The Stanford Encyclopedia of Philosophy.
3375:
is not directly accessible; only the ordinal property (being the
5954:
Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013).
3643:
8696:
8660:
8624:
8588:
8548:
8173:
8062:
7788:
7703:
7658:
7535:
7225:
7172:
7124:
7058:
7010:
6948:
6852:
6813:
6125:
5356:
Discrete and Combinatorial Mathematics: An applied introduction
5092:]. pp. 2:5–23. (The quote is on p. 19). Archived from
363:
A much later advance was the development of the idea that
95:
The natural numbers can be used for counting (as in "there are
5187:. In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.).
3860:, p. 1): "Numbers make up the foundation of mathematics."
3651:
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
653:
4637:"Earliest Known Uses of Some of the Words of Mathematics (N)"
399:
in 525 CE, without being denoted by a numeral. Standard
115:. Natural numbers are sometimes used as labels—also known as
6064:(3rd ed.). Harvard University Press. pp. 346–354.
5957:
Mathematics for Elementary Teachers: A contemporary approach
5067:
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
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
6028:[On the Introduction of the Transfinite Numbers].
5050:
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
4726:. United States Armed Forces Institute. 1958. p. 12.
3407: – Representation of a number as a product of primes
105:. They may also be used for ordering (as in "this is the
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
4789:
4761:
From Great Discoveries in Number Theory to Applications
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
5090:
Annual report of the German Mathematicians Association
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
6593:
6516:
6477:
6439:
6411:
6383:
6355:
6268:
6235:
6207:
6179:
5383:(5th ed.). Boston: Addison-Wesley. p. 133.
4806:. Princeton: Princeton university press. p. 17.
4498:. Mineola, New York: Dover Publications. p. 58.
4493:
Philosophy of mathematics and deductive structure in
4375:
1491: New Revelations of the Americas before Columbus
4046:
An Axiomatization for the System of Positive Integers
3957:
3901:
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:
5102:
Jahresbericht der Deutschen Mathematiker-Vereinigung
5085:
Jahresbericht der Deutschen Mathematiker-Vereinigung
4288:"The Ishango Bone, Democratic Republic of the Congo"
1368:{\displaystyle S\colon \mathbb {N} \to \mathbb {N} }
576:
saw a need to improve upon the logical rigor in the
8520:
8474:
8434:
8385:
8359:
8292:
8276:
8255:
8222:
8187:
8027:
7994:
7951:
7928:
7805:
7493:
7484:
7462:
7419:
7381:
7372:
7305:
7247:
7238:
6726:
6652:
6554:
6462:
6333:
6160:
5033:]. Translated by Greenstreet, William John. VI.
4923:Brown, Jim (1978). "In defense of index origin 0".
3977:; ...'. They follow that with their version of the
1922:if and only if there exists another natural number
6601:
6524:
6485:
6447:
6419:
6391:
6363:
6276:
6243:
6215:
6187:
5430:Fletcher, Harold; Howell, Arnold A. (9 May 2014).
5381:A review of discrete and combinatorial mathematics
5188:
3965:
3939:
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
5979:Szczepanski, Amy F.; Kositsky, Andrew P. (2008).
4693:(in French). Paris, Gauthier-Villars. p. 39.
4561:Mathematical Thought from Ancient to Modern Times
4065: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.
5796:(Revised ed.). Cambridge University Press.
3892:
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
5894:(3rd ed.). American Mathematical Society.
5651:Clapham, Christopher; Nicholson, James (2014).
3413: – Mathematical set that can be enumerated
2837:inside Peano arithmetic. A probable example is
6006:(Second ed.). ClassicalRealAnalysis.com.
5915:Number Systems and the Foundations of Analysis
5191:Studies in the Logic of Charles Sanders Peirce
5008:International Organization for Standardization
4860:Classic Set Theory: A guided independent study
3940:{\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}
3883:, 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
6825:
6137:
5748:An Introduction to the History of Mathematics
5615:(Second ed.). McGraw-Hill Professional.
4144: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
8:
6111:"Axioms and construction of natural numbers"
6058:"On the introduction of transfinite numbers"
5654:The Concise Oxford Dictionary of Mathematics
5195:. Indiana University Press. pp. 43–52.
4676:(in Latin). Fratres Bocca. 1889. p. 12.
3934:
3910:
1293:
1287:
1183:
1157:
1151:
1127:
1073:
1047:
1041:
1017:
937:
931:
880:
850:
812:
806:
755:
731:
6077:
5657:(Fifth ed.). Oxford University Press.
5578:
5468:Addition of natural numbers is associative.
5269:Bulletin of the London Mathematical Society
5053:. Princeton University Press. p. 153.
4989:
4987:
4985:
4631:
4629:
4627:
4625:
4623:
4621:
4619:
4617:
4305:Royal Belgian Institute of Natural Sciences
4270:Royal Belgian Institute of Natural Sciences
4244:Royal Belgian Institute of Natural Sciences
3836: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
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
8693:
8657:
8621:
8585:
8545:
8219:
8184:
8170:
8059:
7802:
7785:
7700:
7655:
7532:
7490:
7378:
7244:
7235:
7222:
7169:
7126:Possessing a specific set of other numbers
7121:
7055:
7007:
6945:
6849:
6832:
6818:
6810:
6794:
6144:
6130:
6122:
5185:"3. Peirce's Axiomatization of Arithmetic"
4083:
4081:
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,
6595:
6594:
6592:
6518:
6517:
6515:
6479:
6478:
6476:
6441:
6440:
6438:
6413:
6412:
6410:
6385:
6384:
6382:
6357:
6356:
6354:
6270:
6269:
6267:
6237:
6236:
6234:
6209:
6208:
6206:
6181:
6180:
6178:
5982:The Complete Idiot's Guide to Pre-algebra
5775:. Springer Science & Business Media.
5524:
5514:
5023:"On the nature of mathematical reasoning"
4785:
4753:
4751:
3959:
3958:
3956:
3903:
3902:
3900:
3879:This convention is used, for example, in
3845:
3623:
3622:
3616:
3585:
3584:
3578:
3547:
3546:
3540:
3509:
3508:
3502:
3471:
3470:
3464:
3042:elements. So, it seems natural to define
2995:
2975:
2944:
2924:
2904:
2884:
2866:The five Peano axioms are the following:
2752:
2751:
2749:
2734:. Other generalizations are discussed in
2635:
2577:
2119:
2099:
1852:
1851:
1848:
1846:
1841:
1778:
1777:
1775:
1752:
1751:
1749:
1661:
1657:
1656:
1650:
1601:
1480:
1479:
1474:
1361:
1360:
1353:
1352:
1344:
1322:
1321:
1319:
1278:
1274:
1273:
1263:
1259:
1258:
1250:
1249:
1247:
1215:
1211:
1210:
1200:
1195:
1191:
1190:
1167:
1166:
1125:
1100:
1096:
1095:
1085:
1081:
1080:
1057:
1056:
1015:
992:
971:
970:
968:
922:
918:
917:
907:
903:
902:
892:
888:
887:
847:
824:
820:
819:
797:
793:
792:
782:
778:
777:
767:
763:
762:
729:
690:
689:
687:
420:The first systematic study of numbers as
210:
205:
151:
150:
148:
6026:"Zur Einführung der transfiniten Zahlen"
5939:(Second ed.). Mcgraw-Hill College.
5486:. Vol. 8. Laidlaw Bros. p. 25.
5358:(5th ed.). Pearson Addison Wesley.
4092:. New York: Academic Press. p. 66.
4012:
3825:
3308:on the natural numbers. This order is a
5588:
5586:
4077:
3817:
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
5844:(Third ed.). Chelsea Publishing.
5820:(Fifth ed.). Chapman & Hall.
4804:The Princeton companion to mathematics
4790:Thomson, Bruckner & Bruckner (2008
4295:'s Portal to the Heritage of Astronomy
4041:
4021:
3857:
3222:= {{ }, {{ }}, ..., {{ }, {{ }}, ...}}
2801:. More precisely, each natural number
5866:. Springer-Verlag Berlin Heidelberg.
5333:. New York: McGraw-Hill. p. 25.
4904:from the original on 13 December 2019
4590:Le Triparty en la science des nombres
3328:does not accept the axiom of infinity
248:, by adjoining to the real numbers a
7:
5592:
5560:from the original on 13 October 2014
5221:(in German). F. Vieweg. 1893. 71-73.
5120:
4995:"Standard number sets and intervals"
4970:from the original on 20 October 2015
4673:Arithmetices principia: nova methodo
4657:Eléments de la géométrie de l'infini
4448:from the original on 15 January 2019
4353:from the original on 19 January 2013
4113:
4111:
4109:
4000:
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^{*}} ,+)}
6308:Set-theoretically definable numbers
5330:Principles of Mathematical Analysis
5218:Was sind und was sollen die Zahlen?
4440:Deckers, Michael (25 August 2003).
4039: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
5460:Davisson, Schuyler Colfax (1910).
4837:from the original on 14 March 2015
4166:from the original on 29 March 2017
3317:von Neumann definition of ordinals
2793:The second definition is based on
2637:
2579:
716:Since natural numbers may contain
66:, while others define them as the
25:
6060:. In van Heijenoort, Jean (ed.).
5466:. Macmillian Company. p. 2.
4215:Brilliant Math & Science Wiki
2670:The least ordinal of cardinality
2623:Natural numbers are also used as
1242:This section uses the convention
8727:
8335:Perfect digit-to-digit invariant
6793:
5078:Weber, Heinrich L. (1891–1892).
4960:"Is index origin 0 a hindrance?"
4392:from the original on 14 May 2015
4319:The Universal History of Numbers
3389:
3030:Intuitively, the natural number
2720:non-standard model of arithmetic
1496:{\displaystyle (\mathbb {N} ,+)}
189:for each nonzero natural number
132:, commonly symbolized as a bold
5723:Essays on the Theory of Numbers
5702:Essays on the Theory of Numbers
5678:Essays on the Theory of Numbers
5526:10.14321/realanalexch.42.2.0193
5142:American Journal of Mathematics
4654:Fontenelle, Bernard de (1727).
4472:"Book VII, definitions 1 and 2"
4347:MacTutor History of Mathematics
3630:{\displaystyle :\;\mathbb {N} }
3592:{\displaystyle :\;\mathbb {Z} }
3554:{\displaystyle :\;\mathbb {Q} }
3516:{\displaystyle :\;\mathbb {R} }
3478:{\displaystyle :\;\mathbb {C} }
2654:countably infinite set without
2217:), and ideas in number theory.
632:). This approach is now called
475:Bernard Le Bovier de Fontenelle
5713:– via Project Gutenberg.
5636:. Cambridge University Press.
5433:Mathematics with Understanding
4764:. Springer Nature. p. 6.
4303:, on permanent display at the
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:
1:
7174:Expressible via specific sums
6642:Plane-based geometric algebra
5725:. Kessinger Publishing, LLC.
5554:European Mathematical Society
4704:Fine, Henry Burchard (1904).
4088:Enderton, Herbert B. (1977).
3973:contains an "initial" number
3951:may be described as follows:
3893: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
700:{\displaystyle \mathbb {N} .}
648:with several weak systems of
513:have preferred to include 0.
6602:{\displaystyle \mathbb {S} }
6525:{\displaystyle \mathbb {C} }
6486:{\displaystyle \mathbb {R} }
6448:{\displaystyle \mathbb {O} }
6420:{\displaystyle \mathbb {H} }
6392:{\displaystyle \mathbb {C} }
6364:{\displaystyle \mathbb {R} }
6277:{\displaystyle \mathbb {A} }
6244:{\displaystyle \mathbb {Q} }
6216:{\displaystyle \mathbb {Z} }
6188:{\displaystyle \mathbb {N} }
5234:Mathematical Logic Quarterly
4856:Goldrei, Derek (1998). "3".
4690:Formulaire des mathematiques
4057:In some set theories, e.g.,
3966:{\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} }
8263:Multiplicative digital root
6098:Encyclopedia of Mathematics
5546:Encyclopedia of Mathematics
5379:Grimaldi, Ralph P. (2003).
5354:Grimaldi, Ralph P. (2004).
4563:. Oxford University Press.
4150:Embedded Systems Dictionary
2401:: for every natural number
2213:), algorithms (such as the
2197:are uniquely determined by
1946:in the following sense: if
1578:is simply the successor of
1336:of natural numbers and the
424:is usually credited to the
128:The natural numbers form a
8790:
5933:Morash, Ronald P. (1991).
5552:, in cooperation with the
5483:Laidlaw mathematics series
3826:§ 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:
29:
8723:
8706:
8692:
8670:
8656:
8634:
8620:
8598:
8584:
8557:
8544:
8340:Perfect digital invariant
8183:
8169:
8077:
8058:
7915:Superior highly composite
7801:
7784:
7712:
7699:
7667:
7654:
7542:
7531:
7234:
7221:
7179:
7168:
7131:
7120:
7068:
7054:
7017:
7006:
6959:
6944:
6862:
6848:
6784:
6632:Algebra of physical space
6076:– English translation of
6036:: 199–208. Archived from
6016:– via Google Books.
5995:– via Google Books.
5974:– via Google Books.
5949:– via Google Books.
5928:– via Google Books.
5904:– via Google Books.
5854:– via Google Books.
5830:– via Google Books.
5806:– via Google Books.
5785:– via Google Books.
5761:– via Google Books.
5751:(6th ed.). Thomson.
5667:– via Google Books.
5646:– via Google Books.
5625:– via Google Books.
5436:. Elsevier. p. 116.
5027:La Science et l'hypothèse
5021:Poincaré, Henri (1905) .
5010:. 19 May 2020. p. 4.
4925:ACM SIGAPL APL Quote Quad
4604:Emerson, William (1763).
4532:"Book VII, definition 22"
4429:– via Google Books.
4419:. John Wiley & Sons.
4400:– via Google Books.
4372:Mann, Charles C. (2005).
4174:– 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
7953:Euler's totient function
7737:Euler–Jacobi pseudoprime
7012:Other polynomial numbers
6688:Extended complex numbers
6671:Extended natural numbers
6003:Elementary Real Analysis
5793:Logic for Mathematicians
5691:– via Archive.org.
5630:Carothers, N.L. (2000).
5246:10.1002/malq.19930390138
5138:"On the Logic of Number"
4802:Gowers, Timothy (2008).
4687:Peano, Giuseppe (1901).
4607:The method of increments
3020:Set-theoretic definition
2899:equals the successor of
1964:are natural numbers and
266:and the distribution of
27:Number used for counting
7767:Somer–Lucas pseudoprime
7757:Lucas–Carmichael number
7592:Lazy caterer's sequence
5841:Foundations of Analysis
5814:; James, Glenn (1992).
5790:Hamilton, A.G. (1988).
5612:Pre-Algebra DeMYSTiFieD
5579:von Neumann (1923)
4660:(in French). p. 3.
4559:Kline, Morris (1990) .
4317:Ifrah, Georges (2000).
4019:0 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:
8759:Elementary mathematics
7642:Wedderburn–Etherington
7042:Lucky numbers of Euler
6744:Transcendental numbers
6603:
6580:Hyperbolic quaternions
6526:
6487:
6449:
6421:
6393:
6365:
6278:
6245:
6217:
6189:
6054:von Neumann, John
5962:Wiley Global Education
5918:. Dover Publications.
5817:Mathematics Dictionary
5609:Bluman, Allan (2010).
5503:Real Analysis Exchange
5183:Shields, Paul (1997).
5031:Science and Hypothesis
4827:Bagaria, Joan (2017).
4534:. In Joyce, D. (ed.).
4474:. In Joyce, D. (ed.).
4090:Elements of set theory
3967:
3941:
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:
7930:Prime omega functions
7747:Frobenius pseudoprime
7537:Combinatorial numbers
7406:Centered dodecahedral
7199:Primary pseudoperfect
6676:Extended real numbers
6604:
6527:
6497:Split-complex numbers
6488:
6450:
6422:
6394:
6366:
6279:
6255:Constructible numbers
6246:
6218:
6190:
6078:von Neumann 1923
5410:mathworld.wolfram.com
5307:functions.wolfram.com
5281:10.1112/blms/14.4.285
5275:(4). Wiley: 285–293.
4937:10.1145/586050.586053
4491:Mueller, Ian (2006).
4411:Evans, Brian (2014).
4378:. Knopf. p. 19.
4268:. Brussels, Belgium:
4242:. Brussels, Belgium:
4124:mathworld.wolfram.com
3968:
3942:
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:
8774:Sets of real numbers
8389:-composition related
8189:Arithmetic functions
7791:Arithmetic functions
7727:Elliptic pseudoprime
7411:Centered icosahedral
7391:Centered tetrahedral
6708:Supernatural numbers
6618:Multicomplex numbers
6591:
6575:Dual-complex numbers
6514:
6475:
6437:
6409:
6381:
6353:
6335:Composition algebras
6303:Arithmetical numbers
6266:
6233:
6205:
6177:
4549:is a perfect number.
4540:. Clark University.
4444:. Hbar.phys.msu.ru.
4307:, Brussels, Belgium.
4301:on 10 November 2014.
4262:"Flash presentation"
3955:
3899:
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
8315:Kaprekar's constant
7835:Colossally abundant
7722:Catalan pseudoprime
7622:Schröder–Hipparchus
7401:Centered octahedral
7277:Centered heptagonal
7267:Centered pentagonal
7257:Centered triangular
6857:and related numbers
6613:Split-biquaternions
6325:Eisenstein integers
6288:Closed-form numbers
6040:on 18 December 2014
5540:Mints, G.E. (ed.).
5404:Weisstein, Eric W.
4894:Merriam-Webster.com
4742:archive.lib.msu.edu
4480:. Clark University.
4343:"A history of Zero"
4118:Weisstein, Eric W.
4031:(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}
8733:Mathematics portal
8675:Aronson's sequence
8421:Smarandache–Wellin
8178:-dependent numbers
7885:Primitive abundant
7772:Strong pseudoprime
7762:Perrin pseudoprime
7742:Fermat pseudoprime
7682:Wolstenholme prime
7506:Squared triangular
7292:Centered decagonal
7287:Centered nonagonal
7282:Centered octagonal
7272:Centered hexagonal
6771:Profinite integers
6734:Irrational numbers
6599:
6522:
6483:
6445:
6417:
6389:
6361:
6318:Gaussian rationals
6298:Computable numbers
6274:
6241:
6213:
6185:
5910:Mendelson, Elliott
5882:Mac Lane, Saunders
5327:Rudin, W. (1976).
5134:Peirce, C. S.
5110:on 20 August 2017.
4784:See, for example,
4710:. Ginn. p. 6.
4187:Weisstein, Eric W.
3963:
3937:
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:
8741:
8740:
8719:
8718:
8688:
8687:
8652:
8651:
8616:
8615:
8580:
8579:
8540:
8539:
8536:
8535:
8355:
8354:
8165:
8164:
8054:
8053:
8050:
8049:
7996:Aliquot sequences
7807:Divisor functions
7780:
7779:
7752:Lucas pseudoprime
7732:Euler pseudoprime
7717:Carmichael number
7695:
7694:
7650:
7649:
7527:
7526:
7523:
7522:
7519:
7518:
7480:
7479:
7368:
7367:
7325:Square triangular
7217:
7216:
7164:
7163:
7116:
7115:
7050:
7049:
7002:
7001:
6940:
6939:
6807:
6806:
6718:Superreal numbers
6698:Levi-Civita field
6693:Hyperreal numbers
6637:Spacetime algebra
6623:Geometric algebra
6536:Bicomplex numbers
6502:Split-quaternions
6343:Division algebras
6313:Gaussian integers
6260:Algebraic numbers
6163:definable numbers
6071:978-0-674-32449-7
6056:(January 2002) .
6022:von Neumann, John
6013:978-1-4348-4367-8
5992:978-1-59257-772-9
5985:. Penguin Group.
5971:978-1-118-45744-3
5960:(10th ed.).
5946:978-0-07-043043-3
5925:978-0-486-45792-5
5901:978-0-8218-1646-2
5886:Birkhoff, Garrett
5873:978-3-662-02310-5
5851:978-0-8218-2693-5
5827:978-0-412-99041-0
5803:978-0-521-36865-0
5782:978-0-387-90092-6
5758:978-0-03-029558-4
5732:978-0-548-08985-9
5719:Dedekind, Richard
5697:Dedekind, Richard
5688:978-0-486-21010-0
5673:Dedekind, Richard
5664:978-0-19-967959-1
5643:978-0-521-49756-5
5622:978-0-07-174251-1
5443:978-1-4832-8079-0
5390:978-0-201-72634-3
5365:978-0-201-72634-3
5340:978-0-07-054235-8
5096:on 9 August 2018;
5060:978-1-4008-2904-0
4875:978-0-412-60610-6
4813:978-0-691-11880-2
4771:978-3-030-83899-7
4707:A College Algebra
4505:978-0-486-45300-2
4495:Euclid's Elements
4426:978-1-118-85397-9
4385:978-1-4000-4006-3
4211:"Natural Numbers"
4190:"Counting Number"
4159:978-1-57820-120-4
3881:Euclid's Elements
3809:
3808:
3805:
3804:
3801:
3800:
3797:
3796:
3786:
3785:
3782:
3781:
3778:
3777:
3774:
3773:
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
16:(Redirected from
8781:
8754:Cardinal numbers
8731:
8694:
8663:Natural language
8658:
8622:
8590:Generated via a
8586:
8546:
8451:Digit-reassembly
8416:Self-descriptive
8220:
8185:
8171:
8122:Lucas–Carmichael
8112:Harmonic divisor
8060:
7986:Sparsely totient
7961:Highly cototient
7870:Multiply perfect
7860:Highly composite
7803:
7786:
7701:
7656:
7637:Telephone number
7533:
7491:
7472:Square pyramidal
7454:Stella octangula
7379:
7245:
7236:
7228:Figurate numbers
7223:
7170:
7122:
7056:
7008:
6946:
6850:
6834:
6827:
6820:
6811:
6797:
6796:
6764:
6754:
6666:Cardinal numbers
6627:Clifford algebra
6608:
6606:
6605:
6600:
6598:
6570:Dual quaternions
6531:
6529:
6528:
6523:
6521:
6492:
6490:
6489:
6484:
6482:
6454:
6452:
6451:
6446:
6444:
6426:
6424:
6423:
6418:
6416:
6398:
6396:
6395:
6390:
6388:
6370:
6368:
6367:
6362:
6360:
6283:
6281:
6280:
6275:
6273:
6250:
6248:
6247:
6242:
6240:
6227:Rational numbers
6222:
6220:
6219:
6214:
6212:
6194:
6192:
6191:
6186:
6184:
6146:
6139:
6132:
6123:
6118:
6106:
6093:"Natural number"
6075:
6049:
6047:
6045:
6017:
5996:
5975:
5950:
5929:
5905:
5877:
5864:Basic Set Theory
5855:
5831:
5812:James, Robert C.
5807:
5786:
5772:Naive Set Theory
5762:
5736:
5714:
5712:
5710:
5692:
5668:
5647:
5626:
5596:
5590:
5581:
5576:
5570:
5569:
5567:
5565:
5537:
5531:
5530:
5528:
5518:
5494:
5488:
5487:
5477:
5471:
5470:
5457:
5451:
5450:
5427:
5421:
5420:
5418:
5416:
5406:"Multiplication"
5401:
5395:
5394:
5376:
5370:
5369:
5351:
5345:
5344:
5324:
5318:
5317:
5315:
5313:
5299:
5293:
5292:
5264:
5258:
5257:
5229:
5223:
5222:
5213:
5207:
5206:
5194:
5180:
5174:
5173:
5130:
5124:
5118:
5112:
5111:
5106:. Archived from
5097:
5075:
5069:
5068:
5041:
5035:
5034:
5018:
5012:
5011:
5003:ISO 80000-2:2019
4999:
4991:
4980:
4979:
4977:
4975:
4955:
4949:
4948:
4920:
4914:
4913:
4911:
4909:
4890:"natural number"
4886:
4880:
4879:
4863:
4853:
4847:
4846:
4844:
4842:
4824:
4818:
4817:
4799:
4793:
4788:, p. 3) or
4782:
4776:
4775:
4755:
4746:
4745:
4738:"Natural Number"
4734:
4728:
4727:
4718:
4712:
4711:
4701:
4695:
4694:
4684:
4678:
4677:
4668:
4662:
4661:
4651:
4645:
4644:
4633:
4612:
4611:
4601:
4595:
4594:
4585:Chuquet, Nicolas
4581:
4575:
4574:
4556:
4550:
4548:
4544:
4524:
4518:
4517:
4488:
4482:
4481:
4464:
4458:
4457:
4455:
4453:
4437:
4431:
4430:
4408:
4402:
4401:
4399:
4397:
4369:
4363:
4362:
4360:
4358:
4339:
4333:
4332:
4314:
4308:
4302:
4297:. Archived from
4284:
4278:
4277:
4272:. Archived from
4258:
4252:
4251:
4250:on 4 March 2016.
4246:. Archived from
4232:
4226:
4225:
4223:
4221:
4207:
4201:
4200:
4199:
4182:
4176:
4175:
4173:
4171:
4141:
4135:
4134:
4132:
4130:
4120:"Natural Number"
4115:
4104:
4103:
4085:
4066:
4055:
4049:
4038:
4034:
4030:
4027:
4018:
4010:
4004:
3997:
3991:
3988:
3982:
3976:
3972:
3970:
3969:
3964:
3962:
3946:
3944:
3943:
3938:
3906:
3890:
3884:
3877:
3871:
3867:
3861:
3855:
3849:
3843:
3837:
3834:
3828:
3822:
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:
8789:
8788:
8784:
8783:
8782:
8780:
8779:
8778:
8744:
8743:
8742:
8737:
8715:
8711:Strobogrammatic
8702:
8684:
8666:
8648:
8630:
8612:
8594:
8576:
8553:
8532:
8516:
8475:Divisor-related
8470:
8430:
8381:
8351:
8288:
8272:
8251:
8218:
8191:
8179:
8161:
8073:
8072:related numbers
8046:
8023:
7990:
7981:Perfect totient
7947:
7924:
7855:Highly abundant
7797:
7776:
7708:
7691:
7663:
7646:
7632:Stirling second
7538:
7515:
7476:
7458:
7415:
7364:
7301:
7262:Centered square
7230:
7213:
7175:
7160:
7127:
7112:
7064:
7063:defined numbers
7046:
7013:
6998:
6969:Double Mersenne
6955:
6936:
6858:
6844:
6842:natural numbers
6838:
6808:
6803:
6780:
6759:
6749:
6722:
6713:Surreal numbers
6703:Ordinal numbers
6648:
6589:
6588:
6550:
6512:
6511:
6509:
6507:Split-octonions
6473:
6472:
6464:
6458:
6435:
6434:
6407:
6406:
6379:
6378:
6375:Complex numbers
6351:
6350:
6329:
6264:
6263:
6231:
6230:
6203:
6202:
6175:
6174:
6171:Natural numbers
6156:
6150:
6109:
6091:
6088:
6083:
6072:
6052:
6043:
6041:
6020:
6014:
5999:
5993:
5978:
5972:
5953:
5947:
5932:
5926:
5908:
5902:
5880:
5874:
5858:
5852:
5834:
5828:
5810:
5804:
5789:
5783:
5765:
5759:
5741:
5733:
5717:
5708:
5706:
5695:
5689:
5671:
5665:
5650:
5644:
5629:
5623:
5608:
5604:
5599:
5591:
5584:
5577:
5573:
5563:
5561:
5539:
5538:
5534:
5496:
5495:
5491:
5479:
5478:
5474:
5463:College Algebra
5459:
5458:
5454:
5444:
5429:
5428:
5424:
5414:
5412:
5403:
5402:
5398:
5391:
5378:
5377:
5373:
5366:
5353:
5352:
5348:
5341:
5326:
5325:
5321:
5311:
5309:
5301:
5300:
5296:
5266:
5265:
5261:
5231:
5230:
5226:
5215:
5214:
5210:
5203:
5182:
5181:
5177:
5154:10.2307/2369151
5132:
5131:
5127:
5119:
5115:
5098:
5077:
5076:
5072:
5061:
5043:
5042:
5038:
5020:
5019:
5015:
4997:
4993:
4992:
4983:
4973:
4971:
4957:
4956:
4952:
4922:
4921:
4917:
4907:
4905:
4898:Merriam-Webster
4888:
4887:
4883:
4876:
4855:
4854:
4850:
4840:
4838:
4826:
4825:
4821:
4814:
4801:
4800:
4796:
4786:Carothers (2000
4783:
4779:
4772:
4757:
4756:
4749:
4736:
4735:
4731:
4720:
4719:
4715:
4703:
4702:
4698:
4686:
4685:
4681:
4670:
4669:
4665:
4653:
4652:
4648:
4635:
4634:
4615:
4603:
4602:
4598:
4583:
4582:
4578:
4571:
4558:
4557:
4553:
4546:
4526:
4525:
4521:
4506:
4490:
4489:
4485:
4466:
4465:
4461:
4451:
4449:
4439:
4438:
4434:
4427:
4410:
4409:
4405:
4395:
4393:
4386:
4371:
4370:
4366:
4356:
4354:
4341:
4340:
4336:
4329:
4316:
4315:
4311:
4286:
4285:
4281:
4276:on 27 May 2016.
4260:
4259:
4255:
4234:
4233:
4229:
4219:
4217:
4209:
4208:
4204:
4185:
4184:
4183:
4179:
4169:
4167:
4160:
4143:
4142:
4138:
4128:
4126:
4117:
4116:
4107:
4100:
4087:
4086:
4079:
4075:
4070:
4069:
4059:New Foundations
4056:
4052:
4040:
4036:
4032:
4028:
4025:
4020:
4016:
4011:
4007:
4003:, p. 606)
3998:
3994:
3989:
3985:
3974:
3953:
3952:
3949:natural numbers
3897:
3896:
3891:
3887:
3878:
3874:
3868:
3864:
3856:
3852:
3846:Mendelson (2008
3844:
3840:
3835:
3831:
3823:
3819:
3814:
3613:
3612:
3575:
3574:
3537:
3536:
3499:
3498:
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:
2992:
2991:
2972:
2971:
2941:
2940:
2921:
2920:
2901:
2900:
2881:
2880:
2864:
2858:
2822:
2818:
2810:
2806:
2802:
2772:
2746:
2745:
2700:
2697:
2693:
2689:
2686:
2682:
2679:initial ordinal
2675:
2671:
2663:
2632:
2631:
2618:
2614:
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
2096:
2095:
2086:
2080:
2073:
2067:
2061:
2036:
2030:
2027:
2014:
1993:
1975:
1965:
1959:
1953:
1947:
1929:
1923:
1913:
1892:
1886:
1883:
1847:
1838:
1837:
1836:. Furthermore,
1827:
1812:
1772:
1771:
1746:
1745:
1707:
1700:
1655:
1647:
1646:
1624:
1617:
1598:
1597:
1590:
1579:
1572:
1541:
1534:
1471:
1470:
1448:
1429:
1423:
1417:
1390:
1380:
1341:
1340:
1316:
1315:
1312:
1272:
1257:
1244:
1243:
1240:
1209:
1122:
1121:
1094:
1079:
1012:
1011:
989:
988:
965:
964:
962:
916:
901:
886:
844:
843:
818:
791:
776:
761:
726:
725:
717:
708:
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:
202:
201:
190:
183:
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:
18:Counting number
15:
12:
11:
5:
8787:
8785:
8777:
8776:
8771:
8766:
8761:
8756:
8746:
8745:
8739:
8738:
8736:
8735:
8724:
8721:
8720:
8717:
8716:
8714:
8713:
8707:
8704:
8703:
8697:
8690:
8689:
8686:
8685:
8683:
8682:
8677:
8671:
8668:
8667:
8661:
8654:
8653:
8650:
8649:
8647:
8646:
8644:Sorting number
8641:
8639:Pancake number
8635:
8632:
8631:
8625:
8618:
8617:
8614:
8613:
8611:
8610:
8605:
8599:
8596:
8595:
8589:
8582:
8581:
8578:
8577:
8575:
8574:
8569:
8564:
8558:
8555:
8554:
8551:Binary numbers
8549:
8542:
8541:
8538:
8537:
8534:
8533:
8531:
8530:
8524:
8522:
8518:
8517:
8515:
8514:
8509:
8504:
8499:
8494:
8489:
8484:
8478:
8476:
8472:
8471:
8469:
8468:
8463:
8458:
8453:
8448:
8442:
8440:
8432:
8431:
8429:
8428:
8423:
8418:
8413:
8408:
8403:
8398:
8392:
8390:
8383:
8382:
8380:
8379:
8378:
8377:
8366:
8364:
8361:P-adic numbers
8357:
8356:
8353:
8352:
8350:
8349:
8348:
8347:
8337:
8332:
8327:
8322:
8317:
8312:
8307:
8302:
8296:
8294:
8290:
8289:
8287:
8286:
8280:
8278:
8277:Coding-related
8274:
8273:
8271:
8270:
8265:
8259:
8257:
8253:
8252:
8250:
8249:
8244:
8239:
8234:
8228:
8226:
8217:
8216:
8215:
8214:
8212:Multiplicative
8209:
8198:
8196:
8181:
8180:
8176:Numeral system
8174:
8167:
8166:
8163:
8162:
8160:
8159:
8154:
8149:
8144:
8139:
8134:
8129:
8124:
8119:
8114:
8109:
8104:
8099:
8094:
8089:
8084:
8078:
8075:
8074:
8063:
8056:
8055:
8052:
8051:
8048:
8047:
8045:
8044:
8039:
8033:
8031:
8025:
8024:
8022:
8021:
8016:
8011:
8006:
8000:
7998:
7992:
7991:
7989:
7988:
7983:
7978:
7973:
7968:
7966:Highly totient
7963:
7957:
7955:
7949:
7948:
7946:
7945:
7940:
7934:
7932:
7926:
7925:
7923:
7922:
7917:
7912:
7907:
7902:
7897:
7892:
7887:
7882:
7877:
7872:
7867:
7862:
7857:
7852:
7847:
7842:
7837:
7832:
7827:
7822:
7820:Almost perfect
7817:
7811:
7809:
7799:
7798:
7789:
7782:
7781:
7778:
7777:
7775:
7774:
7769:
7764:
7759:
7754:
7749:
7744:
7739:
7734:
7729:
7724:
7719:
7713:
7710:
7709:
7704:
7697:
7696:
7693:
7692:
7690:
7689:
7684:
7679:
7674:
7668:
7665:
7664:
7659:
7652:
7651:
7648:
7647:
7645:
7644:
7639:
7634:
7629:
7627:Stirling first
7624:
7619:
7614:
7609:
7604:
7599:
7594:
7589:
7584:
7579:
7574:
7569:
7564:
7559:
7554:
7549:
7543:
7540:
7539:
7536:
7529:
7528:
7525:
7524:
7521:
7520:
7517:
7516:
7514:
7513:
7508:
7503:
7497:
7495:
7488:
7482:
7481:
7478:
7477:
7475:
7474:
7468:
7466:
7460:
7459:
7457:
7456:
7451:
7446:
7441:
7436:
7431:
7425:
7423:
7417:
7416:
7414:
7413:
7408:
7403:
7398:
7393:
7387:
7385:
7376:
7370:
7369:
7366:
7365:
7363:
7362:
7357:
7352:
7347:
7342:
7337:
7332:
7327:
7322:
7317:
7311:
7309:
7303:
7302:
7300:
7299:
7294:
7289:
7284:
7279:
7274:
7269:
7264:
7259:
7253:
7251:
7242:
7232:
7231:
7226:
7219:
7218:
7215:
7214:
7212:
7211:
7206:
7201:
7196:
7191:
7186:
7180:
7177:
7176:
7173:
7166:
7165:
7162:
7161:
7159:
7158:
7153:
7148:
7143:
7138:
7132:
7129:
7128:
7125:
7118:
7117:
7114:
7113:
7111:
7110:
7105:
7100:
7095:
7090:
7085:
7080:
7075:
7069:
7066:
7065:
7059:
7052:
7051:
7048:
7047:
7045:
7044:
7039:
7034:
7029:
7024:
7018:
7015:
7014:
7011:
7004:
7003:
7000:
6999:
6997:
6996:
6991:
6986:
6981:
6976:
6971:
6966:
6960:
6957:
6956:
6949:
6942:
6941:
6938:
6937:
6935:
6934:
6929:
6924:
6919:
6914:
6909:
6904:
6899:
6894:
6889:
6884:
6879:
6874:
6869:
6863:
6860:
6859:
6853:
6846:
6845:
6839:
6837:
6836:
6829:
6822:
6814:
6805:
6804:
6802:
6801:
6791:
6789:Classification
6785:
6782:
6781:
6779:
6778:
6776:Normal numbers
6773:
6768:
6746:
6741:
6736:
6730:
6728:
6724:
6723:
6721:
6720:
6715:
6710:
6705:
6700:
6695:
6690:
6685:
6684:
6683:
6673:
6668:
6662:
6660:
6658:infinitesimals
6650:
6649:
6647:
6646:
6645:
6644:
6639:
6634:
6620:
6615:
6610:
6597:
6582:
6577:
6572:
6567:
6561:
6559:
6552:
6551:
6549:
6548:
6543:
6538:
6533:
6520:
6504:
6499:
6494:
6481:
6468:
6466:
6460:
6459:
6457:
6456:
6443:
6428:
6415:
6400:
6387:
6372:
6359:
6339:
6337:
6331:
6330:
6328:
6327:
6322:
6321:
6320:
6310:
6305:
6300:
6295:
6290:
6285:
6272:
6257:
6252:
6239:
6224:
6211:
6196:
6183:
6167:
6165:
6158:
6157:
6151:
6149:
6148:
6141:
6134:
6126:
6120:
6119:
6107:
6087:
6086:External links
6084:
6082:
6081:
6070:
6050:
6018:
6012:
5997:
5991:
5976:
5970:
5951:
5945:
5930:
5924:
5906:
5900:
5878:
5872:
5856:
5850:
5836:Landau, Edmund
5832:
5826:
5808:
5802:
5787:
5781:
5763:
5757:
5739:
5738:
5737:
5731:
5715:
5687:
5669:
5663:
5648:
5642:
5627:
5621:
5605:
5603:
5600:
5598:
5597:
5582:
5571:
5542:"Peano axioms"
5532:
5509:(2): 193–253.
5489:
5472:
5452:
5442:
5422:
5396:
5389:
5371:
5364:
5346:
5339:
5319:
5294:
5259:
5240:(3): 338–352.
5224:
5208:
5201:
5175:
5125:
5113:
5070:
5059:
5036:
5013:
4981:
4950:
4915:
4881:
4874:
4848:
4819:
4812:
4794:
4777:
4770:
4747:
4729:
4713:
4696:
4679:
4663:
4646:
4613:
4610:. p. 113.
4596:
4576:
4569:
4551:
4519:
4504:
4483:
4459:
4432:
4425:
4403:
4384:
4364:
4334:
4327:
4309:
4279:
4253:
4236:"Introduction"
4227:
4202:
4177:
4158:
4136:
4105:
4098:
4076:
4074:
4071:
4068:
4067:
4050:
4013:Hamilton (1988
4005:
3992:
3983:
3979:Peano's axioms
3961:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3915:
3912:
3909:
3905:
3885:
3872:
3862:
3850:
3838:
3829:
3816:
3815:
3813:
3810:
3807:
3806:
3803:
3802:
3799:
3798:
3795:
3794:
3788:
3787:
3784:
3783:
3780:
3779:
3776:
3775:
3772:
3771:
3769:Transcendental
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8786:
8775:
8772:
8770:
8769:Number theory
8767:
8765:
8762:
8760:
8757:
8755:
8752:
8751:
8749:
8734:
8730:
8726:
8725:
8722:
8712:
8709:
8708:
8705:
8700:
8695:
8691:
8681:
8678:
8676:
8673:
8672:
8669:
8664:
8659:
8655:
8645:
8642:
8640:
8637:
8636:
8633:
8628:
8623:
8619:
8609:
8606:
8604:
8601:
8600:
8597:
8593:
8587:
8583:
8573:
8570:
8568:
8565:
8563:
8560:
8559:
8556:
8552:
8547:
8543:
8529:
8526:
8525:
8523:
8519:
8513:
8510:
8508:
8505:
8503:
8502:Polydivisible
8500:
8498:
8495:
8493:
8490:
8488:
8485:
8483:
8480:
8479:
8477:
8473:
8467:
8464:
8462:
8459:
8457:
8454:
8452:
8449:
8447:
8444:
8443:
8441:
8438:
8433:
8427:
8424:
8422:
8419:
8417:
8414:
8412:
8409:
8407:
8404:
8402:
8399:
8397:
8394:
8393:
8391:
8388:
8384:
8376:
8373:
8372:
8371:
8368:
8367:
8365:
8362:
8358:
8346:
8343:
8342:
8341:
8338:
8336:
8333:
8331:
8328:
8326:
8323:
8321:
8318:
8316:
8313:
8311:
8308:
8306:
8303:
8301:
8298:
8297:
8295:
8291:
8285:
8282:
8281:
8279:
8275:
8269:
8266:
8264:
8261:
8260:
8258:
8256:Digit product
8254:
8248:
8245:
8243:
8240:
8238:
8235:
8233:
8230:
8229:
8227:
8225:
8221:
8213:
8210:
8208:
8205:
8204:
8203:
8200:
8199:
8197:
8195:
8190:
8186:
8182:
8177:
8172:
8168:
8158:
8155:
8153:
8150:
8148:
8145:
8143:
8140:
8138:
8135:
8133:
8130:
8128:
8125:
8123:
8120:
8118:
8115:
8113:
8110:
8108:
8105:
8103:
8100:
8098:
8095:
8093:
8092:Erdős–Nicolas
8090:
8088:
8085:
8083:
8080:
8079:
8076:
8071:
8067:
8061:
8057:
8043:
8040:
8038:
8035:
8034:
8032:
8030:
8026:
8020:
8017:
8015:
8012:
8010:
8007:
8005:
8002:
8001:
7999:
7997:
7993:
7987:
7984:
7982:
7979:
7977:
7974:
7972:
7969:
7967:
7964:
7962:
7959:
7958:
7956:
7954:
7950:
7944:
7941:
7939:
7936:
7935:
7933:
7931:
7927:
7921:
7918:
7916:
7913:
7911:
7910:Superabundant
7908:
7906:
7903:
7901:
7898:
7896:
7893:
7891:
7888:
7886:
7883:
7881:
7878:
7876:
7873:
7871:
7868:
7866:
7863:
7861:
7858:
7856:
7853:
7851:
7848:
7846:
7843:
7841:
7838:
7836:
7833:
7831:
7828:
7826:
7823:
7821:
7818:
7816:
7813:
7812:
7810:
7808:
7804:
7800:
7796:
7792:
7787:
7783:
7773:
7770:
7768:
7765:
7763:
7760:
7758:
7755:
7753:
7750:
7748:
7745:
7743:
7740:
7738:
7735:
7733:
7730:
7728:
7725:
7723:
7720:
7718:
7715:
7714:
7711:
7707:
7702:
7698:
7688:
7685:
7683:
7680:
7678:
7675:
7673:
7670:
7669:
7666:
7662:
7657:
7653:
7643:
7640:
7638:
7635:
7633:
7630:
7628:
7625:
7623:
7620:
7618:
7615:
7613:
7610:
7608:
7605:
7603:
7600:
7598:
7595:
7593:
7590:
7588:
7585:
7583:
7580:
7578:
7575:
7573:
7570:
7568:
7565:
7563:
7560:
7558:
7555:
7553:
7550:
7548:
7545:
7544:
7541:
7534:
7530:
7512:
7509:
7507:
7504:
7502:
7499:
7498:
7496:
7492:
7489:
7487:
7486:4-dimensional
7483:
7473:
7470:
7469:
7467:
7465:
7461:
7455:
7452:
7450:
7447:
7445:
7442:
7440:
7437:
7435:
7432:
7430:
7427:
7426:
7424:
7422:
7418:
7412:
7409:
7407:
7404:
7402:
7399:
7397:
7396:Centered cube
7394:
7392:
7389:
7388:
7386:
7384:
7380:
7377:
7375:
7374:3-dimensional
7371:
7361:
7358:
7356:
7353:
7351:
7348:
7346:
7343:
7341:
7338:
7336:
7333:
7331:
7328:
7326:
7323:
7321:
7318:
7316:
7313:
7312:
7310:
7308:
7304:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7278:
7275:
7273:
7270:
7268:
7265:
7263:
7260:
7258:
7255:
7254:
7252:
7250:
7246:
7243:
7241:
7240:2-dimensional
7237:
7233:
7229:
7224:
7220:
7210:
7207:
7205:
7202:
7200:
7197:
7195:
7192:
7190:
7187:
7185:
7184:Nonhypotenuse
7182:
7181:
7178:
7171:
7167:
7157:
7154:
7152:
7149:
7147:
7144:
7142:
7139:
7137:
7134:
7133:
7130:
7123:
7119:
7109:
7106:
7104:
7101:
7099:
7096:
7094:
7091:
7089:
7086:
7084:
7081:
7079:
7076:
7074:
7071:
7070:
7067:
7062:
7057:
7053:
7043:
7040:
7038:
7035:
7033:
7030:
7028:
7025:
7023:
7020:
7019:
7016:
7009:
7005:
6995:
6992:
6990:
6987:
6985:
6982:
6980:
6977:
6975:
6972:
6970:
6967:
6965:
6962:
6961:
6958:
6953:
6947:
6943:
6933:
6930:
6928:
6925:
6923:
6922:Perfect power
6920:
6918:
6915:
6913:
6912:Seventh power
6910:
6908:
6905:
6903:
6900:
6898:
6895:
6893:
6890:
6888:
6885:
6883:
6880:
6878:
6875:
6873:
6870:
6868:
6865:
6864:
6861:
6856:
6851:
6847:
6843:
6835:
6830:
6828:
6823:
6821:
6816:
6815:
6812:
6800:
6792:
6790:
6787:
6786:
6783:
6777:
6774:
6772:
6769:
6766:
6762:
6756:
6752:
6747:
6745:
6742:
6740:
6739:Fuzzy numbers
6737:
6735:
6732:
6731:
6729:
6725:
6719:
6716:
6714:
6711:
6709:
6706:
6704:
6701:
6699:
6696:
6694:
6691:
6689:
6686:
6682:
6679:
6678:
6677:
6674:
6672:
6669:
6667:
6664:
6663:
6661:
6659:
6655:
6651:
6643:
6640:
6638:
6635:
6633:
6630:
6629:
6628:
6624:
6621:
6619:
6616:
6614:
6611:
6586:
6583:
6581:
6578:
6576:
6573:
6571:
6568:
6566:
6563:
6562:
6560:
6558:
6553:
6547:
6544:
6542:
6541:Biquaternions
6539:
6537:
6534:
6508:
6505:
6503:
6500:
6498:
6495:
6470:
6469:
6467:
6461:
6432:
6429:
6404:
6401:
6376:
6373:
6348:
6344:
6341:
6340:
6338:
6336:
6332:
6326:
6323:
6319:
6316:
6315:
6314:
6311:
6309:
6306:
6304:
6301:
6299:
6296:
6294:
6291:
6289:
6286:
6261:
6258:
6256:
6253:
6228:
6225:
6200:
6197:
6172:
6169:
6168:
6166:
6164:
6159:
6154:
6147:
6142:
6140:
6135:
6133:
6128:
6127:
6124:
6116:
6112:
6108:
6104:
6100:
6099:
6094:
6090:
6089:
6085:
6079:
6073:
6067:
6063:
6059:
6055:
6051:
6039:
6035:
6031:
6027:
6023:
6019:
6015:
6009:
6005:
6004:
5998:
5994:
5988:
5984:
5983:
5977:
5973:
5967:
5963:
5959:
5958:
5952:
5948:
5942:
5938:
5937:
5931:
5927:
5921:
5917:
5916:
5911:
5907:
5903:
5897:
5893:
5892:
5887:
5883:
5879:
5875:
5869:
5865:
5861:
5857:
5853:
5847:
5843:
5842:
5837:
5833:
5829:
5823:
5819:
5818:
5813:
5809:
5805:
5799:
5795:
5794:
5788:
5784:
5778:
5774:
5773:
5768:
5764:
5760:
5754:
5750:
5749:
5744:
5740:
5734:
5728:
5724:
5720:
5716:
5704:
5703:
5698:
5694:
5693:
5690:
5684:
5680:
5679:
5674:
5670:
5666:
5660:
5656:
5655:
5649:
5645:
5639:
5635:
5634:
5633:Real Analysis
5628:
5624:
5618:
5614:
5613:
5607:
5606:
5601:
5594:
5589:
5587:
5583:
5580:
5575:
5572:
5559:
5555:
5551:
5547:
5543:
5536:
5533:
5527:
5522:
5517:
5512:
5508:
5504:
5500:
5493:
5490:
5485:
5484:
5476:
5473:
5469:
5465:
5464:
5456:
5453:
5449:
5445:
5439:
5435:
5434:
5426:
5423:
5411:
5407:
5400:
5397:
5392:
5386:
5382:
5375:
5372:
5367:
5361:
5357:
5350:
5347:
5342:
5336:
5332:
5331:
5323:
5320:
5308:
5304:
5298:
5295:
5290:
5286:
5282:
5278:
5274:
5270:
5263:
5260:
5255:
5251:
5247:
5243:
5239:
5235:
5228:
5225:
5220:
5219:
5212:
5209:
5204:
5202:0-253-33020-3
5198:
5193:
5192:
5186:
5179:
5176:
5171:
5167:
5163:
5159:
5155:
5151:
5147:
5143:
5139:
5135:
5129:
5126:
5122:
5117:
5114:
5109:
5105:
5103:
5095:
5091:
5087:
5084:
5081:
5074:
5071:
5066:
5062:
5056:
5052:
5051:
5046:
5040:
5037:
5032:
5028:
5024:
5017:
5014:
5009:
5005:
5004:
4996:
4990:
4988:
4986:
4982:
4969:
4965:
4964:jsoftware.com
4961:
4954:
4951:
4946:
4942:
4938:
4934:
4930:
4926:
4919:
4916:
4903:
4899:
4895:
4891:
4885:
4882:
4877:
4871:
4867:
4862:
4861:
4852:
4849:
4836:
4832:
4831:
4823:
4820:
4815:
4809:
4805:
4798:
4795:
4791:
4787:
4781:
4778:
4773:
4767:
4763:
4762:
4754:
4752:
4748:
4743:
4739:
4733:
4730:
4725:
4724:
4717:
4714:
4709:
4708:
4700:
4697:
4692:
4691:
4683:
4680:
4675:
4674:
4667:
4664:
4659:
4658:
4650:
4647:
4642:
4641:Maths History
4638:
4632:
4630:
4628:
4626:
4624:
4622:
4620:
4618:
4614:
4609:
4608:
4600:
4597:
4592:
4591:
4586:
4580:
4577:
4572:
4570:0-19-506135-7
4566:
4562:
4555:
4552:
4547:6 = 1 + 2 + 3
4543:
4539:
4538:
4533:
4529:
4523:
4520:
4515:
4511:
4507:
4501:
4497:
4496:
4487:
4484:
4479:
4478:
4473:
4469:
4463:
4460:
4447:
4443:
4436:
4433:
4428:
4422:
4418:
4414:
4407:
4404:
4391:
4387:
4381:
4377:
4376:
4368:
4365:
4352:
4348:
4344:
4338:
4335:
4330:
4328:0-471-37568-3
4324:
4320:
4313:
4310:
4306:
4300:
4296:
4294:
4289:
4283:
4280:
4275:
4271:
4267:
4263:
4257:
4254:
4249:
4245:
4241:
4237:
4231:
4228:
4216:
4212:
4206:
4203:
4197:
4196:
4191:
4188:
4181:
4178:
4165:
4161:
4155:
4151:
4147:
4140:
4137:
4125:
4121:
4114:
4112:
4110:
4106:
4101:
4095:
4091:
4084:
4082:
4078:
4072:
4064:
4063:universal set
4060:
4054:
4051:
4047:
4043:
4042:Morash (1991)
4023:
4014:
4009:
4006:
4002:
3996:
3993:
3987:
3984:
3980:
3950:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3907:
3894:
3889:
3886:
3882:
3876:
3873:
3866:
3863:
3859:
3854:
3851:
3847:
3842:
3839:
3833:
3830:
3827:
3821:
3818:
3811:
3793:
3790:
3789:
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:
3382:
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:
3019:
3017:
3003:
3000:
2997:
2977:
2965:
2961:
2946:
2926:
2906:
2886:
2878:
2875:
2872:
2869:
2868:
2867:
2863:
2855:
2853:
2851:
2847:
2842:
2840:
2836:
2831:
2826:
2816:
2800:
2796:
2791:
2789:
2785:
2781:
2777:
2769:
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:
2551:
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:
2220:
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:
2024:
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:
1880:
1878:
1861:
1858:
1834:
1830:
1823:
1819:
1815:
1809:
1807:
1803:
1799:
1795:
1769:
1743:
1740:
1734:
1730:
1726:
1722:
1718:
1714:
1710:
1705:
1697:
1695:
1693:
1692:prime numbers
1689:
1670:
1667:
1662:
1645:. This turns
1643:
1639:
1635:
1631:
1627:
1620:
1603:
1595:
1587:
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:
1309:
1307:
1290:
1284:
1279:
1269:
1264:
1254:
1237:
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:
667:
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:
556:
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:
458:
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
294:
289:
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:
8466:Transposable
8330:Narcissistic
8237:Digital root
8157:Super-Poulet
8117:Jordan–Pólya
8066:prime factor
7971:Noncototient
7938:Almost prime
7920:Superperfect
7895:Refactorable
7890:Quasiperfect
7865:Hyperperfect
7706:Pseudoprimes
7677:Wall–Sun–Sun
7612:Ordered Bell
7582:Fuss–Catalan
7494:non-centered
7444:Dodecahedral
7421:non-centered
7307:non-centered
7209:Wolstenholme
6954:× 2 ± 1
6951:
6950:Of the form
6917:Eighth power
6897:Fourth power
6841:
6760:
6750:
6565:Dual numbers
6557:hypercomplex
6347:Real numbers
6170:
6114:
6096:
6061:
6044:15 September
6042:. Retrieved
6038:the original
6033:
6029:
6002:
5981:
5956:
5935:
5914:
5890:
5863:
5860:Levy, Azriel
5840:
5816:
5792:
5771:
5767:Halmos, Paul
5747:
5743:Eves, Howard
5722:
5707:. Retrieved
5701:
5677:
5653:
5632:
5611:
5602:Bibliography
5595:, p. 52
5574:
5562:. Retrieved
5545:
5535:
5506:
5502:
5492:
5482:
5475:
5467:
5462:
5455:
5447:
5432:
5425:
5413:. Retrieved
5409:
5399:
5380:
5374:
5355:
5349:
5329:
5322:
5310:. Retrieved
5306:
5297:
5272:
5268:
5262:
5237:
5233:
5227:
5217:
5211:
5190:
5178:
5148:(1): 85–95.
5145:
5141:
5128:
5123:, Chapter 15
5116:
5108:the original
5101:
5094:the original
5089:
5086:
5083:
5073:
5049:
5045:Gray, Jeremy
5039:
5030:
5026:
5016:
5002:
4972:. Retrieved
4963:
4958:Hui, Roger.
4953:
4928:
4924:
4918:
4906:. Retrieved
4893:
4884:
4859:
4851:
4839:. Retrieved
4829:
4822:
4803:
4797:
4792:, p. 2)
4780:
4760:
4741:
4732:
4722:
4716:
4706:
4699:
4689:
4682:
4672:
4666:
4656:
4649:
4640:
4606:
4599:
4593:(in French).
4589:
4579:
4560:
4554:
4541:
4535:
4522:
4492:
4486:
4475:
4462:
4450:. Retrieved
4435:
4416:
4406:
4394:. Retrieved
4374:
4367:
4355:. Retrieved
4346:
4337:
4318:
4312:
4299:the original
4291:
4282:
4274:the original
4266:Ishango bone
4256:
4248:the original
4240:Ishango bone
4230:
4218:. Retrieved
4214:
4205:
4193:
4180:
4168:. Retrieved
4149:
4139:
4127:. Retrieved
4123:
4089:
4053:
4045:
4026:
4022:Halmos (1960
4017:
4008:
3995:
3986:
3948:
3888:
3875:
3865:
3858:Bluman (2010
3853:
3841:
3832:
3820:
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:
8487:Extravagant
8482:Equidigital
8437:permutation
8396:Palindromic
8370:Automorphic
8268:Sum-product
8247:Sum-product
8202:Persistence
8097:Erdős–Woods
8019:Untouchable
7900:Semiperfect
7850:Hemiperfect
7511:Tesseractic
7449:Icosahedral
7429:Tetrahedral
7360:Dodecagonal
7061:Recursively
6932:Prime power
6907:Sixth power
6902:Fifth power
6882:Power of 10
6840:Classes of
6727:Other types
6546:Bioctonions
6403:Quaternions
6115:apronus.com
5593:Levy (1979)
5100:"access to
5080:"Kronecker"
4841:13 February
4452:13 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
8748:Categories
8699:Graphemics
8572:Pernicious
8426:Undulating
8401:Pandigital
8375:Trimorphic
7976:Nontotient
7825:Arithmetic
7439:Octahedral
7340:Heptagonal
7330:Pentagonal
7315:Triangular
7156:Sierpiński
7078:Jacobsthal
6877:Power of 3
6872:Power of 2
6681:Projective
6654:Infinities
5516:1703.00425
4974:19 January
4830:Set Theory
4396:3 February
4357:23 January
4099:0122384407
4073: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
8456:Parasitic
8305:Factorion
8232:Digit sum
8224:Digit sum
8042:Fortunate
8029:Primorial
7943:Semiprime
7880:Practical
7845:Descartes
7840:Deficient
7830:Betrothed
7672:Wieferich
7501:Pentatope
7464:pyramidal
7355:Decagonal
7350:Nonagonal
7345:Octagonal
7335:Hexagonal
7194:Practical
7141:Congruent
7073:Fibonacci
7037:Loeschian
6765:solenoids
6585:Sedenions
6431:Octonions
6103:EMS Press
5912:(2008) .
5721:(2007) .
5709:13 August
5675:(1963) .
5564:8 October
5289:0024-6093
5121:Eves 1990
4908:4 October
4587:(1881) .
4321:. Wiley.
4220:11 August
4195:MathWorld
4146:"integer"
4129:11 August
4001:Eves 1990
3932:…
3792: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
8764:Integers
8528:Friedman
8461:Primeval
8406:Repdigit
8363:-related
8310:Kaprekar
8284:Meertens
8207:Additive
8194:dynamics
8102:Friendly
8014:Sociable
8004:Amicable
7815:Abundant
7795:dynamics
7617:Schröder
7607:Narayana
7577:Eulerian
7567:Delannoy
7562:Dedekind
7383:centered
7249:centered
7136:Amenable
7093:Narayana
7083:Leonardo
6979:Mersenne
6927:Powerful
6867:Achilles
6199:Integers
6161:Sets of
6024:(1923).
5888:(1999).
5862:(1979).
5838:(1966).
5769:(1960).
5745:(1990).
5699:(1901).
5558:Archived
5550:Springer
5136:(1881).
5065:Archived
5047:(2008).
4968:Archived
4945:40187000
4931:(2): 7.
4902:Archived
4835:Archived
4537:Elements
4514:69792712
4477:Elements
4446:Archived
4390:Archived
4351:Archived
4170:28 March
4164: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
8701:related
8665:related
8629:related
8627:Sorting
8512:Vampire
8497:Harshad
8439:related
8411:Repunit
8325:Lychrel
8300:Dudeney
8152:Størmer
8147:Sphenic
8132:Regular
8070:divisor
8009:Perfect
7905:Sublime
7875:Perfect
7602:Motzkin
7557:Catalan
7098:Padovan
7032:Leyland
7027:Idoneal
7022:Hilbert
6994:Woodall
6755:numbers
6587: (
6433: (
6405: (
6377: (
6349: (
6293:Periods
6262: (
6229: (
6201: (
6173: (
6155:systems
6105:, 2001
5891:Algebra
5415:27 July
5312:27 July
5254:1270381
5170:1507856
5162:2369151
3947: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
8567:Odious
8492:Frugal
8446:Cyclic
8435:Digit-
8142:Smooth
8127:Pronic
8087:Cyclic
8064:Other
8037:Euclid
7687:Wilson
7661:Primes
7320:Square
7189:Polite
7151:Riesel
7146:Knödel
7108:Perrin
6989:Thabit
6974:Fermat
6964:Cullen
6887:Square
6855:Powers
6555:Other
6153:Number
6068:
6010:
5989:
5968:
5943:
5922:
5898:
5870:
5848:
5824:
5800:
5779:
5755:
5729:
5685:
5661:
5640:
5619:
5440:
5387:
5362:
5337:
5287:
5252:
5199:
5168:
5160:
5057:
4943:
4872:
4810:
4768:
4567:
4528:Euclid
4512:
4502:
4468:Euclid
4423:
4382:
4325:
4293:UNESCO
4156:
4096:
3870: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
8608:Prime
8603:Lucky
8592:sieve
8521:Other
8507:Smith
8387:Digit
8345:Happy
8320:Keith
8293:Other
8137:Rough
8107:Giuga
7572:Euler
7434:Cubic
7088:Lucas
6984:Proth
6763:-adic
6753:-adic
6510:Over
6471:Over
6465:types
6463:Split
5511:arXiv
5158:JSTOR
5088:[
5029:[
4998:(PDF)
4941:S2CID
4033:0 = ∅
4029:0 ∈ ω
3812:Notes
3358:) = {
3290:is a
3258:from
3096:Call
2846:model
2688:) is
2510:: if
2493:) + (
2485:) = (
2327:) = (
2301:) = (
2072:with
1881:Order
1727:) + (
1719:) = (
1632:) = (
1524:group
1510:with
1503:is a
624:Latin
597:Frege
485:and N
449:India
426:Greek
405:nulla
373:Olmec
369:digit
107:third
8562:Evil
8242:Self
8192:and
8082:Blum
7793:and
7597:Lobb
7552:Cake
7547:Bell
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