3330:
40:
245:
8691:
6757:
455:
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
5001:
2649:
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,
527:
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.
2597:. 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
777:
5962:
2507:
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
1171:
379:, 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
2771:
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
1056:
2791:(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.
887:
542:. 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.
3782:
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
1243:
5974:
409:(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.
1312:
538:. 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
504:
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.
659:
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:
3858:
1683:. 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
299:
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.
310:
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
2083:
1623:
666:
2905:: 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.
1814:
268:
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.
2909:
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
1440:
1062:
3574:
3536:
3498:
3460:
3422:
6254:
644:
6569:
6492:
6453:
6415:
6387:
6359:
6331:
6219:
6186:
6158:
6130:
3884:
2703:
1729:
1703:
1273:
922:
163:
2587:
2529:
952:
1553:
4224:
926:
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 "
4150:, 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."
2953:
784:
3269:, 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.
2927:
2896:
2876:
2856:
2836:
2601:(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
944:
3903:
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."
5016:
6793:
4172:
3254:
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, ...
557:
proposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of
Dedekind's axioms in his book
4542:
4198:
4985:
456:
books including 0. There are exceptions to each of these tendencies and as of 2023 no formal survey has been conducted. Arguments raised include
4944:
3343:
531:
4287:
3912:"Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (
1184:
1750:
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
567:
92:
are another term for the natural numbers, particularly in primary school education, and are ambiguous as well although typically start at 1.
5036:
4931:
4838:
4241:
4206:
4180:
3373:
2964:
2744:
is defined as an explicitly defined set, whose elements allow counting the elements of other sets, in the sense that the sentence "a set
253:
4326:
5494:
3318:
th element of a sequence) is immediate. Unlike von
Neumann's construction, the Zermelo ordinals do not extend to infinite ordinals.
2566:: "first", "second", "third", and so forth. The numbering of ordinals usually begins at zero, to accommodate the order type of the
6786:
6080:
1991:
While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of
4658:
5398:
4382:
488:
standard included 0 in the natural numbers in its first edition in 1978 and this has continued through its present edition as
375:. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.
6006:
5948:
5927:
5906:
5881:
5860:
5836:
5826:
5808:
5786:
5762:
5738:
5717:
5693:
5667:
5623:
5599:
5578:
5557:
5378:
5325:
5300:
5275:
4995:
4904:
4810:
4748:
4706:
4440:
4361:
4320:
4070:
17:
4696:
4076:
3937:, p. 46) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I)
6269:
5368:
4771:
500:
In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers.
4642:
4235:
2164:
The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:
1281:
291:
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
7593:
6779:
416:
wrote that his notions of distance and element led to defining the natural numbers as including or excluding 0. In 1889,
6264:
1709:
under subtraction (that is, subtracting one natural from another does not always result in another natural), means that
7588:
5967:
Acta
Litterarum AC Scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio Scientiarum Mathematicarum
583:: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is
7603:
5240:"Listing of the Mathematical Notations used in the Mathematical Functions Website: Numbers, variables, and functions"
5137:
4505:
4263:
7583:
3811:
8296:
7876:
5486:
5418:
5030:
2658:
772:{\displaystyle \{1,2,...\}=\mathbb {N} ^{*}=\mathbb {N} ^{+}=\mathbb {N} _{0}\smallsetminus \{0\}=\mathbb {N} _{1}}
6224:
1948:: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an
8720:
6642:
4482:
In definition VII.3 a "part" was defined as a number, but here 1 is considered to be a part, so that for example
2617:
2589:. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any
2036:
599:
replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include
413:
6720:
1587:
7598:
5707:
5385:...the set of natural numbers is closed under addition... set of natural numbers is closed under multiplication
4184:
4010:
8735:
8382:
6603:
6039:
5568:
5490:
2563:
59:
0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the
5728:
8048:
7698:
7367:
7160:
6073:
512:
5434:
Fletcher, Peter; Hrbacek, Karel; Kanovei, Vladimir; Katz, Mikhail G.; Lobry, Claude; Sanders, Sam (2017).
5169:
Baratella, Stefano; Ferro, Ruggero (1993). "A theory of sets with the negation of the axiom of infinity".
5153:
4608:
3016:, although Levy attributes the idea to unpublished work of Zermelo in 1916. As this definition extends to
1778:
8224:
8083:
7914:
7728:
7718:
7372:
7352:
6229:
6034:
4210:
3707:
1166:{\displaystyle \{0,1,2,\dots \}=\{x\in \mathbb {Z} :x\geq 0\}=\mathbb {Z} _{0}^{+}=\mathbb {Z} _{\geq 0}}
8053:
946:" or "+" is added in the former case, and a subscript (or superscript) "0" is added in the latter case:
8173:
7796:
7638:
7553:
7362:
7344:
7238:
7228:
7218:
7054:
4468:
2784:
516:
8078:
4625:
4408:
3276:
provided a construction that is nowadays only of historical interest, and is sometimes referred to as
2369:
If the natural numbers are taken as "excluding 0", and "starting at 1", then for every natural number
1411:
8715:
8301:
7846:
7467:
7253:
7248:
7243:
7233:
7210:
6637:
6593:
3553:
3515:
3477:
3439:
3401:
2990:
2753:
477:
222:
218:
8058:
6235:
6029:
4065:. Taylor & Francis. pp. 138 (integer), 247 (signed integer), & 276 (unsigned integer).
275:
to represent numbers. This allowed systems to be developed for recording large numbers. The ancient
7723:
7633:
7286:
6760:
6632:
4279:
2777:
2669:
numbers are an uncountable model that can be constructed from the ordinary natural numbers via the
2495:
Two important generalizations of natural numbers arise from the two uses of counting and ordering:
1051:{\displaystyle \{1,2,3,\dots \}=\{x\in \mathbb {Z} :x>0\}=\mathbb {Z} ^{+}=\mathbb {Z} _{>0}}
624:
276:
6552:
6475:
6436:
6398:
6370:
6342:
6314:
6202:
6169:
6141:
6113:
3867:
2686:
1712:
1686:
1256:
905:
550:
146:
8412:
8377:
8163:
8073:
7947:
7922:
7831:
7821:
7543:
7433:
7415:
7335:
6066:
5044:
2670:
1882:
1626:
2572:
2514:
8730:
8672:
7942:
7816:
7447:
7223:
7003:
6930:
5898:
5070:
3262:, including the infinite ones: "each ordinal is the well-ordered set of all smaller ordinals."
2713:
There are two standard methods for formally defining natural numbers. The first one, named for
600:
546:
4802:
4354:
The
Development of Mathematics Throughout the Centuries: A brief history in a cultural context
2539:
between them. The set of natural numbers itself, and any bijective image of it, is said to be
8636:
8276:
7927:
7781:
7708:
6863:
6705:
6541:
5994:
5121:
4826:
4349:
4310:
3714:
3091:
2168:
1706:
1538:
1458:
4794:
4525:
1676:. These properties of addition and multiplication make the natural numbers an instance of a
882:{\displaystyle \;\{0,1,2,...\}=\mathbb {N} _{0}=\mathbb {N} ^{0}=\mathbb {N} ^{*}\cup \{0\}}
8569:
8463:
8427:
8168:
7891:
7871:
7688:
7357:
7145:
6458:
6191:
5478:
5204:
Kirby, Laurie; Paris, Jeff (1982). "Accessible
Independence Results for Peano Arithmetic".
5190:
5106:
4674:
4573:
4473:
4431:
4413:
3793:
3025:
2998:
539:
524:
481:
469:
280:
239:
7648:
7117:
2787:
of Peano arithmetic inside set theory. An important consequence is that, if set theory is
283:
for 1, 10, and all powers of 10 up to over 1 million. A stone carving from
260:
The most primitive method of representing a natural number is to use one's fingers, as in
8:
8291:
8155:
8150:
8118:
7881:
7856:
7851:
7826:
7756:
7752:
7683:
7573:
7405:
7201:
7170:
6669:
6579:
6536:
6518:
6296:
4592:
3255:
2932:
2674:
2661:
satisfying the Peano
Arithmetic (that is, the first-order Peano axioms) was developed by
2153:
1985:
1841:
1462:
1406:
445:
429:
296:
8694:
8448:
8443:
8357:
8331:
8229:
8208:
7980:
7861:
7811:
7733:
7703:
7643:
7410:
7390:
7321:
7034:
6574:
6286:
5613:
5447:
5417:
Brandon, Bertha (M.); Brown, Kenneth E.; Gundlach, Bernard H.; Cooke, Ralph J. (1962).
5094:
4877:
3335:
2912:
2902:
2881:
2861:
2841:
2821:
1996:
1736:
1276:
929:
449:
326:. The use of a numeral 0 in modern times originated with the Indian mathematician
7578:
6047:
2549:
8725:
8690:
8588:
8533:
8387:
8362:
8336:
7791:
7786:
7713:
7693:
7678:
7400:
7382:
7301:
7291:
7276:
7039:
6732:
6695:
6659:
6598:
6584:
6279:
6259:
6002:
5944:
5923:
5902:
5877:
5856:
5846:
5832:
5818:
5804:
5782:
5758:
5734:
5713:
5689:
5663:
5619:
5595:
5574:
5553:
5374:
5321:
5296:
5271:
5221:
5133:
5126:
4991:
4806:
4795:
4744:
4702:
4501:
4446:
4436:
4357:
4316:
4259:
4099:
4066:
4006:
3891:
3691:
3662:
3329:
3087:
2986:
2974:
2737:
2598:
612:
596:
520:
505:
465:
335:
315:
129:
8113:
5462:
5435:
5342:
4881:
4102:
2767:
The sets used to define natural numbers satisfy Peano axioms. It follows that every
501:
256:) is believed to have been used 20,000 years ago for natural number arithmetic.
8624:
8417:
8003:
7975:
7965:
7957:
7841:
7806:
7801:
7768:
7462:
7425:
7316:
7311:
7306:
7296:
7268:
7155:
7102:
7059:
6998:
6750:
6679:
6654:
6588:
6497:
6463:
6304:
6274:
6196:
6099:
5990:
5958:
5822:
5655:
5642:. Translated by Beman, Wooster Woodruff. Chicago, IL: Open Court Publishing Company
5633:
5609:
5457:
5436:"Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others"
5213:
5178:
5086:
4869:
4378:
4032:
3957:
gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1:
3737:
3700:
3620:
3605:
3013:
2722:
2718:
2337:
1450:
572:
554:
457:
437:
433:
7107:
4896:
428:. Historically, most definitions have excluded 0, but many mathematicians such as
420:
used N for the positive integers and started at 1, but he later changed to using N
8600:
8489:
8422:
8348:
8271:
8245:
8063:
7776:
7568:
7538:
7528:
7523:
7189:
7097:
7044:
6888:
6828:
6627:
6531:
6163:
5938:
5917:
5892:
5871:
5850:
5776:
5752:
5748:
5683:
5589:
5547:
5186:
5102:
4834:
4521:
4058:
3971:
3655:
3472:
3367:
2496:
2385:. However, the "existence of additive identity element" property is not satisfied
1954:
402:
307:
261:
176:
139:
116:
101:
4801:(1. ed., 1. print ed.). Boca Raton, Fla. : Chapman & Hall/CRC. p.
4765:
4350:"Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations"
8605:
8473:
8458:
8322:
8286:
8261:
8137:
8108:
8093:
7970:
7866:
7836:
7563:
7518:
7395:
6993:
6988:
6983:
6955:
6940:
6853:
6838:
6816:
6674:
6664:
6649:
6468:
6336:
3648:
3396:
3361:
3259:
3021:
2714:
2532:
2500:
2390:
1949:
1642:
1532:
584:
580:
576:
562:
508:
summarized his belief as "God made the integers, all else is the work of man".
417:
380:
364:
339:
272:
188:
184:
121:
111:
4959:
2531:. This concept of "size" relies on maps between sets, such that two sets have
76:
Some authors acknowledge both definitions whenever convenient. Sometimes, the
8709:
8528:
8512:
8453:
8407:
8103:
8088:
7998:
7281:
7150:
7112:
7069:
6950:
6935:
6925:
6883:
6873:
6848:
6771:
6737:
6710:
6619:
5772:
5239:
5225:
5182:
3975:
3389:
3349:
3311:
3307:
3273:
3240:
2541:
2282:
2206:
1315:
535:
489:
214:
202:
5217:
5074:
4660:
Advanced
Algebra: A Study Guide to be Used with USAFI Course MC 166 Or CC166
4450:
1629:
with identity element 1; a generator set for this monoid is the set of
39:
8564:
8553:
8468:
8306:
8281:
8198:
8098:
8068:
8043:
8027:
7932:
7899:
7622:
7533:
7472:
7049:
6945:
6878:
6858:
6833:
6700:
6502:
4202:
4176:
3598:
3173:
3017:
2800:
2726:
2680:
2666:
2594:
2446:
2149:
1630:
383:
which comes shortly afterward, Euclid treats 1 as a number like any other.
249:
210:
206:
4938:
4873:
8523:
8398:
8203:
7667:
7558:
7513:
7508:
7258:
7165:
7064:
6893:
6868:
6843:
6526:
6308:
5796:
5703:
5679:
5637:
5618:. Translated by Beman, Wooster Woodruff (reprint ed.). Dover Books.
5265:
4981:
4123:
3808:, p. 15) include zero in the natural numbers: 'Intuitively, the set
3434:
3244:
2783:
The definition of the integers as sets satisfying Peano axioms provide a
2546:
1848:
1677:
1454:
1443:
441:
391:
360:
327:
323:
306:
can be considered as a number, with its own numeral. The use of a 0
244:
180:
48:
3109:
It follows that the natural numbers are defined iteratively as follows:
2159:
1238:{\displaystyle \mathbb {N} =\mathbb {N} _{0}=\mathbb {N} ^{*}\cup \{0\}}
386:
Independent studies on numbers also occurred at around the same time in
8660:
8641:
7937:
7548:
6507:
6364:
5098:
3248:
2994:
2788:
2733:
2646:
2605:; this is also the ordinal number of the set of natural numbers itself.
2590:
1945:
588:
372:
368:
303:
265:
168:
3928:, pp. 117 ff) calls them "Peano's Postulates" and begins with "1.
1641:
Addition and multiplication are compatible, which is expressed in the
8266:
8193:
8185:
7990:
7904:
7022:
4695:
Křížek, Michal; Somer, Lawrence; Šolcová, Alena (21 September 2021).
4107:
3194:
3040:
2567:
2536:
2509:
2108:
485:
461:
330:
in 628 CE. However, 0 had been used as a number in the medieval
292:
195:
5937:
Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008).
5090:
4479:
A perfect number is that which is equal to the sum of its own parts.
4379:"Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius"
2812:
Every natural number has a successor which is also a natural number.
356:, the Latin word for "none", was employed to denote a 0 value.
8367:
6615:
6546:
6392:
5999:
From Frege to Gödel: A source book in mathematical logic, 1879–1931
5452:
3639:
3355:
3266:
3098:. The intersection of all inductive sets is still an inductive set.
2683:
used to claim provocatively that "The naïve integers don't fill up
2651:
2096:
1740:
1680:
1636:
473:
331:
2148:. This Euclidean division is key to the several other properties (
412:
Starting at 0 or 1 has long been a matter of definition. In 1727,
95:
The natural numbers are used for counting things, like "there are
8372:
8031:
8025:
6135:
6058:
5873:
Bridge to
Abstract Mathematics: Mathematical proof and structures
4770:(Winter 2014 ed.). The Stanford Encyclopedia of Philosophy.
3510:
3314:
is not directly accessible; only the ordinal property (being the
2768:
1466:
897:
172:
85:
31:
5891:
Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013).
6089:
5293:
Discrete and Combinatorial Mathematics: An applied introduction
5029:]. pp. 2:5–23. (The quote is on p. 19). Archived from
4464:
4404:
4229:
3230:
2662:
1446:
893:
376:
302:
A much later advance was the development of the idea that
288:
284:
105:. They are also used to put things in order, like "this is the
56:
7087:
5124:. In Houser, Nathan; Roberts, Don D.; Van Evra, James (eds.).
4162:, p. 1): "Numbers make up the foundation of mathematics."
3272:
There are other set theoretical constructions. In particular,
2997:, as such an equivalence class would not be a set (because of
1944:
An important property of the natural numbers is that they are
4574:"Earliest Known Uses of Some of the Words of Mathematics (N)"
387:
338:
in 525 CE, without being denoted by a numeral. Standard
311:
6001:(3rd ed.). Harvard University Press. pp. 346–354.
5894:
Mathematics for Elementary Teachers: A contemporary approach
5004:
from the original on 29 March 2017 – via Google Books.
3001:). The standard solution is to define a particular set with
1314:
sending each natural number to the next one, one can define
484:. Including 0 began to rise in popularity in the 1960s. The
3582:
2171:
under addition and multiplication: for all natural numbers
1465:. The smallest group containing the natural numbers is the
5965:[On the Introduction of the Transfinite Numbers].
5433:
4987:
Plato's Ghost: The modernist transformation of mathematics
1999:
is available as a substitute: for any two natural numbers
892:
Alternatively, since the natural numbers naturally form a
5416:
4663:. United States Armed Forces Institute. 1958. p. 12.
3590:
3346: – Representation of a number as a product of primes
592:
217:
studies counting and arranging numbered objects, such as
4097:
2393:
of multiplication over addition for all natural numbers
1457:
on one generator. This commutative monoid satisfies the
334:(the calculation of the date of Easter), beginning with
201:
Natural numbers are studied in different areas of math.
5936:
5509:
4726:
4698:
From Great Discoveries in Number Theory to Applications
4091:
3310:. So, the property of the natural numbers to represent
3172:
It can be checked that the natural numbers satisfy the
3094:
under the successor function. Such sets are said to be
287:, dating back from around 1500 BCE and now at the
80:
are the natural numbers plus zero. In other cases, the
5027:
Annual report of the German Mathematicians Association
3799:
3358: – Function of the natural numbers in another set
3306:
With this definition each nonzero natural number is a
559:
The principles of arithmetic presented by a new method
279:
developed a powerful system of numerals with distinct
271:
The first major advance in abstraction was the use of
7751:
6555:
6478:
6439:
6401:
6373:
6345:
6317:
6238:
6205:
6172:
6144:
6116:
5320:(5th ed.). Boston: Addison-Wesley. p. 133.
4743:. Princeton: Princeton university press. p. 17.
4435:. Mineola, New York: Dover Publications. p. 58.
4430:
Philosophy of mathematics and deductive structure in
4312:
1491: New Revelations of the Americas before Columbus
3959:
An Axiomatization for the System of Positive Integers
3870:
3814:
3556:
3518:
3480:
3442:
3404:
3193:
elements" can be formally defined as "there exists a
2935:
2915:
2884:
2864:
2844:
2824:
2689:
2575:
2517:
2160:
Algebraic properties satisfied by the natural numbers
2039:
1781:
1715:
1689:
1590:
1541:
1531:
Analogously, given that addition has been defined, a
1414:
1284:
1259:
1187:
1065:
955:
932:
908:
787:
669:
627:
171:
are built from the natural numbers. For example, the
149:
8136:
5890:
5039:
Jahresbericht der Deutschen Mathematiker-Vereinigung
5022:
Jahresbericht der Deutschen Mathematiker-Vereinigung
4225:"The Ishango Bone, Democratic Republic of the Congo"
3325:
1307:{\displaystyle S\colon \mathbb {N} \to \mathbb {N} }
515:
saw a need to improve upon the logical rigor in the
5915:
4970:]. Translated by Greenstreet, William John. VI.
4860:Brown, Jim (1978). "In defense of index origin 0".
3890:; ...'. They follow that with their version of the
1861:if and only if there exists another natural number
99:coins on the table", in which case they are called
6563:
6486:
6447:
6409:
6381:
6353:
6325:
6248:
6213:
6180:
6152:
6124:
5367:Fletcher, Harold; Howell, Arnold A. (9 May 2014).
5318:A review of discrete and combinatorial mathematics
5125:
4694:
4421:
3878:
3852:
3568:
3530:
3492:
3454:
3416:
3364: – Generalization of "n-th" to infinite cases
2947:
2921:
2890:
2870:
2850:
2830:
2697:
2581:
2523:
2077:
1808:
1723:
1697:
1617:
1547:
1434:
1306:
1267:
1237:
1165:
1050:
938:
916:
881:
771:
638:
157:
66:, while others start with 1, defining them as the
7135:
5916:Szczepanski, Amy F.; Kositsky, Andrew P. (2008).
4630:(in French). Paris, Gauthier-Villars. p. 39.
4498:Mathematical Thought from Ancient to Modern Times
3978:exists and Russel's paradox cannot be formulated.
3024:, the sets considered below are sometimes called
318:used 0 as a separate number as early as the
198:the natural numbers in the other number systems.
8707:
5817:
5733:(Revised ed.). Cambridge University Press.
5587:
3805:
3012:The following definition was first published by
3005:elements that will be called the natural number
2631:but many well-ordered sets with cardinal number
1637:Relationship between addition and multiplication
205:looks at things like how numbers divide evenly (
115:. Natural numbers are also used as labels, like
7021:
5831:(3rd ed.). American Mathematical Society.
5588:Clapham, Christopher; Nicholson, James (2014).
3352: – Mathematical set that can be enumerated
2776:inside Peano arithmetic. A probable example is
175:are made by adding 0 and negative numbers. The
109:largest city in the country", which are called
6815:
6801:
5943:(Second ed.). ClassicalRealAnalysis.com.
5852:Number Systems and the Foundations of Analysis
5366:
5168:
5128:Studies in the Logic of Charles Sanders Peirce
4945:International Organization for Standardization
4797:Classic Set Theory: A guided independent study
4397:
3853:{\displaystyle \mathbb {N} =\{0,1,2,\ldots \}}
3796:, see D. Joyce's web edition of Book VII.
3344:Canonical representation of a positive integer
1968:In this section, juxtaposed variables such as
1952:; for the natural numbers, this is denoted as
1824:In this section, juxtaposed variables such as
532:Set-theoretical definitions of natural numbers
6787:
6074:
5685:An Introduction to the History of Mathematics
5552:(Second ed.). McGraw-Hill Professional.
4057:Ganssle, Jack G. & Barr, Michael (2003).
3179:With this definition, given a natural number
2815:0 is not the successor of any natural number.
2736:. It defines the natural numbers as specific
1851:on the natural numbers is defined by letting
615:of all natural numbers is standardly denoted
568:Arithmetices principia, nova methodo exposita
8623:
6973:
6048:"Axioms and construction of natural numbers"
5995:"On the introduction of transfinite numbers"
5591:The Concise Oxford Dictionary of Mathematics
5132:. Indiana University Press. pp. 43–52.
4613:(in Latin). Fratres Bocca. 1889. p. 12.
3847:
3823:
1232:
1226:
1122:
1096:
1090:
1066:
1012:
986:
980:
956:
876:
870:
819:
789:
751:
745:
694:
670:
6014:
5989:
5957:
5594:(Fifth ed.). Oxford University Press.
5515:
5405:Addition of natural numbers is associative.
5206:Bulletin of the London Mathematical Society
4990:. Princeton University Press. p. 153.
4926:
4924:
4922:
4568:
4566:
4564:
4562:
4560:
4558:
4556:
4554:
4242:Royal Belgian Institute of Natural Sciences
4207:Royal Belgian Institute of Natural Sciences
4191:
4181:Royal Belgian Institute of Natural Sciences
4056:
3374:Set-theoretic definition of natural numbers
3090:, there exist sets which contain 0 and are
2965:Set-theoretic definition of natural numbers
2958:
2078:{\displaystyle a=bq+r{\text{ and }}r<b.}
351:
254:Royal Belgian Institute of Natural Sciences
7088:Possessing a specific set of other numbers
6911:
6794:
6780:
6756:
6081:
6067:
5747:
5203:
5122:"3. Peirce's Axiomatization of Arithmetic"
5014:
4590:
4165:
3996:
3994:
3560:
3522:
3484:
3446:
3408:
1618:{\displaystyle (\mathbb {N} ^{*},\times )}
1318:of natural numbers recursively by setting
788:
342:do not have a symbol for 0; instead,
8551:
7498:
6557:
6480:
6441:
6403:
6375:
6347:
6319:
6207:
6174:
6146:
6118:
5919:The Complete Idiot's Guide to Pre-algebra
5845:
5712:. Springer Science & Business Media.
5566:
5461:
5451:
4960:"On the nature of mathematical reasoning"
4722:
4690:
4688:
4147:
3872:
3816:
3792:This convention is used, for example, in
3562:
3524:
3486:
3448:
3410:
2981:elements. So, it seems natural to define
2805:The five Peano axioms are the following:
2691:
2673:. Other generalizations are discussed in
1791:
1787:
1717:
1691:
1596:
1419:
1300:
1292:
1261:
1213:
1198:
1189:
1150:
1130:
1106:
1035:
1020:
996:
910:
857:
842:
827:
759:
732:
717:
702:
629:
359:The first systematic study of numbers as
151:
125:and do not have mathematical properties.
5963:"Zur Einführung der transfiniten Zahlen"
5876:(Second ed.). Mcgraw-Hill College.
5726:
5654:
5632:
5608:
5423:. Vol. 8. Laidlaw Bros. p. 25.
5396:
5315:
5295:(5th ed.). Pearson Addison Wesley.
5290:
5284:
4957:
4919:
4551:
4005:. New York: Academic Press. p. 66.
4000:
3925:
3771:
3247:on the natural numbers. This order is a
243:
38:
5525:
5523:
5119:
4792:
4763:
4540:
4520:
4427:
4376:
3991:
3370: – Size of a possibly infinite set
2675:Number § Extensions of the concept
646:Older texts have occasionally employed
553:of natural-number arithmetic. In 1888,
322:, but this usage did not spread beyond
194:. This chain of extensions canonically
14:
8708:
8659:
5869:
5781:(Third ed.). Chelsea Publishing.
5771:
5757:(Fifth ed.). Chapman & Hall.
5702:
5545:
5069:
4741:The Princeton companion to mathematics
4738:
4727:Thomson, Bruckner & Bruckner (2008
4685:
4232:'s Portal to the Heritage of Astronomy
4217:
4159:
3954:
3934:
3161:= {{ }, {{ }}, ..., {{ }, {{ }}, ...}}
2740:. More precisely, each natural number
495:
397:
119:on a sports team, where they serve as
8658:
8622:
8586:
8550:
8510:
8135:
8024:
7750:
7665:
7620:
7497:
7187:
7134:
7086:
7020:
6972:
6910:
6814:
6775:
6062:
5803:. Springer-Verlag Berlin Heidelberg.
5340:
5270:. New York: McGraw-Hill. p. 25.
5263:
4859:
4841:from the original on 13 December 2019
4623:
4527:Le Triparty en la science des nombres
4495:
4347:
4253:
4098:
4030:
3267:does not accept the axiom of infinity
2708:
7188:
5795:
5678:
5529:
5520:
5497:from the original on 13 October 2014
5158:(in German). F. Vieweg. 1893. 71-73.
5057:
4980:
4932:"Standard number sets and intervals"
4907:from the original on 20 October 2015
4640:
4610:Arithmetices principia: nova methodo
4594:Eléments de la géométrie de l'infini
4385:from the original on 15 January 2019
4308:
4290:from the original on 19 January 2013
4026:
4024:
4022:
3913:
3101:This intersection is the set of the
3031:The definition proceeds as follows:
2752:elements" means that there exists a
2638:have an ordinal number greater than
1881:. This order is compatible with the
1809:{\displaystyle (\mathbb {N^{*}} ,+)}
8587:
6270:Set-theoretically definable numbers
5267:Principles of Mathematical Analysis
5155:Was sind und was sollen die Zahlen?
4377:Deckers, Michael (25 August 2003).
3952:is the set of all natural numbers).
3205:. This formalizes the operation of
3128:2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}}
2989:under the relation "can be made in
88:, including negative integers. The
24:
8511:
6241:
6088:
5397:Davisson, Schuyler Colfax (1910).
4774:from the original on 14 March 2015
4463:
4403:
4079:from the original on 29 March 2017
3256:von Neumann definition of ordinals
2732:The second definition is based on
2576:
2518:
2490:
655:Since natural numbers may contain
25:
8747:
6022:
5997:. In van Heijenoort, Jean (ed.).
5476:
5403:. Macmillian Company. p. 2.
4128:Brilliant Math & Science Wiki
4019:
2609:The least ordinal of cardinality
2562:Natural numbers are also used as
1526:
1181:This section uses the convention
8689:
8297:Perfect digit-to-digit invariant
7666:
6755:
5015:Weber, Heinrich L. (1891–1892).
4897:"Is index origin 0 a hindrance?"
4329:from the original on 14 May 2015
4256:The Universal History of Numbers
3328:
2969:Intuitively, the natural number
2659:non-standard model of arithmetic
1435:{\displaystyle (\mathbb {N} ,+)}
233:
132:, commonly symbolized as a bold
5660:Essays on the Theory of Numbers
5639:Essays on the Theory of Numbers
5615:Essays on the Theory of Numbers
5538:
5470:
5463:10.14321/realanalexch.42.2.0193
5427:
5410:
5390:
5360:
5334:
5309:
5257:
5232:
5197:
5162:
5146:
5113:
5079:American Journal of Mathematics
5063:
5051:
5008:
4974:
4951:
4894:
4888:
4853:
4819:
4786:
4757:
4732:
4715:
4667:
4651:
4634:
4617:
4601:
4591:Fontenelle, Bernard de (1727).
4584:
4534:
4514:
4489:
4457:
4409:"Book VII, definitions 1 and 2"
4370:
4341:
4302:
4284:MacTutor History of Mathematics
4272:
4247:
3964:
3919:
3906:
3897:
3786:
3776:
3569:{\displaystyle :\;\mathbb {N} }
3531:{\displaystyle :\;\mathbb {Z} }
3493:{\displaystyle :\;\mathbb {Q} }
3455:{\displaystyle :\;\mathbb {R} }
3417:{\displaystyle :\;\mathbb {C} }
2794:
2593:countably infinite set without
2156:), and ideas in number theory.
571:). This approach is now called
414:Bernard Le Bovier de Fontenelle
6249:{\displaystyle {\mathcal {P}}}
5650:– via Project Gutenberg.
5573:. Cambridge University Press.
5370:Mathematics with Understanding
4701:. Springer Nature. p. 6.
4240:, on permanent display at the
4153:
4141:
4116:
4050:
3764:
3376: – Axiom(s) of Set Theory
2973:is the common property of all
2461:are natural numbers such that
1803:
1782:
1612:
1591:
1429:
1415:
1296:
13:
1:
7136:Expressible via specific sums
6604:Plane-based geometric algebra
5662:. Kessinger Publishing, LLC.
5491:European Mathematical Society
4641:Fine, Henry Burchard (1904).
4001:Enderton, Herbert B. (1977).
3985:
3886:contains an "initial" number
3864:may be described as follows:
3806:Mac Lane & Birkhoff (1999
2993:". This does not work in all
2725:, based on few axioms called
1461:, so it can be embedded in a
1176:
639:{\displaystyle \mathbb {N} .}
587:with several weak systems of
452:have preferred to include 0.
6564:{\displaystyle \mathbb {S} }
6487:{\displaystyle \mathbb {C} }
6448:{\displaystyle \mathbb {R} }
6410:{\displaystyle \mathbb {O} }
6382:{\displaystyle \mathbb {H} }
6354:{\displaystyle \mathbb {C} }
6326:{\displaystyle \mathbb {R} }
6214:{\displaystyle \mathbb {A} }
6181:{\displaystyle \mathbb {Q} }
6153:{\displaystyle \mathbb {Z} }
6125:{\displaystyle \mathbb {N} }
5171:Mathematical Logic Quarterly
4793:Goldrei, Derek (1998). "3".
4627:Formulaire des mathematiques
3970:In some set theories, e.g.,
3879:{\displaystyle \mathbb {N} }
3137:= {{ }, {{ }}, {{ }, {{ }}}}
2717:, consists of an autonomous
2698:{\displaystyle \mathbb {N} }
2535:, exactly if there exists a
1724:{\displaystyle \mathbb {N} }
1698:{\displaystyle \mathbb {N} }
1268:{\displaystyle \mathbb {N} }
917:{\displaystyle \mathbb {Z} }
652:as the symbol for this set.
472:when enumerating items like
158:{\displaystyle \mathbb {N} }
7:
8225:Multiplicative digital root
6035:Encyclopedia of Mathematics
5483:Encyclopedia of Mathematics
5316:Grimaldi, Ralph P. (2003).
5291:Grimaldi, Ralph P. (2004).
4500:. Oxford University Press.
4063:Embedded Systems Dictionary
3321:
2340:: for every natural number
2152:), algorithms (such as the
2136:are uniquely determined by
1963:
1885:in the following sense: if
1517:is simply the successor of
1275:of natural numbers and the
1248:
606:
363:is usually credited to the
128:The natural numbers form a
10:
8752:
7621:
5870:Morash, Ronald P. (1991).
5489:, in cooperation with the
5420:Laidlaw mathematics series
3772:§ Emergence as a term
3284:. It consists in defining
2962:
2798:
2582:{\displaystyle \emptyset }
2564:linguistic ordinal numbers
2524:{\displaystyle \emptyset }
2285:: for all natural numbers
2209:: for all natural numbers
2018:there are natural numbers
517:foundations of mathematics
237:
228:
29:
8685:
8668:
8654:
8632:
8618:
8596:
8582:
8560:
8546:
8519:
8506:
8482:
8436:
8396:
8347:
8321:
8302:Perfect digital invariant
8254:
8238:
8217:
8184:
8149:
8145:
8131:
8039:
8020:
7989:
7956:
7913:
7890:
7877:Superior highly composite
7767:
7763:
7746:
7674:
7661:
7629:
7616:
7504:
7493:
7455:
7446:
7424:
7381:
7343:
7334:
7267:
7209:
7200:
7196:
7183:
7141:
7130:
7093:
7082:
7030:
7016:
6979:
6968:
6921:
6906:
6824:
6810:
6746:
6688:
6614:
6594:Algebra of physical space
6516:
6424:
6295:
6097:
6013:– English translation of
5973:: 199–208. Archived from
5953:– via Google Books.
5932:– via Google Books.
5911:– via Google Books.
5886:– via Google Books.
5865:– via Google Books.
5841:– via Google Books.
5791:– via Google Books.
5767:– via Google Books.
5743:– via Google Books.
5722:– via Google Books.
5698:– via Google Books.
5688:(6th ed.). Thomson.
5604:– via Google Books.
5583:– via Google Books.
5562:– via Google Books.
5373:. Elsevier. p. 116.
4964:La Science et l'hypothèse
4958:Poincaré, Henri (1905) .
4947:. 19 May 2020. p. 4.
4862:ACM SIGAPL APL Quote Quad
4541:Emerson, William (1763).
4469:"Book VII, definition 22"
4366:– via Google Books.
4356:. John Wiley & Sons.
4337:– via Google Books.
4309:Mann, Charles C. (2005).
4087:– via Google Books.
3122:1 = 0 ∪ {0} = {0} = {{ }}
2991:one to one correspondence
2754:one to one correspondence
1816:has no identity element.
7915:Euler's totient function
7699:Euler–Jacobi pseudoprime
6974:Other polynomial numbers
6650:Extended complex numbers
6633:Extended natural numbers
5940:Elementary Real Analysis
5730:Logic for Mathematicians
5628:– via Archive.org.
5567:Carothers, N.L. (2000).
5183:10.1002/malq.19930390138
5075:"On the Logic of Number"
4739:Gowers, Timothy (2008).
4624:Peano, Giuseppe (1901).
4544:The method of increments
3757:
2959:Set-theoretic definition
2838:equals the successor of
1903:are natural numbers and
1819:
27:Number used for counting
7729:Somer–Lucas pseudoprime
7719:Lucas–Carmichael number
7554:Lazy caterer's sequence
5778:Foundations of Analysis
5751:; James, Glenn (1992).
5727:Hamilton, A.G. (1988).
5549:Pre-Algebra DeMYSTiFieD
5516:von Neumann (1923)
4597:(in French). p. 3.
4496:Kline, Morris (1990) .
4254:Ifrah, Georges (2000).
3932:0 is a natural number."
3134:3 = 2 ∪ {2} = {0, 1, 2}
2671:ultrapower construction
1993:division with remainder
1883:arithmetical operations
1627:free commutative monoid
1548:{\displaystyle \times }
663:Naturals without zero:
183:add infinite decimals.
179:add fractions, and the
8721:Elementary mathematics
7604:Wedderburn–Etherington
7004:Lucky numbers of Euler
6706:Transcendental numbers
6565:
6542:Hyperbolic quaternions
6488:
6449:
6411:
6383:
6355:
6327:
6250:
6215:
6182:
6154:
6126:
5991:von Neumann, John
5899:Wiley Global Education
5855:. Dover Publications.
5754:Mathematics Dictionary
5546:Bluman, Allan (2010).
5440:Real Analysis Exchange
5120:Shields, Paul (1997).
4968:Science and Hypothesis
4764:Bagaria, Joan (2017).
4471:. In Joyce, D. (ed.).
4411:. In Joyce, D. (ed.).
4003:Elements of set theory
3880:
3854:
3656:Dyadic (finite binary)
3570:
3532:
3494:
3456:
3418:
3288:as the empty set, and
3239:. In other words, the
3185:, the sentence "a set
2949:
2923:
2892:
2872:
2852:
2832:
2809:0 is a natural number.
2699:
2583:
2525:
2079:
1810:
1725:
1699:
1619:
1549:
1436:
1308:
1269:
1239:
1167:
1052:
940:
918:
883:
773:
640:
566:
547:Charles Sanders Peirce
352:
346:(or the genitive form
257:
252:(on exhibition at the
159:
44:
7892:Prime omega functions
7709:Frobenius pseudoprime
7499:Combinatorial numbers
7368:Centered dodecahedral
7161:Primary pseudoperfect
6638:Extended real numbers
6566:
6489:
6459:Split-complex numbers
6450:
6412:
6384:
6356:
6328:
6251:
6216:
6192:Constructible numbers
6183:
6155:
6127:
6015:von Neumann 1923
5347:mathworld.wolfram.com
5244:functions.wolfram.com
5218:10.1112/blms/14.4.285
5212:(4). Wiley: 285–293.
4874:10.1145/586050.586053
4428:Mueller, Ian (2006).
4348:Evans, Brian (2014).
4315:. Knopf. p. 19.
4205:. Brussels, Belgium:
4179:. Brussels, Belgium:
4037:mathworld.wolfram.com
3881:
3855:
3571:
3533:
3495:
3457:
3419:
2950:
2924:
2893:
2873:
2853:
2833:
2778:Fermat's Last Theorem
2756:between the two sets
2700:
2584:
2526:
2080:
1974:indicate the product
1830:indicate the product
1811:
1726:
1700:
1620:
1550:
1459:cancellation property
1437:
1309:
1270:
1240:
1168:
1053:
941:
919:
884:
774:
641:
591:. One such system is
579:of the properties of
407:progression naturelle
247:
238:Further information:
160:
61:non-negative integers
42:
18:Non-negative integers
8736:Sets of real numbers
8351:-composition related
8151:Arithmetic functions
7753:Arithmetic functions
7689:Elliptic pseudoprime
7373:Centered icosahedral
7353:Centered tetrahedral
6670:Supernatural numbers
6580:Multicomplex numbers
6553:
6537:Dual-complex numbers
6476:
6437:
6399:
6371:
6343:
6315:
6297:Composition algebras
6265:Arithmetical numbers
6236:
6203:
6170:
6142:
6114:
4486:is a perfect number.
4477:. Clark University.
4381:. Hbar.phys.msu.ru.
4244:, Brussels, Belgium.
4238:on 10 November 2014.
4199:"Flash presentation"
3868:
3812:
3701:Algebraic irrational
3554:
3516:
3478:
3440:
3402:
3026:von Neumann ordinals
2933:
2913:
2882:
2862:
2842:
2822:
2818:If the successor of
2687:
2573:
2515:
2203:are natural numbers.
2037:
1779:
1713:
1687:
1588:
1539:
1412:
1282:
1257:
1185:
1063:
953:
930:
906:
785:
781:Naturals with zero:
667:
625:
575:. It is based on an
525:recursive definition
460:and the size of the
240:Prehistoric counting
147:
84:refer to all of the
8277:Kaprekar's constant
7797:Colossally abundant
7684:Catalan pseudoprime
7584:Schröder–Hipparchus
7363:Centered octahedral
7239:Centered heptagonal
7229:Centered pentagonal
7219:Centered triangular
6819:and related numbers
6575:Split-biquaternions
6287:Eisenstein integers
6225:Closed-form numbers
5977:on 18 December 2014
5477:Mints, G.E. (ed.).
5341:Weisstein, Eric W.
4831:Merriam-Webster.com
4679:archive.lib.msu.edu
4417:. Clark University.
4280:"A history of Zero"
4031:Weisstein, Eric W.
3944:(where, of course,
3392:
3020:as a definition of
2948:{\displaystyle x+1}
2154:Euclidean algorithm
2112:of the division of
1986:order of operations
1984:, and the standard
1842:order of operations
1840:, and the standard
1555:can be defined via
1472:If 1 is defined as
1407:algebraic structure
1144:
601:Goodstein's theorem
549:provided the first
496:Formal construction
446:Stephen Cole Kleene
430:George A. Wentworth
398:Emergence as a term
8695:Mathematics portal
8637:Aronson's sequence
8383:Smarandache–Wellin
8140:-dependent numbers
7847:Primitive abundant
7734:Strong pseudoprime
7724:Perrin pseudoprime
7704:Fermat pseudoprime
7644:Wolstenholme prime
7468:Squared triangular
7254:Centered decagonal
7249:Centered nonagonal
7244:Centered octagonal
7234:Centered hexagonal
6733:Profinite integers
6696:Irrational numbers
6561:
6484:
6445:
6407:
6379:
6351:
6323:
6280:Gaussian rationals
6260:Computable numbers
6246:
6211:
6178:
6150:
6122:
5847:Mendelson, Elliott
5819:Mac Lane, Saunders
5264:Rudin, W. (1976).
5071:Peirce, C. S.
5047:on 20 August 2017.
4721:See, for example,
4647:. Ginn. p. 6.
4100:Weisstein, Eric W.
3876:
3850:
3566:
3528:
3490:
3452:
3414:
3388:
3336:Mathematics portal
3243:defines the usual
3154:−1} = {0, 1, ...,
2945:
2919:
2903:axiom of induction
2888:
2868:
2848:
2828:
2709:Formal definitions
2695:
2579:
2542:countably infinite
2521:
2075:
1997:Euclidean division
1806:
1739:; instead it is a
1721:
1695:
1615:
1545:
1432:
1304:
1277:successor function
1265:
1235:
1163:
1128:
1048:
936:
914:
879:
769:
636:
534:were initiated by
466:Computer languages
450:John Horton Conway
316:Maya civilizations
258:
155:
45:
8703:
8702:
8681:
8680:
8650:
8649:
8614:
8613:
8578:
8577:
8542:
8541:
8502:
8501:
8498:
8497:
8317:
8316:
8127:
8126:
8016:
8015:
8012:
8011:
7958:Aliquot sequences
7769:Divisor functions
7742:
7741:
7714:Lucas pseudoprime
7694:Euler pseudoprime
7679:Carmichael number
7657:
7656:
7612:
7611:
7489:
7488:
7485:
7484:
7481:
7480:
7442:
7441:
7330:
7329:
7287:Square triangular
7179:
7178:
7126:
7125:
7078:
7077:
7012:
7011:
6964:
6963:
6902:
6901:
6769:
6768:
6680:Superreal numbers
6660:Levi-Civita field
6655:Hyperreal numbers
6599:Spacetime algebra
6585:Geometric algebra
6498:Bicomplex numbers
6464:Split-quaternions
6305:Division algebras
6275:Gaussian integers
6197:Algebraic numbers
6100:definable numbers
6008:978-0-674-32449-7
5993:(January 2002) .
5959:von Neumann, John
5950:978-1-4348-4367-8
5929:978-1-59257-772-9
5922:. Penguin Group.
5908:978-1-118-45744-3
5897:(10th ed.).
5883:978-0-07-043043-3
5862:978-0-486-45792-5
5838:978-0-8218-1646-2
5823:Birkhoff, Garrett
5810:978-3-662-02310-5
5788:978-0-8218-2693-5
5764:978-0-412-99041-0
5740:978-0-521-36865-0
5719:978-0-387-90092-6
5695:978-0-03-029558-4
5669:978-0-548-08985-9
5656:Dedekind, Richard
5634:Dedekind, Richard
5625:978-0-486-21010-0
5610:Dedekind, Richard
5601:978-0-19-967959-1
5580:978-0-521-49756-5
5559:978-0-07-174251-1
5380:978-1-4832-8079-0
5327:978-0-201-72634-3
5302:978-0-201-72634-3
5277:978-0-07-054235-8
5033:on 9 August 2018;
4997:978-1-4008-2904-0
4812:978-0-412-60610-6
4750:978-0-691-11880-2
4708:978-3-030-83899-7
4644:A College Algebra
4442:978-0-486-45300-2
4432:Euclid's Elements
4363:978-1-118-85397-9
4322:978-1-4000-4006-3
4124:"Natural Numbers"
4103:"Counting Number"
4072:978-1-57820-120-4
3794:Euclid's Elements
3755:
3754:
3751:
3750:
3747:
3746:
3743:
3742:
3732:
3731:
3728:
3727:
3724:
3723:
3720:
3719:
3708:Irrational period
3682:
3681:
3678:
3677:
3674:
3673:
3670:
3669:
3663:Repeating decimal
3630:
3629:
3626:
3625:
3621:Negative integers
3615:
3614:
3611:
3610:
3606:Composite numbers
3258:for defining all
3088:axiom of infinity
2999:Russell's paradox
2987:equivalence class
2922:{\displaystyle x}
2891:{\displaystyle y}
2871:{\displaystyle x}
2851:{\displaystyle y}
2831:{\displaystyle x}
2599:order isomorphism
2338:identity elements
2061:
1743:(also known as a
1453: 0. It is a
1405:, and so on. The
939:{\displaystyle *}
597:axiom of infinity
540:Russell's paradox
521:Hermann Grassmann
506:Leopold Kronecker
336:Dionysius Exiguus
264:. Putting down a
68:positive integers
16:(Redirected from
8743:
8716:Cardinal numbers
8693:
8656:
8655:
8625:Natural language
8620:
8619:
8584:
8583:
8552:Generated via a
8548:
8547:
8508:
8507:
8413:Digit-reassembly
8378:Self-descriptive
8182:
8181:
8147:
8146:
8133:
8132:
8084:Lucas–Carmichael
8074:Harmonic divisor
8022:
8021:
7948:Sparsely totient
7923:Highly cototient
7832:Multiply perfect
7822:Highly composite
7765:
7764:
7748:
7747:
7663:
7662:
7618:
7617:
7599:Telephone number
7495:
7494:
7453:
7452:
7434:Square pyramidal
7416:Stella octangula
7341:
7340:
7207:
7206:
7198:
7197:
7190:Figurate numbers
7185:
7184:
7132:
7131:
7084:
7083:
7018:
7017:
6970:
6969:
6908:
6907:
6812:
6811:
6796:
6789:
6782:
6773:
6772:
6759:
6758:
6726:
6716:
6628:Cardinal numbers
6589:Clifford algebra
6570:
6568:
6567:
6562:
6560:
6532:Dual quaternions
6493:
6491:
6490:
6485:
6483:
6454:
6452:
6451:
6446:
6444:
6416:
6414:
6413:
6408:
6406:
6388:
6386:
6385:
6380:
6378:
6360:
6358:
6357:
6352:
6350:
6332:
6330:
6329:
6324:
6322:
6255:
6253:
6252:
6247:
6245:
6244:
6220:
6218:
6217:
6212:
6210:
6187:
6185:
6184:
6179:
6177:
6164:Rational numbers
6159:
6157:
6156:
6151:
6149:
6131:
6129:
6128:
6123:
6121:
6083:
6076:
6069:
6060:
6059:
6055:
6043:
6030:"Natural number"
6012:
5986:
5984:
5982:
5954:
5933:
5912:
5887:
5866:
5842:
5814:
5801:Basic Set Theory
5792:
5768:
5749:James, Robert C.
5744:
5723:
5709:Naive Set Theory
5699:
5673:
5651:
5649:
5647:
5629:
5605:
5584:
5563:
5533:
5527:
5518:
5513:
5507:
5506:
5504:
5502:
5474:
5468:
5467:
5465:
5455:
5431:
5425:
5424:
5414:
5408:
5407:
5394:
5388:
5387:
5364:
5358:
5357:
5355:
5353:
5343:"Multiplication"
5338:
5332:
5331:
5313:
5307:
5306:
5288:
5282:
5281:
5261:
5255:
5254:
5252:
5250:
5236:
5230:
5229:
5201:
5195:
5194:
5166:
5160:
5159:
5150:
5144:
5143:
5131:
5117:
5111:
5110:
5067:
5061:
5055:
5049:
5048:
5043:. Archived from
5034:
5012:
5006:
5005:
4978:
4972:
4971:
4955:
4949:
4948:
4940:ISO 80000-2:2019
4936:
4928:
4917:
4916:
4914:
4912:
4892:
4886:
4885:
4857:
4851:
4850:
4848:
4846:
4827:"natural number"
4823:
4817:
4816:
4800:
4790:
4784:
4783:
4781:
4779:
4761:
4755:
4754:
4736:
4730:
4725:, p. 3) or
4719:
4713:
4712:
4692:
4683:
4682:
4675:"Natural Number"
4671:
4665:
4664:
4655:
4649:
4648:
4638:
4632:
4631:
4621:
4615:
4614:
4605:
4599:
4598:
4588:
4582:
4581:
4570:
4549:
4548:
4538:
4532:
4531:
4522:Chuquet, Nicolas
4518:
4512:
4511:
4493:
4487:
4485:
4481:
4461:
4455:
4454:
4425:
4419:
4418:
4401:
4395:
4394:
4392:
4390:
4374:
4368:
4367:
4345:
4339:
4338:
4336:
4334:
4306:
4300:
4299:
4297:
4295:
4276:
4270:
4269:
4251:
4245:
4239:
4234:. Archived from
4221:
4215:
4214:
4209:. Archived from
4195:
4189:
4188:
4187:on 4 March 2016.
4183:. Archived from
4169:
4163:
4157:
4151:
4145:
4139:
4138:
4136:
4134:
4120:
4114:
4113:
4112:
4095:
4089:
4088:
4086:
4084:
4054:
4048:
4047:
4045:
4043:
4033:"Natural Number"
4028:
4017:
4016:
3998:
3979:
3968:
3962:
3951:
3947:
3943:
3940:
3931:
3923:
3917:
3910:
3904:
3901:
3895:
3889:
3885:
3883:
3882:
3877:
3875:
3859:
3857:
3856:
3851:
3819:
3803:
3797:
3790:
3784:
3780:
3774:
3768:
3697:
3696:
3688:
3687:
3645:
3644:
3636:
3635:
3579:
3578:
3575:
3573:
3572:
3567:
3565:
3545:
3544:
3541:
3540:
3537:
3535:
3534:
3529:
3527:
3507:
3506:
3503:
3502:
3499:
3497:
3496:
3491:
3489:
3469:
3468:
3465:
3464:
3461:
3459:
3458:
3453:
3451:
3431:
3430:
3427:
3426:
3423:
3421:
3420:
3415:
3413:
3393:
3387:
3384:
3383:
3380:
3379:
3338:
3333:
3332:
3317:
3302:
3287:
3282:
3281:
3280:Zermelo ordinals
3238:
3228:
3222:
3212:
3209:the elements of
3204:
3200:
3192:
3188:
3184:
3162:
3159:
3138:
3135:
3129:
3123:
3117:
3082:
3063:
3059:
3038:
3014:John von Neumann
3008:
3004:
2984:
2980:
2972:
2954:
2952:
2951:
2946:
2928:
2926:
2925:
2920:
2897:
2895:
2894:
2889:
2877:
2875:
2874:
2869:
2857:
2855:
2854:
2849:
2837:
2835:
2834:
2829:
2763:
2759:
2751:
2747:
2743:
2723:Peano arithmetic
2719:axiomatic theory
2704:
2702:
2701:
2696:
2694:
2641:
2637:
2630:
2626:
2615:
2604:
2588:
2586:
2585:
2580:
2558:
2530:
2528:
2527:
2522:
2497:cardinal numbers
2485:
2478:
2471:
2460:
2454:
2441:
2410:
2404:
2398:
2384:
2374:
2365:
2355:
2345:
2332:
2314:
2296:
2290:
2278:
2252:
2226:
2220:
2214:
2202:
2192:
2182:
2176:
2147:
2141:
2135:
2129:
2123:
2117:
2105:
2093:
2084:
2082:
2081:
2076:
2062:
2059:
2029:
2023:
2017:
2010:
2004:
1983:
1973:
1959:
1940:
1930:
1912:
1902:
1896:
1890:
1880:
1866:
1860:
1839:
1829:
1815:
1813:
1812:
1807:
1796:
1795:
1794:
1774:
1764:
1730:
1728:
1727:
1722:
1720:
1704:
1702:
1701:
1696:
1694:
1675:
1643:distribution law
1624:
1622:
1621:
1616:
1605:
1604:
1599:
1583:
1561:
1554:
1552:
1551:
1546:
1522:
1516:
1509:
1478:
1451:identity element
1441:
1439:
1438:
1433:
1422:
1404:
1385:
1366:
1360:
1354:
1327:
1313:
1311:
1310:
1305:
1303:
1295:
1274:
1272:
1271:
1266:
1264:
1244:
1242:
1241:
1236:
1222:
1221:
1216:
1207:
1206:
1201:
1192:
1172:
1170:
1169:
1164:
1162:
1161:
1153:
1143:
1138:
1133:
1109:
1057:
1055:
1054:
1049:
1047:
1046:
1038:
1029:
1028:
1023:
999:
945:
943:
942:
937:
925:
923:
921:
920:
915:
913:
888:
886:
885:
880:
866:
865:
860:
851:
850:
845:
836:
835:
830:
778:
776:
775:
770:
768:
767:
762:
741:
740:
735:
726:
725:
720:
711:
710:
705:
658:
651:
645:
643:
642:
637:
632:
620:
573:Peano arithmetic
555:Richard Dedekind
519:. In the 1860s,
458:division by zero
438:Nicolas Bourbaki
434:Bertrand Russell
355:
321:
213:are spread out.
192:
177:rational numbers
166:
164:
162:
161:
156:
154:
137:
102:cardinal numbers
90:counting numbers
75:
73:
65:
21:
8751:
8750:
8746:
8745:
8744:
8742:
8741:
8740:
8706:
8705:
8704:
8699:
8677:
8673:Strobogrammatic
8664:
8646:
8628:
8610:
8592:
8574:
8556:
8538:
8515:
8494:
8478:
8437:Divisor-related
8432:
8392:
8343:
8313:
8250:
8234:
8213:
8180:
8153:
8141:
8123:
8035:
8034:related numbers
8008:
7985:
7952:
7943:Perfect totient
7909:
7886:
7817:Highly abundant
7759:
7738:
7670:
7653:
7625:
7608:
7594:Stirling second
7500:
7477:
7438:
7420:
7377:
7326:
7263:
7224:Centered square
7192:
7175:
7137:
7122:
7089:
7074:
7026:
7025:defined numbers
7008:
6975:
6960:
6931:Double Mersenne
6917:
6898:
6820:
6806:
6804:natural numbers
6800:
6770:
6765:
6742:
6721:
6711:
6684:
6675:Surreal numbers
6665:Ordinal numbers
6610:
6556:
6554:
6551:
6550:
6512:
6479:
6477:
6474:
6473:
6471:
6469:Split-octonions
6440:
6438:
6435:
6434:
6426:
6420:
6402:
6400:
6397:
6396:
6374:
6372:
6369:
6368:
6346:
6344:
6341:
6340:
6337:Complex numbers
6318:
6316:
6313:
6312:
6291:
6240:
6239:
6237:
6234:
6233:
6206:
6204:
6201:
6200:
6173:
6171:
6168:
6167:
6145:
6143:
6140:
6139:
6117:
6115:
6112:
6111:
6108:Natural numbers
6093:
6087:
6046:
6028:
6025:
6020:
6009:
5980:
5978:
5951:
5930:
5909:
5884:
5863:
5839:
5811:
5789:
5765:
5741:
5720:
5696:
5670:
5645:
5643:
5626:
5602:
5581:
5560:
5541:
5536:
5528:
5521:
5514:
5510:
5500:
5498:
5475:
5471:
5432:
5428:
5415:
5411:
5400:College Algebra
5395:
5391:
5381:
5365:
5361:
5351:
5349:
5339:
5335:
5328:
5314:
5310:
5303:
5289:
5285:
5278:
5262:
5258:
5248:
5246:
5238:
5237:
5233:
5202:
5198:
5167:
5163:
5152:
5151:
5147:
5140:
5118:
5114:
5091:10.2307/2369151
5068:
5064:
5056:
5052:
5035:
5013:
5009:
4998:
4979:
4975:
4956:
4952:
4934:
4930:
4929:
4920:
4910:
4908:
4893:
4889:
4858:
4854:
4844:
4842:
4835:Merriam-Webster
4825:
4824:
4820:
4813:
4791:
4787:
4777:
4775:
4762:
4758:
4751:
4737:
4733:
4723:Carothers (2000
4720:
4716:
4709:
4693:
4686:
4673:
4672:
4668:
4657:
4656:
4652:
4639:
4635:
4622:
4618:
4607:
4606:
4602:
4589:
4585:
4572:
4571:
4552:
4539:
4535:
4519:
4515:
4508:
4494:
4490:
4483:
4462:
4458:
4443:
4426:
4422:
4402:
4398:
4388:
4386:
4375:
4371:
4364:
4346:
4342:
4332:
4330:
4323:
4307:
4303:
4293:
4291:
4278:
4277:
4273:
4266:
4252:
4248:
4223:
4222:
4218:
4213:on 27 May 2016.
4197:
4196:
4192:
4171:
4170:
4166:
4158:
4154:
4148:Mendelson (2008
4146:
4142:
4132:
4130:
4122:
4121:
4117:
4096:
4092:
4082:
4080:
4073:
4055:
4051:
4041:
4039:
4029:
4020:
4013:
3999:
3992:
3988:
3983:
3982:
3972:New Foundations
3969:
3965:
3953:
3949:
3945:
3941:
3938:
3933:
3929:
3924:
3920:
3916:, p. 606)
3911:
3907:
3902:
3898:
3887:
3871:
3869:
3866:
3865:
3862:natural numbers
3815:
3813:
3810:
3809:
3804:
3800:
3791:
3787:
3781:
3777:
3769:
3765:
3760:
3561:
3555:
3552:
3551:
3523:
3517:
3514:
3513:
3485:
3479:
3476:
3475:
3447:
3441:
3438:
3437:
3409:
3403:
3400:
3399:
3368:Cardinal number
3334:
3327:
3324:
3315:
3289:
3285:
3279:
3278:
3260:ordinal numbers
3234:
3224:
3223:if and only if
3214:
3210:
3202:
3198:
3190:
3186:
3180:
3160:
3142:
3136:
3133:
3127:
3121:
3115:
3103:natural numbers
3065:
3061:
3050:
3036:
3006:
3002:
2982:
2978:
2970:
2967:
2961:
2934:
2931:
2930:
2914:
2911:
2910:
2883:
2880:
2879:
2863:
2860:
2859:
2843:
2840:
2839:
2823:
2820:
2819:
2803:
2797:
2761:
2757:
2749:
2745:
2741:
2711:
2690:
2688:
2685:
2684:
2639:
2636:
2632:
2628:
2625:
2621:
2618:initial ordinal
2614:
2610:
2602:
2574:
2571:
2570:
2557:
2553:
2516:
2513:
2512:
2501:ordinal numbers
2493:
2491:Generalizations
2480:
2473:
2462:
2456:
2450:
2412:
2406:
2400:
2394:
2376:
2370:
2357:
2347:
2341:
2316:
2298:
2292:
2286:
2254:
2228:
2222:
2216:
2210:
2194:
2184:
2178:
2172:
2162:
2143:
2137:
2131:
2125:
2119:
2113:
2101:
2089:
2060: and
2058:
2038:
2035:
2034:
2025:
2019:
2012:
2006:
2000:
1975:
1969:
1966:
1953:
1932:
1914:
1904:
1898:
1892:
1886:
1868:
1862:
1852:
1831:
1825:
1822:
1790:
1786:
1785:
1780:
1777:
1776:
1775:. Furthermore,
1766:
1751:
1716:
1714:
1711:
1710:
1690:
1688:
1685:
1684:
1646:
1639:
1600:
1595:
1594:
1589:
1586:
1585:
1563:
1556:
1540:
1537:
1536:
1529:
1518:
1511:
1480:
1473:
1418:
1413:
1410:
1409:
1387:
1368:
1362:
1356:
1329:
1319:
1299:
1291:
1283:
1280:
1279:
1260:
1258:
1255:
1254:
1251:
1217:
1212:
1211:
1202:
1197:
1196:
1188:
1186:
1183:
1182:
1179:
1154:
1149:
1148:
1139:
1134:
1129:
1105:
1064:
1061:
1060:
1039:
1034:
1033:
1024:
1019:
1018:
995:
954:
951:
950:
931:
928:
927:
909:
907:
904:
903:
901:
861:
856:
855:
846:
841:
840:
831:
826:
825:
786:
783:
782:
763:
758:
757:
736:
731:
730:
721:
716:
715:
706:
701:
700:
668:
665:
664:
656:
647:
628:
626:
623:
622:
616:
609:
581:ordinal numbers
513:constructivists
498:
470:start from zero
427:
423:
403:Nicolas Chuquet
400:
320:1st century BCE
319:
262:finger counting
242:
236:
231:
190:
189:square root of
185:Complex numbers
150:
148:
145:
144:
142:
140:blackboard bold
133:
122:nominal numbers
112:ordinal numbers
71:
70:
64:0, 1, 2, 3, ...
63:
53:natural numbers
35:
28:
23:
22:
15:
12:
11:
5:
8749:
8739:
8738:
8733:
8728:
8723:
8718:
8701:
8700:
8698:
8697:
8686:
8683:
8682:
8679:
8678:
8676:
8675:
8669:
8666:
8665:
8652:
8651:
8648:
8647:
8645:
8644:
8639:
8633:
8630:
8629:
8616:
8615:
8612:
8611:
8609:
8608:
8606:Sorting number
8603:
8601:Pancake number
8597:
8594:
8593:
8580:
8579:
8576:
8575:
8573:
8572:
8567:
8561:
8558:
8557:
8544:
8543:
8540:
8539:
8537:
8536:
8531:
8526:
8520:
8517:
8516:
8513:Binary numbers
8504:
8503:
8500:
8499:
8496:
8495:
8493:
8492:
8486:
8484:
8480:
8479:
8477:
8476:
8471:
8466:
8461:
8456:
8451:
8446:
8440:
8438:
8434:
8433:
8431:
8430:
8425:
8420:
8415:
8410:
8404:
8402:
8394:
8393:
8391:
8390:
8385:
8380:
8375:
8370:
8365:
8360:
8354:
8352:
8345:
8344:
8342:
8341:
8340:
8339:
8328:
8326:
8323:P-adic numbers
8319:
8318:
8315:
8314:
8312:
8311:
8310:
8309:
8299:
8294:
8289:
8284:
8279:
8274:
8269:
8264:
8258:
8256:
8252:
8251:
8249:
8248:
8242:
8240:
8239:Coding-related
8236:
8235:
8233:
8232:
8227:
8221:
8219:
8215:
8214:
8212:
8211:
8206:
8201:
8196:
8190:
8188:
8179:
8178:
8177:
8176:
8174:Multiplicative
8171:
8160:
8158:
8143:
8142:
8138:Numeral system
8129:
8128:
8125:
8124:
8122:
8121:
8116:
8111:
8106:
8101:
8096:
8091:
8086:
8081:
8076:
8071:
8066:
8061:
8056:
8051:
8046:
8040:
8037:
8036:
8018:
8017:
8014:
8013:
8010:
8009:
8007:
8006:
8001:
7995:
7993:
7987:
7986:
7984:
7983:
7978:
7973:
7968:
7962:
7960:
7954:
7953:
7951:
7950:
7945:
7940:
7935:
7930:
7928:Highly totient
7925:
7919:
7917:
7911:
7910:
7908:
7907:
7902:
7896:
7894:
7888:
7887:
7885:
7884:
7879:
7874:
7869:
7864:
7859:
7854:
7849:
7844:
7839:
7834:
7829:
7824:
7819:
7814:
7809:
7804:
7799:
7794:
7789:
7784:
7782:Almost perfect
7779:
7773:
7771:
7761:
7760:
7744:
7743:
7740:
7739:
7737:
7736:
7731:
7726:
7721:
7716:
7711:
7706:
7701:
7696:
7691:
7686:
7681:
7675:
7672:
7671:
7659:
7658:
7655:
7654:
7652:
7651:
7646:
7641:
7636:
7630:
7627:
7626:
7614:
7613:
7610:
7609:
7607:
7606:
7601:
7596:
7591:
7589:Stirling first
7586:
7581:
7576:
7571:
7566:
7561:
7556:
7551:
7546:
7541:
7536:
7531:
7526:
7521:
7516:
7511:
7505:
7502:
7501:
7491:
7490:
7487:
7486:
7483:
7482:
7479:
7478:
7476:
7475:
7470:
7465:
7459:
7457:
7450:
7444:
7443:
7440:
7439:
7437:
7436:
7430:
7428:
7422:
7421:
7419:
7418:
7413:
7408:
7403:
7398:
7393:
7387:
7385:
7379:
7378:
7376:
7375:
7370:
7365:
7360:
7355:
7349:
7347:
7338:
7332:
7331:
7328:
7327:
7325:
7324:
7319:
7314:
7309:
7304:
7299:
7294:
7289:
7284:
7279:
7273:
7271:
7265:
7264:
7262:
7261:
7256:
7251:
7246:
7241:
7236:
7231:
7226:
7221:
7215:
7213:
7204:
7194:
7193:
7181:
7180:
7177:
7176:
7174:
7173:
7168:
7163:
7158:
7153:
7148:
7142:
7139:
7138:
7128:
7127:
7124:
7123:
7121:
7120:
7115:
7110:
7105:
7100:
7094:
7091:
7090:
7080:
7079:
7076:
7075:
7073:
7072:
7067:
7062:
7057:
7052:
7047:
7042:
7037:
7031:
7028:
7027:
7014:
7013:
7010:
7009:
7007:
7006:
7001:
6996:
6991:
6986:
6980:
6977:
6976:
6966:
6965:
6962:
6961:
6959:
6958:
6953:
6948:
6943:
6938:
6933:
6928:
6922:
6919:
6918:
6904:
6903:
6900:
6899:
6897:
6896:
6891:
6886:
6881:
6876:
6871:
6866:
6861:
6856:
6851:
6846:
6841:
6836:
6831:
6825:
6822:
6821:
6808:
6807:
6799:
6798:
6791:
6784:
6776:
6767:
6766:
6764:
6763:
6753:
6751:Classification
6747:
6744:
6743:
6741:
6740:
6738:Normal numbers
6735:
6730:
6708:
6703:
6698:
6692:
6690:
6686:
6685:
6683:
6682:
6677:
6672:
6667:
6662:
6657:
6652:
6647:
6646:
6645:
6635:
6630:
6624:
6622:
6620:infinitesimals
6612:
6611:
6609:
6608:
6607:
6606:
6601:
6596:
6582:
6577:
6572:
6559:
6544:
6539:
6534:
6529:
6523:
6521:
6514:
6513:
6511:
6510:
6505:
6500:
6495:
6482:
6466:
6461:
6456:
6443:
6430:
6428:
6422:
6421:
6419:
6418:
6405:
6390:
6377:
6362:
6349:
6334:
6321:
6301:
6299:
6293:
6292:
6290:
6289:
6284:
6283:
6282:
6272:
6267:
6262:
6257:
6243:
6227:
6222:
6209:
6194:
6189:
6176:
6161:
6148:
6133:
6120:
6104:
6102:
6095:
6094:
6086:
6085:
6078:
6071:
6063:
6057:
6056:
6044:
6024:
6023:External links
6021:
6019:
6018:
6007:
5987:
5955:
5949:
5934:
5928:
5913:
5907:
5888:
5882:
5867:
5861:
5843:
5837:
5815:
5809:
5793:
5787:
5773:Landau, Edmund
5769:
5763:
5745:
5739:
5724:
5718:
5700:
5694:
5676:
5675:
5674:
5668:
5652:
5624:
5606:
5600:
5585:
5579:
5564:
5558:
5542:
5540:
5537:
5535:
5534:
5519:
5508:
5479:"Peano axioms"
5469:
5446:(2): 193–253.
5426:
5409:
5389:
5379:
5359:
5333:
5326:
5308:
5301:
5283:
5276:
5256:
5231:
5196:
5177:(3): 338–352.
5161:
5145:
5138:
5112:
5062:
5050:
5007:
4996:
4973:
4950:
4918:
4887:
4852:
4818:
4811:
4785:
4756:
4749:
4731:
4714:
4707:
4684:
4666:
4650:
4633:
4616:
4600:
4583:
4550:
4547:. p. 113.
4533:
4513:
4506:
4488:
4456:
4441:
4420:
4396:
4369:
4362:
4340:
4321:
4301:
4271:
4264:
4246:
4216:
4190:
4173:"Introduction"
4164:
4152:
4140:
4115:
4090:
4071:
4049:
4018:
4011:
3989:
3987:
3984:
3981:
3980:
3963:
3926:Hamilton (1988
3918:
3905:
3896:
3892:Peano's axioms
3874:
3849:
3846:
3843:
3840:
3837:
3834:
3831:
3828:
3825:
3822:
3818:
3798:
3785:
3775:
3762:
3761:
3759:
3756:
3753:
3752:
3749:
3748:
3745:
3744:
3741:
3740:
3734:
3733:
3730:
3729:
3726:
3725:
3722:
3721:
3718:
3717:
3715:Transcendental
3711:
3710:
3704:
3703:
3694:
3684:
3683:
3680:
3679:
3676:
3675:
3672:
3671:
3668:
3667:
3665:
3659:
3658:
3652:
3651:
3649:Finite decimal
3642:
3632:
3631:
3628:
3627:
3624:
3623:
3617:
3616:
3613:
3612:
3609:
3608:
3602:
3601:
3595:
3594:
3587:
3586:
3576:
3564:
3559:
3538:
3526:
3521:
3500:
3488:
3483:
3462:
3450:
3445:
3424:
3412:
3407:
3390:Number systems
3378:
3377:
3371:
3365:
3362:Ordinal number
3359:
3353:
3347:
3340:
3339:
3323:
3320:
3170:
3169:
3168:
3167:
3164:
3140:
3131:
3125:
3119:
3107:
3106:
3099:
3084:
3044:
3022:ordinal number
2963:Main article:
2960:
2957:
2944:
2941:
2938:
2918:
2907:
2906:
2899:
2887:
2867:
2847:
2827:
2816:
2813:
2810:
2799:Main article:
2796:
2793:
2715:Giuseppe Peano
2710:
2707:
2693:
2634:
2623:
2616:(that is, the
2612:
2607:
2606:
2578:
2560:
2555:
2520:
2492:
2489:
2488:
2487:
2443:
2391:Distributivity
2388:
2387:
2386:
2334:
2280:
2204:
2161:
2158:
2124:. The numbers
2106:is called the
2094:is called the
2086:
2085:
2074:
2071:
2068:
2065:
2057:
2054:
2051:
2048:
2045:
2042:
1965:
1962:
1950:ordinal number
1821:
1818:
1805:
1802:
1799:
1793:
1789:
1784:
1719:
1693:
1638:
1635:
1614:
1611:
1608:
1603:
1598:
1593:
1544:
1533:multiplication
1528:
1527:Multiplication
1525:
1431:
1428:
1425:
1421:
1417:
1302:
1298:
1294:
1290:
1287:
1263:
1253:Given the set
1250:
1247:
1234:
1231:
1228:
1225:
1220:
1215:
1210:
1205:
1200:
1195:
1191:
1178:
1175:
1174:
1173:
1160:
1157:
1152:
1147:
1142:
1137:
1132:
1127:
1124:
1121:
1118:
1115:
1112:
1108:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1080:
1077:
1074:
1071:
1068:
1058:
1045:
1042:
1037:
1032:
1027:
1022:
1017:
1014:
1011:
1008:
1005:
1002:
998:
994:
991:
988:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
935:
912:
890:
889:
878:
875:
872:
869:
864:
859:
854:
849:
844:
839:
834:
829:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
779:
766:
761:
756:
753:
750:
747:
744:
739:
734:
729:
724:
719:
714:
709:
704:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
635:
631:
608:
605:
585:equiconsistent
577:axiomatization
551:axiomatization
502:Henri Poincaré
497:
494:
482:array-elements
425:
421:
418:Giuseppe Peano
405:used the term
399:
396:
381:perfect number
340:Roman numerals
235:
232:
230:
227:
153:
117:jersey numbers
26:
9:
6:
4:
3:
2:
8748:
8737:
8734:
8732:
8731:Number theory
8729:
8727:
8724:
8722:
8719:
8717:
8714:
8713:
8711:
8696:
8692:
8688:
8687:
8684:
8674:
8671:
8670:
8667:
8662:
8657:
8653:
8643:
8640:
8638:
8635:
8634:
8631:
8626:
8621:
8617:
8607:
8604:
8602:
8599:
8598:
8595:
8590:
8585:
8581:
8571:
8568:
8566:
8563:
8562:
8559:
8555:
8549:
8545:
8535:
8532:
8530:
8527:
8525:
8522:
8521:
8518:
8514:
8509:
8505:
8491:
8488:
8487:
8485:
8481:
8475:
8472:
8470:
8467:
8465:
8464:Polydivisible
8462:
8460:
8457:
8455:
8452:
8450:
8447:
8445:
8442:
8441:
8439:
8435:
8429:
8426:
8424:
8421:
8419:
8416:
8414:
8411:
8409:
8406:
8405:
8403:
8400:
8395:
8389:
8386:
8384:
8381:
8379:
8376:
8374:
8371:
8369:
8366:
8364:
8361:
8359:
8356:
8355:
8353:
8350:
8346:
8338:
8335:
8334:
8333:
8330:
8329:
8327:
8324:
8320:
8308:
8305:
8304:
8303:
8300:
8298:
8295:
8293:
8290:
8288:
8285:
8283:
8280:
8278:
8275:
8273:
8270:
8268:
8265:
8263:
8260:
8259:
8257:
8253:
8247:
8244:
8243:
8241:
8237:
8231:
8228:
8226:
8223:
8222:
8220:
8218:Digit product
8216:
8210:
8207:
8205:
8202:
8200:
8197:
8195:
8192:
8191:
8189:
8187:
8183:
8175:
8172:
8170:
8167:
8166:
8165:
8162:
8161:
8159:
8157:
8152:
8148:
8144:
8139:
8134:
8130:
8120:
8117:
8115:
8112:
8110:
8107:
8105:
8102:
8100:
8097:
8095:
8092:
8090:
8087:
8085:
8082:
8080:
8077:
8075:
8072:
8070:
8067:
8065:
8062:
8060:
8057:
8055:
8054:Erdős–Nicolas
8052:
8050:
8047:
8045:
8042:
8041:
8038:
8033:
8029:
8023:
8019:
8005:
8002:
8000:
7997:
7996:
7994:
7992:
7988:
7982:
7979:
7977:
7974:
7972:
7969:
7967:
7964:
7963:
7961:
7959:
7955:
7949:
7946:
7944:
7941:
7939:
7936:
7934:
7931:
7929:
7926:
7924:
7921:
7920:
7918:
7916:
7912:
7906:
7903:
7901:
7898:
7897:
7895:
7893:
7889:
7883:
7880:
7878:
7875:
7873:
7872:Superabundant
7870:
7868:
7865:
7863:
7860:
7858:
7855:
7853:
7850:
7848:
7845:
7843:
7840:
7838:
7835:
7833:
7830:
7828:
7825:
7823:
7820:
7818:
7815:
7813:
7810:
7808:
7805:
7803:
7800:
7798:
7795:
7793:
7790:
7788:
7785:
7783:
7780:
7778:
7775:
7774:
7772:
7770:
7766:
7762:
7758:
7754:
7749:
7745:
7735:
7732:
7730:
7727:
7725:
7722:
7720:
7717:
7715:
7712:
7710:
7707:
7705:
7702:
7700:
7697:
7695:
7692:
7690:
7687:
7685:
7682:
7680:
7677:
7676:
7673:
7669:
7664:
7660:
7650:
7647:
7645:
7642:
7640:
7637:
7635:
7632:
7631:
7628:
7624:
7619:
7615:
7605:
7602:
7600:
7597:
7595:
7592:
7590:
7587:
7585:
7582:
7580:
7577:
7575:
7572:
7570:
7567:
7565:
7562:
7560:
7557:
7555:
7552:
7550:
7547:
7545:
7542:
7540:
7537:
7535:
7532:
7530:
7527:
7525:
7522:
7520:
7517:
7515:
7512:
7510:
7507:
7506:
7503:
7496:
7492:
7474:
7471:
7469:
7466:
7464:
7461:
7460:
7458:
7454:
7451:
7449:
7448:4-dimensional
7445:
7435:
7432:
7431:
7429:
7427:
7423:
7417:
7414:
7412:
7409:
7407:
7404:
7402:
7399:
7397:
7394:
7392:
7389:
7388:
7386:
7384:
7380:
7374:
7371:
7369:
7366:
7364:
7361:
7359:
7358:Centered cube
7356:
7354:
7351:
7350:
7348:
7346:
7342:
7339:
7337:
7336:3-dimensional
7333:
7323:
7320:
7318:
7315:
7313:
7310:
7308:
7305:
7303:
7300:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7278:
7275:
7274:
7272:
7270:
7266:
7260:
7257:
7255:
7252:
7250:
7247:
7245:
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7216:
7214:
7212:
7208:
7205:
7203:
7202:2-dimensional
7199:
7195:
7191:
7186:
7182:
7172:
7169:
7167:
7164:
7162:
7159:
7157:
7154:
7152:
7149:
7147:
7146:Nonhypotenuse
7144:
7143:
7140:
7133:
7129:
7119:
7116:
7114:
7111:
7109:
7106:
7104:
7101:
7099:
7096:
7095:
7092:
7085:
7081:
7071:
7068:
7066:
7063:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7032:
7029:
7024:
7019:
7015:
7005:
7002:
7000:
6997:
6995:
6992:
6990:
6987:
6985:
6982:
6981:
6978:
6971:
6967:
6957:
6954:
6952:
6949:
6947:
6944:
6942:
6939:
6937:
6934:
6932:
6929:
6927:
6924:
6923:
6920:
6915:
6909:
6905:
6895:
6892:
6890:
6887:
6885:
6884:Perfect power
6882:
6880:
6877:
6875:
6874:Seventh power
6872:
6870:
6867:
6865:
6862:
6860:
6857:
6855:
6852:
6850:
6847:
6845:
6842:
6840:
6837:
6835:
6832:
6830:
6827:
6826:
6823:
6818:
6813:
6809:
6805:
6797:
6792:
6790:
6785:
6783:
6778:
6777:
6774:
6762:
6754:
6752:
6749:
6748:
6745:
6739:
6736:
6734:
6731:
6728:
6724:
6718:
6714:
6709:
6707:
6704:
6702:
6701:Fuzzy numbers
6699:
6697:
6694:
6693:
6691:
6687:
6681:
6678:
6676:
6673:
6671:
6668:
6666:
6663:
6661:
6658:
6656:
6653:
6651:
6648:
6644:
6641:
6640:
6639:
6636:
6634:
6631:
6629:
6626:
6625:
6623:
6621:
6617:
6613:
6605:
6602:
6600:
6597:
6595:
6592:
6591:
6590:
6586:
6583:
6581:
6578:
6576:
6573:
6548:
6545:
6543:
6540:
6538:
6535:
6533:
6530:
6528:
6525:
6524:
6522:
6520:
6515:
6509:
6506:
6504:
6503:Biquaternions
6501:
6499:
6496:
6470:
6467:
6465:
6462:
6460:
6457:
6432:
6431:
6429:
6423:
6394:
6391:
6366:
6363:
6338:
6335:
6310:
6306:
6303:
6302:
6300:
6298:
6294:
6288:
6285:
6281:
6278:
6277:
6276:
6273:
6271:
6268:
6266:
6263:
6261:
6258:
6231:
6228:
6226:
6223:
6198:
6195:
6193:
6190:
6165:
6162:
6137:
6134:
6109:
6106:
6105:
6103:
6101:
6096:
6091:
6084:
6079:
6077:
6072:
6070:
6065:
6064:
6061:
6053:
6049:
6045:
6041:
6037:
6036:
6031:
6027:
6026:
6016:
6010:
6004:
6000:
5996:
5992:
5988:
5976:
5972:
5968:
5964:
5960:
5956:
5952:
5946:
5942:
5941:
5935:
5931:
5925:
5921:
5920:
5914:
5910:
5904:
5900:
5896:
5895:
5889:
5885:
5879:
5875:
5874:
5868:
5864:
5858:
5854:
5853:
5848:
5844:
5840:
5834:
5830:
5829:
5824:
5820:
5816:
5812:
5806:
5802:
5798:
5794:
5790:
5784:
5780:
5779:
5774:
5770:
5766:
5760:
5756:
5755:
5750:
5746:
5742:
5736:
5732:
5731:
5725:
5721:
5715:
5711:
5710:
5705:
5701:
5697:
5691:
5687:
5686:
5681:
5677:
5671:
5665:
5661:
5657:
5653:
5641:
5640:
5635:
5631:
5630:
5627:
5621:
5617:
5616:
5611:
5607:
5603:
5597:
5593:
5592:
5586:
5582:
5576:
5572:
5571:
5570:Real Analysis
5565:
5561:
5555:
5551:
5550:
5544:
5543:
5531:
5526:
5524:
5517:
5512:
5496:
5492:
5488:
5484:
5480:
5473:
5464:
5459:
5454:
5449:
5445:
5441:
5437:
5430:
5422:
5421:
5413:
5406:
5402:
5401:
5393:
5386:
5382:
5376:
5372:
5371:
5363:
5348:
5344:
5337:
5329:
5323:
5319:
5312:
5304:
5298:
5294:
5287:
5279:
5273:
5269:
5268:
5260:
5245:
5241:
5235:
5227:
5223:
5219:
5215:
5211:
5207:
5200:
5192:
5188:
5184:
5180:
5176:
5172:
5165:
5157:
5156:
5149:
5141:
5139:0-253-33020-3
5135:
5130:
5129:
5123:
5116:
5108:
5104:
5100:
5096:
5092:
5088:
5084:
5080:
5076:
5072:
5066:
5059:
5054:
5046:
5042:
5040:
5032:
5028:
5024:
5021:
5018:
5011:
5003:
4999:
4993:
4989:
4988:
4983:
4977:
4969:
4965:
4961:
4954:
4946:
4942:
4941:
4933:
4927:
4925:
4923:
4906:
4902:
4901:jsoftware.com
4898:
4891:
4883:
4879:
4875:
4871:
4867:
4863:
4856:
4840:
4836:
4832:
4828:
4822:
4814:
4808:
4804:
4799:
4798:
4789:
4773:
4769:
4768:
4760:
4752:
4746:
4742:
4735:
4728:
4724:
4718:
4710:
4704:
4700:
4699:
4691:
4689:
4680:
4676:
4670:
4662:
4661:
4654:
4646:
4645:
4637:
4629:
4628:
4620:
4612:
4611:
4604:
4596:
4595:
4587:
4579:
4578:Maths History
4575:
4569:
4567:
4565:
4563:
4561:
4559:
4557:
4555:
4546:
4545:
4537:
4529:
4528:
4523:
4517:
4509:
4507:0-19-506135-7
4503:
4499:
4492:
4484:6 = 1 + 2 + 3
4480:
4476:
4475:
4470:
4466:
4460:
4452:
4448:
4444:
4438:
4434:
4433:
4424:
4416:
4415:
4410:
4406:
4400:
4384:
4380:
4373:
4365:
4359:
4355:
4351:
4344:
4328:
4324:
4318:
4314:
4313:
4305:
4289:
4285:
4281:
4275:
4267:
4265:0-471-37568-3
4261:
4257:
4250:
4243:
4237:
4233:
4231:
4226:
4220:
4212:
4208:
4204:
4200:
4194:
4186:
4182:
4178:
4174:
4168:
4161:
4156:
4149:
4144:
4129:
4125:
4119:
4110:
4109:
4104:
4101:
4094:
4078:
4074:
4068:
4064:
4060:
4053:
4038:
4034:
4027:
4025:
4023:
4014:
4008:
4004:
3997:
3995:
3990:
3977:
3976:universal set
3973:
3967:
3960:
3956:
3955:Morash (1991)
3936:
3927:
3922:
3915:
3909:
3900:
3893:
3863:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3820:
3807:
3802:
3795:
3789:
3779:
3773:
3767:
3763:
3739:
3736:
3735:
3716:
3713:
3712:
3709:
3706:
3705:
3702:
3699:
3698:
3695:
3693:
3690:
3689:
3686:
3685:
3666:
3664:
3661:
3660:
3657:
3654:
3653:
3650:
3647:
3646:
3643:
3641:
3638:
3637:
3634:
3633:
3622:
3619:
3618:
3607:
3604:
3603:
3600:
3599:Prime numbers
3597:
3596:
3592:
3589:
3588:
3584:
3581:
3580:
3577:
3557:
3550:
3547:
3546:
3543:
3542:
3539:
3519:
3512:
3509:
3508:
3505:
3504:
3501:
3481:
3474:
3471:
3470:
3467:
3466:
3463:
3443:
3436:
3433:
3432:
3429:
3428:
3425:
3405:
3398:
3395:
3394:
3391:
3386:
3385:
3382:
3381:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3350:Countable set
3348:
3345:
3342:
3341:
3337:
3331:
3326:
3319:
3313:
3312:cardinalities
3309:
3308:singleton set
3304:
3300:
3296:
3292:
3283:
3275:
3274:Ernst Zermelo
3270:
3268:
3263:
3261:
3257:
3252:
3250:
3246:
3242:
3241:set inclusion
3237:
3232:
3227:
3221:
3217:
3208:
3196:
3183:
3177:
3175:
3165:
3157:
3153:
3149:
3145:
3141:
3132:
3126:
3120:
3114:
3113:
3112:
3111:
3110:
3104:
3100:
3097:
3093:
3089:
3085:
3080:
3076:
3072:
3068:
3057:
3053:
3049:
3045:
3042:
3034:
3033:
3032:
3029:
3027:
3023:
3019:
3015:
3010:
3000:
2996:
2992:
2988:
2976:
2966:
2956:
2942:
2939:
2936:
2916:
2904:
2900:
2885:
2865:
2845:
2825:
2817:
2814:
2811:
2808:
2807:
2806:
2802:
2792:
2790:
2786:
2781:
2779:
2775:
2770:
2765:
2755:
2739:
2735:
2730:
2728:
2724:
2720:
2716:
2706:
2682:
2678:
2676:
2672:
2668:
2665:in 1933. The
2664:
2660:
2655:
2653:
2648:
2643:
2619:
2600:
2596:
2592:
2569:
2565:
2561:
2551:
2548:
2544:
2543:
2538:
2534:
2533:the same size
2511:
2506:
2505:
2504:
2502:
2498:
2483:
2476:
2469:
2465:
2459:
2453:
2448:
2447:zero divisors
2444:
2439:
2435:
2431:
2427:
2423:
2419:
2415:
2409:
2403:
2397:
2392:
2389:
2383:
2379:
2373:
2368:
2367:
2364:
2360:
2354:
2350:
2344:
2339:
2336:Existence of
2335:
2331:
2327:
2323:
2319:
2313:
2309:
2305:
2301:
2295:
2289:
2284:
2283:Commutativity
2281:
2277:
2273:
2269:
2265:
2261:
2257:
2251:
2247:
2243:
2239:
2235:
2231:
2225:
2219:
2213:
2208:
2207:Associativity
2205:
2201:
2197:
2191:
2187:
2181:
2175:
2170:
2167:
2166:
2165:
2157:
2155:
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2043:
2040:
2033:
2032:
2031:
2028:
2022:
2015:
2009:
2003:
1998:
1994:
1989:
1987:
1982:
1978:
1972:
1961:
1958:
1957:
1951:
1947:
1942:
1939:
1935:
1929:
1925:
1921:
1917:
1911:
1907:
1901:
1895:
1889:
1884:
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1865:
1859:
1855:
1850:
1845:
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1838:
1834:
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1817:
1800:
1797:
1773:
1769:
1762:
1758:
1754:
1748:
1746:
1742:
1738:
1734:
1708:
1682:
1679:
1673:
1669:
1665:
1661:
1657:
1653:
1649:
1644:
1634:
1632:
1631:prime numbers
1628:
1609:
1606:
1601:
1584:. This turns
1582:
1578:
1574:
1570:
1566:
1559:
1542:
1534:
1524:
1521:
1514:
1507:
1503:
1499:
1495:
1491:
1487:
1483:
1476:
1470:
1468:
1464:
1460:
1456:
1452:
1448:
1445:
1426:
1423:
1408:
1402:
1398:
1394:
1390:
1383:
1379:
1375:
1371:
1365:
1359:
1352:
1348:
1344:
1340:
1336:
1332:
1326:
1322:
1317:
1288:
1285:
1278:
1246:
1229:
1223:
1218:
1208:
1203:
1193:
1158:
1155:
1145:
1140:
1135:
1125:
1119:
1116:
1113:
1110:
1102:
1099:
1093:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1059:
1043:
1040:
1030:
1025:
1015:
1009:
1006:
1003:
1000:
992:
989:
983:
977:
974:
971:
968:
965:
962:
959:
949:
948:
947:
933:
899:
895:
873:
867:
862:
852:
847:
837:
832:
822:
816:
813:
810:
807:
804:
801:
798:
795:
792:
780:
764:
754:
748:
742:
737:
727:
722:
712:
707:
697:
691:
688:
685:
682:
679:
676:
673:
662:
661:
660:
653:
650:
633:
619:
614:
604:
602:
598:
594:
590:
586:
582:
578:
574:
570:
569:
564:
560:
556:
552:
548:
543:
541:
537:
533:
529:
526:
522:
518:
514:
509:
507:
503:
493:
491:
487:
483:
479:
475:
474:loop counters
471:
467:
463:
459:
453:
451:
447:
443:
439:
435:
431:
419:
415:
410:
408:
404:
395:
393:
390:, China, and
389:
384:
382:
378:
374:
370:
367:philosophers
366:
362:
357:
354:
349:
345:
341:
337:
333:
329:
325:
317:
313:
309:
305:
300:
298:
294:
290:
286:
282:
278:
274:
269:
267:
263:
255:
251:
246:
241:
234:Ancient roots
226:
224:
220:
216:
215:Combinatorics
212:
211:prime numbers
208:
204:
203:Number theory
199:
197:
193:
186:
182:
178:
174:
170:
167:. Many other
141:
136:
131:
126:
124:
123:
118:
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:
8428:Transposable
8292:Narcissistic
8199:Digital root
8119:Super-Poulet
8079:Jordan–Pólya
8028:prime factor
7933:Noncototient
7900:Almost prime
7882:Superperfect
7857:Refactorable
7852:Quasiperfect
7827:Hyperperfect
7668:Pseudoprimes
7639:Wall–Sun–Sun
7574:Ordered Bell
7544:Fuss–Catalan
7456:non-centered
7406:Dodecahedral
7383:non-centered
7269:non-centered
7171:Wolstenholme
6916:× 2 ± 1
6913:
6912:Of the form
6879:Eighth power
6859:Fourth power
6803:
6722:
6712:
6527:Dual numbers
6519:hypercomplex
6309:Real numbers
6107:
6051:
6033:
5998:
5981:15 September
5979:. Retrieved
5975:the original
5970:
5966:
5939:
5918:
5893:
5872:
5851:
5827:
5800:
5797:Levy, Azriel
5777:
5753:
5729:
5708:
5704:Halmos, Paul
5684:
5680:Eves, Howard
5659:
5644:. Retrieved
5638:
5614:
5590:
5569:
5548:
5539:Bibliography
5532:, p. 52
5511:
5499:. Retrieved
5482:
5472:
5443:
5439:
5429:
5419:
5412:
5404:
5399:
5392:
5384:
5369:
5362:
5350:. Retrieved
5346:
5336:
5317:
5311:
5292:
5286:
5266:
5259:
5247:. Retrieved
5243:
5234:
5209:
5205:
5199:
5174:
5170:
5164:
5154:
5148:
5127:
5115:
5085:(1): 85–95.
5082:
5078:
5065:
5060:, Chapter 15
5053:
5045:the original
5038:
5031:the original
5026:
5023:
5020:
5010:
4986:
4982:Gray, Jeremy
4976:
4967:
4963:
4953:
4939:
4909:. Retrieved
4900:
4895:Hui, Roger.
4890:
4865:
4861:
4855:
4843:. Retrieved
4830:
4821:
4796:
4788:
4776:. Retrieved
4766:
4759:
4740:
4734:
4729:, p. 2)
4717:
4697:
4678:
4669:
4659:
4653:
4643:
4636:
4626:
4619:
4609:
4603:
4593:
4586:
4577:
4543:
4536:
4530:(in French).
4526:
4516:
4497:
4491:
4478:
4472:
4459:
4429:
4423:
4412:
4399:
4387:. Retrieved
4372:
4353:
4343:
4331:. Retrieved
4311:
4304:
4292:. Retrieved
4283:
4274:
4255:
4249:
4236:the original
4228:
4219:
4211:the original
4203:Ishango bone
4193:
4185:the original
4177:Ishango bone
4167:
4160:Bluman (2010
4155:
4143:
4131:. Retrieved
4127:
4118:
4106:
4093:
4081:. Retrieved
4062:
4052:
4040:. Retrieved
4036:
4002:
3966:
3958:
3939:
3935:Halmos (1960
3930:
3921:
3908:
3899:
3861:
3801:
3788:
3778:
3766:
3548:
3305:
3298:
3294:
3290:
3277:
3271:
3264:
3253:
3235:
3225:
3219:
3215:
3206:
3181:
3178:
3174:Peano axioms
3171:
3155:
3151:
3147:
3143:
3108:
3102:
3095:
3078:
3074:
3070:
3066:
3055:
3051:
3047:
3030:
3018:infinite set
3011:
2995:set theories
2968:
2908:
2804:
2801:Peano axioms
2795:Peano axioms
2782:
2773:
2766:
2731:
2727:Peano axioms
2712:
2681:Georges Reeb
2679:
2667:hypernatural
2657:A countable
2656:
2644:
2608:
2595:limit points
2591:well-ordered
2545:and to have
2540:
2494:
2481:
2474:
2467:
2463:
2457:
2451:
2437:
2433:
2429:
2425:
2421:
2417:
2413:
2407:
2401:
2395:
2381:
2377:
2371:
2362:
2358:
2352:
2348:
2342:
2329:
2325:
2321:
2317:
2311:
2307:
2303:
2299:
2293:
2287:
2275:
2271:
2267:
2263:
2259:
2255:
2249:
2245:
2241:
2237:
2233:
2229:
2223:
2217:
2211:
2199:
2195:
2189:
2185:
2179:
2173:
2163:
2150:divisibility
2144:
2138:
2132:
2126:
2120:
2114:
2107:
2102:
2095:
2090:
2087:
2026:
2020:
2013:
2007:
2001:
1992:
1990:
1988:is assumed.
1980:
1976:
1970:
1967:
1955:
1946:well-ordered
1943:
1937:
1933:
1927:
1923:
1919:
1915:
1909:
1905:
1899:
1893:
1887:
1877:
1873:
1869:
1863:
1857:
1853:
1846:
1844:is assumed.
1836:
1832:
1826:
1823:
1771:
1767:
1760:
1756:
1752:
1749:
1744:
1732:
1671:
1667:
1663:
1659:
1655:
1651:
1647:
1640:
1580:
1576:
1572:
1568:
1564:
1557:
1530:
1519:
1512:
1505:
1501:
1497:
1493:
1489:
1485:
1481:
1474:
1471:
1400:
1396:
1392:
1388:
1381:
1377:
1373:
1369:
1363:
1357:
1350:
1346:
1342:
1338:
1334:
1330:
1324:
1320:
1252:
1180:
891:
654:
648:
617:
610:
558:
544:
530:
523:suggested a
510:
499:
454:
411:
406:
401:
385:
361:abstractions
358:
347:
343:
301:
270:
259:
250:Ishango bone
223:enumerations
207:divisibility
200:
181:real numbers
134:
127:
120:
110:
106:
100:
96:
94:
89:
81:
77:
72:1, 2, 3, ...
67:
60:
52:
46:
36:
8449:Extravagant
8444:Equidigital
8399:permutation
8358:Palindromic
8332:Automorphic
8230:Sum-product
8209:Sum-product
8164:Persistence
8059:Erdős–Woods
7981:Untouchable
7862:Semiperfect
7812:Hemiperfect
7473:Tesseractic
7411:Icosahedral
7391:Tetrahedral
7322:Dodecagonal
7023:Recursively
6894:Prime power
6869:Sixth power
6864:Fifth power
6844:Power of 10
6802:Classes of
6689:Other types
6508:Bioctonions
6365:Quaternions
6052:apronus.com
5530:Levy (1979)
5037:"access to
5017:"Kronecker"
4778:13 February
4389:13 February
3245:total order
3060:of any set
3046:Define the
2547:cardinality
2445:No nonzero
2088:The number
1849:total order
1678:commutative
1510:. That is,
1455:free monoid
1444:commutative
1395:+ S(1) = S(
1376:+ S(0) = S(
490:ISO 80000-2
442:Paul Halmos
392:Mesoamerica
328:Brahmagupta
324:Mesoamerica
297:place-value
293:Babylonians
281:hieroglyphs
169:number sets
49:mathematics
8710:Categories
8661:Graphemics
8534:Pernicious
8388:Undulating
8363:Pandigital
8337:Trimorphic
7938:Nontotient
7787:Arithmetic
7401:Octahedral
7302:Heptagonal
7292:Pentagonal
7277:Triangular
7118:Sierpiński
7040:Jacobsthal
6839:Power of 3
6834:Power of 2
6643:Projective
6616:Infinities
5453:1703.00425
4911:19 January
4767:Set Theory
4333:3 February
4294:23 January
4012:0122384407
3986:References
3692:Irrational
3249:well-order
2977:that have
2789:consistent
2734:set theory
2550:aleph-null
2486:(or both).
2030:such that
1399:+1) = S(S(
1177:Properties
589:set theory
373:Archimedes
369:Pythagoras
266:tally mark
219:partitions
209:), or how
8418:Parasitic
8267:Factorion
8194:Digit sum
8186:Digit sum
8004:Fortunate
7991:Primorial
7905:Semiprime
7842:Practical
7807:Descartes
7802:Deficient
7792:Betrothed
7634:Wieferich
7463:Pentatope
7426:pyramidal
7317:Decagonal
7312:Nonagonal
7307:Octagonal
7297:Hexagonal
7156:Practical
7103:Congruent
7035:Fibonacci
6999:Loeschian
6727:solenoids
6547:Sedenions
6393:Octonions
6040:EMS Press
5849:(2008) .
5658:(2007) .
5646:13 August
5612:(1963) .
5501:8 October
5226:0024-6093
5058:Eves 1990
4845:4 October
4524:(1881) .
4258:. Wiley.
4133:11 August
4108:MathWorld
4059:"integer"
4042:11 August
3914:Eves 1990
3845:…
3738:Imaginary
3195:bijection
3096:inductive
3048:successor
3041:empty set
2577:∅
2568:empty set
2537:bijection
2519:∅
2510:empty set
2142:and
2109:remainder
1960:(omega).
1792:∗
1610:×
1602:∗
1543:×
1535:operator
1297:→
1289::
1224:∪
1219:∗
1156:≥
1117:≥
1103:∈
1088:…
993:∈
978:…
934:∗
868:∪
863:∗
743:∖
708:∗
595:with the
545:In 1881,
486:ISO 31-11
462:empty set
277:Egyptians
8726:Integers
8490:Friedman
8423:Primeval
8368:Repdigit
8325:-related
8272:Kaprekar
8246:Meertens
8169:Additive
8156:dynamics
8064:Friendly
7976:Sociable
7966:Amicable
7777:Abundant
7757:dynamics
7579:Schröder
7569:Narayana
7539:Eulerian
7529:Delannoy
7524:Dedekind
7345:centered
7211:centered
7098:Amenable
7055:Narayana
7045:Leonardo
6941:Mersenne
6889:Powerful
6829:Achilles
6136:Integers
6098:Sets of
5961:(1923).
5825:(1999).
5799:(1979).
5775:(1966).
5706:(1960).
5682:(1990).
5636:(1901).
5495:Archived
5487:Springer
5073:(1881).
5002:Archived
4984:(2008).
4905:Archived
4882:40187000
4868:(2): 7.
4839:Archived
4772:Archived
4474:Elements
4451:69792712
4414:Elements
4383:Archived
4327:Archived
4288:Archived
4083:28 March
4077:Archived
3640:Fraction
3473:Rational
3356:Sequence
3322:See also
3213:. Also,
3207:counting
2774:provable
2652:sequence
2118:by
2097:quotient
1964:Division
1741:semiring
1681:semiring
1467:integers
1380:+0) = S(
1367:. Thus,
1355:for all
1316:addition
1249:Addition
902:denoted
898:integers
607:Notation
332:computus
273:numerals
187:add the
173:integers
86:integers
55:are the
8663:related
8627:related
8591:related
8589:Sorting
8474:Vampire
8459:Harshad
8401:related
8373:Repunit
8287:Lychrel
8262:Dudeney
8114:Størmer
8109:Sphenic
8094:Regular
8032:divisor
7971:Perfect
7867:Sublime
7837:Perfect
7564:Motzkin
7519:Catalan
7060:Padovan
6994:Leyland
6989:Idoneal
6984:Hilbert
6956:Woodall
6717:numbers
6549: (
6395: (
6367: (
6339: (
6311: (
6232: (
6230:Periods
6199: (
6166: (
6138: (
6110: (
6092:systems
6042:, 2001
5828:Algebra
5352:27 July
5249:27 July
5191:1270381
5107:1507856
5099:2369151
3860:of all
3549:Natural
3511:Integer
3397:Complex
3265:If one
3116:0 = { }
3086:By the
3037:0 = { }
2878:equals
2858:, then
2769:theorem
2721:called
2472:, then
2183:, both
2169:Closure
1913:, then
1705:is not
1625:into a
1560:× 0 = 0
1500:+ 0) =
1479:, then
900:(often
896:of the
478:string-
350:) from
229:History
165:
143:
57:numbers
32:Integer
8529:Odious
8454:Frugal
8408:Cyclic
8397:Digit-
8104:Smooth
8089:Pronic
8049:Cyclic
8026:Other
7999:Euclid
7649:Wilson
7623:Primes
7282:Square
7151:Polite
7113:Riesel
7108:Knödel
7070:Perrin
6951:Thabit
6936:Fermat
6926:Cullen
6849:Square
6817:Powers
6517:Other
6090:Number
6005:
5947:
5926:
5905:
5880:
5859:
5835:
5807:
5785:
5761:
5737:
5716:
5692:
5666:
5622:
5598:
5577:
5556:
5377:
5324:
5299:
5274:
5224:
5189:
5136:
5105:
5097:
4994:
4880:
4809:
4747:
4705:
4504:
4465:Euclid
4449:
4439:
4405:Euclid
4360:
4319:
4262:
4230:UNESCO
4069:
4009:
3783:place.
3231:subset
3150:−1 ∪ {
3092:closed
3039:, the
2985:as an
2663:Skolem
2647:finite
2405:, and
2380:× 1 =
2361:× 1 =
2351:+ 0 =
2221:, and
1867:where
1770:× 1 =
1755:+ 1 =
1707:closed
1492:(0) =
1484:+ 1 =
1447:monoid
1391:+ 2 =
1372:+ 1 =
1323:+ 0 =
894:subset
468:often
448:, and
377:Euclid
353:nullus
348:nullae
295:had a
289:Louvre
285:Karnak
196:embeds
51:, the
8570:Prime
8565:Lucky
8554:sieve
8483:Other
8469:Smith
8349:Digit
8307:Happy
8282:Keith
8255:Other
8099:Rough
8069:Giuga
7534:Euler
7396:Cubic
7050:Lucas
6946:Proth
6725:-adic
6715:-adic
6472:Over
6433:Over
6427:types
6425:Split
5448:arXiv
5095:JSTOR
5025:[
4966:[
4935:(PDF)
4878:S2CID
3946:0 = ∅
3942:0 ∈ ω
3758:Notes
3297:) = {
3229:is a
3197:from
3035:Call
2785:model
2627:) is
2449:: if
2432:) + (
2424:) = (
2266:) = (
2240:) = (
2011:with
1820:Order
1666:) + (
1658:) = (
1571:) = (
1463:group
1449:with
1442:is a
563:Latin
536:Frege
424:and N
388:India
365:Greek
344:nulla
312:Olmec
308:digit
107:third
8524:Evil
8204:Self
8154:and
8044:Blum
7755:and
7559:Lobb
7514:Cake
7509:Bell
7259:Star
7166:Ulam
7065:Pell
6854:Cube
6761:List
6618:and
6003:ISBN
5983:2013
5945:ISBN
5924:ISBN
5903:ISBN
5878:ISBN
5857:ISBN
5833:ISBN
5805:ISBN
5783:ISBN
5759:ISBN
5735:ISBN
5714:ISBN
5690:ISBN
5664:ISBN
5648:2020
5620:ISBN
5596:ISBN
5575:ISBN
5554:ISBN
5503:2014
5375:ISBN
5354:2020
5322:ISBN
5297:ISBN
5272:ISBN
5251:2020
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