Seoh 1 frumtraht

{| style="float:right;width:256px;border:2px solid black;margin:0 0 0 10px"
|<center>''' numbers examples'''
: &nbsp;[[One|1]] &nbsp; [[Two|2]] &nbsp; [[Three|3]] &nbsp; [[Four|4]] &nbsp; [[Five|5]] &nbsp; [[Six|6]] &nbsp; [[Seven|7]] &nbsp; [[Eight|8]] &nbsp; [[Nine|9]] &nbsp; [[Ten|10]] &nbsp;
: [[11 (number)|11]] [[12 (number)|12]] [[13 (number)|13]] [[14 (number)|14]] [[15 (number)|15]] [[16 (number)|16]] [[17 (number)|17]] [[18 (number)|18]] [[19 (number)|19]] &nbsp; [[Twenty|20]] &nbsp;
: [[21 (number)|21]] [[22 (number)|22]] [[23 (number)|23]] [[24 (number)|24]] [[25 (number)|25]] [[26 (number)|26]] [[27 (number)|27]] [[28 (number)|28]] [[29 (number)|29]] &nbsp; [[Thirty|30]] &nbsp;
: [[31 (number)|31]] [[32 (number)|32]] [[33 (number)|33]] [[34 (number)|34]] [[35 (number)|35]] [[36 (number)|36]] [[37 (number)|37]] [[38 (number)|38]] [[39 (number)|39]] &nbsp; [[Forty|40]] &nbsp;
: [[41 (number)|41]] [[42 (number)|42]] [[43 (number)|43]] [[44 (number)|44]] [[45 (number)|45]] [[46 (number)|46]] [[47 (number)|47]] [[48 (number)|48]] [[49 (number)|49]] &nbsp; [[Fifty|50]] &nbsp;
: [[51 (number)|51]] [[52 (number)|52]] [[53 (number)|53]] [[54 (number)|54]] [[56 (number)|55]] [[56 (number)|56]] [[57 (number)|57]] [[58 (number)|58]] [[59 (number)|59]] &nbsp; [[Sixty|60]] &nbsp;
: [[61 (number)|61]] [[62 (number)|62]] [[63 (number)|63]] [[64 (number)|64]] [[65 (number)|65]] [[66 (number)|66]] [[67 (number)|67]] [[68 (number)|68]] [[69 (number)|69]] &nbsp; [[Seventy|70]] &nbsp;
: [[71 (number)|71]] [[72 (number)|72]] [[73 (number)|73]] [[74 (number)|74]] [[75 (number)|75]] [[76 (number)|76]] [[77 (number)|77]] [[78 (number)|78]] [[79 (number)|79]] &nbsp; [[Eighty|80]] &nbsp;
: [[81 (number)|81]] [[82 (number)|82]] [[83 (number)|83]] [[84 (number)|84]] [[85 (number)|85]] [[86 (number)|86]] [[87 (number)|87]] [[88 (number)|88]] [[89 (number)|89]] &nbsp; [[Ninety|90]] &nbsp;
: [[91 (number)|91]] [[92 (number)|92]] [[93 (number)|93]] [[94 (number)|94]] [[95 (number)|95]] [[96 (number)|96]] [[97 (number)|97]] [[98 (number)|98]] [[99 (number)|99]] &nbsp;

: [[100 (number)|100]] &nbsp; [[200 (number)|200]] &nbsp; [[300 (number)|300]] &nbsp; [[400 (number)|400]] &nbsp; [[500 (number)|500]] &nbsp;
: [[600 (number)|600]] &nbsp; [[700 (number)|700]] &nbsp; [[800 (number)|800]] &nbsp; [[900 (number)|900]] &nbsp;
: [[Thousand|1000]] &nbsp; [[2000 (number)|2000]] &nbsp; [[3000 (number)|3000]] &nbsp; [[4000 (number)|4000]] &nbsp; [[5000 (number)|5000]] &nbsp;
: [[6000 (number)|6000]] &nbsp; [[7000 (number)|7000]] &nbsp; [[8000 (number)|8000]] &nbsp; [[9000 (number)|9000]] &nbsp;
: [[10000 (number)|10,000]] &nbsp; [[100000 (number)|100,000]] &nbsp; [[Million|1,000,000]] &nbsp;
: [[Billion|1,000,000,000]] &nbsp; [[Trillion|1,000,000,000,000]] &nbsp;

Numbers less than or equal to [[0]] (such as [[−1]]) are not natural numbers (rather [[Integers]]).
</center>
|}

'''Natural numbers''', also called '''counting numbers''', are the [[Number|numbers]] used for counting things. Natural numbers are the numbers small children learn about when they first started to count. Natural numbers are always [[whole numbers]] ([[integers]] excluding [[Negative number|negative numbers]]) and often exclude zero, in which case [[one]] is the smallest natural number. The set of natural numbers can be represented by the symbol <math>\mathbb{N}</math>.<ref name=":0">{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/|access-date=2020-08-11|website=Math Vault|language=en-US}}</ref><ref name=":1">{{Cite web|last=Weisstein|first=Eric W.|title=Natural Number|url=https://mathworld.wolfram.com/NaturalNumber.html|access-date=2020-08-11|website=mathworld.wolfram.com|language=en}}</ref>

There is no largest natural number. The next natural number can be found by adding 1 to the current natural number, producing numbers that go on "forever". There is no natural number that is [[Infinity|infinite]] in size. Any natural number can be reached by adding 1 enough times to the smallest natural number.

== Non-natural numbers ==
The following types of numbers are not natural numbers:

* Numbers less than 0 ([[Negative number|negative numbers]]), for example, −2 and −1
* [[Fraction (mathematics)|Fractions]], for example, ½ and 3¼
* [[Decimal|Decimals]], for example, 7.675
* [[Irrational number|Irrational numbers]], for example, <math>{\sqrt{2}}</math> and  <math>\pi</math> ([[Pi (mathematical constant)|pi]])
* [[Imaginary number|Imaginary numbers]], for example, <math>{\sqrt{-1}}</math> ('''i''')
* [[infinity]], for example, <math>\infty</math> and <math>  \aleph_0</math>
* 0 (if not include as a natural number)

== Basic operations ==

* ''Addition''; The sum of two natural numbers is a natural number. <math> l + m = n</math>
** <math>l +n = n + l</math>
** <math>n + 0 = n</math>
** <math>(l + m) + n = l + (m + n)</math>
* ''Multiplication'': The product of two natural numbers is a natural number. <math> l \times m = n</math>
** <math>l \times m = m \times n</math>
** <math>n \times 0 = 0</math>
** <math>n \times 1 = n</math>
** <math> (l \times m) \times n = l \times (m \times n)</math>
** <math> l \times (m + n) = (l \times m) + (l \times n)</math>

* ''Ordering'': Of two natural numbers, if they are not the same, then one is bigger than the other, and the other is smaller.  m = n or m > n or m < n
** if l > m then l + n > m + n
** if l > m and l > 0 then l x n > l x m
** Zero is the smallest natural number: 0 = n or 0 < n
** There is no largest natural number n < n + 1

* "Subtraction": If ''n'' is smaller than ''m'' then ''m'' minus ''n'' is a natural number. If n < m then m - n = p. 
** if l - m = n then l = n + m
** if ''n'' is greater than ''m'', then ''m'' minus ''n'' is not a natural number
** if l = m - n and p < n then l > m - p

* ''Division'': If <math>l \times m = n</math> then <math>n / m = l</math>

* [[Mathematical induction]]: If these two things are true of any property ''P'' of natural numbers, then ''P'' is true of every natural number
** if ''P'' is true of 1
** and if ''P'' of ''n'' then ''P'' of ''n''+1
** then ''P'' is true of all natural numbers

== Special natural numbers ==

* [[Even number|Even numbers]]: If ''n'' = ''m'' x 2, then ''n'' is an even number
** The even numbers are 0, 2, 4, 6, and so on. Zero is the smallest (or first) even number.
* [[Odd number|Odd numbers]]: If ''n'' = ''m'' x 2 +1, then ''n'' is an odd number
** A number is either even or odd but not both.
** The odd numbers are 1, 3, 5, 7, and so on.
* [[Composite numbers]]: If ''n'' = ''m'' x ''l'', and ''m'' and ''l'' are not 0 or 1, then ''n'' is a composite number.
** The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21 and so on.
* [[Prime number|Prime numbers]]: If a number is not 0, 1, and not a composite number, then it is a prime number.
** The prime numbers are 2, 3, 5, 7, 11, 13, 17 and so on. Two is the smallest (or first) prime number. Two is the only even prime number.
** There is no biggest prime number.
* [[Square number|Square numbers]]: If ''n'' = ''m'' x ''m'', then ''n'' is a square. ''n'' is the square of ''m''.
** The squares are 0, 1, 4, 9, 16, 25, 36, 49 and so on.

== How to write it ==
<math>\mathbf{N}</math> or <math>\mathbb{N}</math> is the way to write the [[set]] of all natural numbers.<ref name=":0" /> Because some people say 0 is a natural number, and some people say it is not, people use the following symbols to talk about the natural numbers:<ref name=":1" />

{| border="0" cellpadding="2" cellspacing="1" style="float:left; margin-right:1em; background:#e3e3e3;"
!Symbol
!Meaning
|- style="background:#fff; text-align:center;"
|<math>\mathbb{N}^+</math>
|Positive numbers, without zero
|- style="background:#fff; text-align:center;"
|<math>\mathbb{N}^*</math>
|Positive numbers without zero
|- style="background:#fff; text-align:center;"
|<math>\mathbb{N}_0</math>
|Positive numbers, with zero
|- style="background:#fff; text-align:center;"
|<math>\mathbb{N}_{>0}</math>
|Positive numbers without zero
|- style="background:#fff; text-align:center;"
|<math>\mathbb{N} \setminus \{0\}</math>
|Positive numbers without zero
|}
{{-}}

== Related pages ==

* [[1 (number)]]
* [[Integer]]
* [[Zero]]

[[Flocc:Number theory]]

== References ==
<references />
{{Math-stub}}