Natural number

2017-01-17T01:12:08Z

73.67.244.70: The number between 68 and 70 is not, in fact, 68

{| style="float:right;width:256px;border:2px solid black;margin:0 0 0 10px"
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|<center>'''Natural numbers example'''
: ([[0]]) [[one|1]]   [[two|2]]   [[three|3]]   [[four|4]]   [[five|5]]   [[six|6]]   [[seven|7]]   [[eight|8]]   [[nine|9]]   [[ten|10]]  
:[[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]]   [[twenty|20]]  
:[[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]]   [[thirty|30]]  
:[[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]]   [[forty|40]]  
:[[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]]   [[fifty|50]]  
:[[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]]   [[sixty|60]]  
:[[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]]   [[seventy|70]]  
:[[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]]   [[eighty|80]]  
:[[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]]   [[ninety|90]]  
:[[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]]  

:[[100 (number)|100]]   [[200 (number)|200]]   [[300 (number)|300]]   [[400 (number)|400]]   [[500 (number)|500]]  
:[[600 (number)|600]]   [[700 (number)|700]]   [[800 (number)|800]]   [[900 (number)|900]]  
:[[thousand|1000]]   [[2000 (number)|2000]]   [[3000 (number)|3000]]   [[4000 (number)|4000]]   [[5000 (number)|5000]]  
:[[6000 (number)|6000]]   [[7000 (number)|7000]]   [[8000 (number)|8000]]   [[9000 (number)|9000]]  
:[[10000 (number)|10,000]]   [[100000 (number)|100,000]]   [[million|1,000,000]]  
:[[billion|1,000,000,000]]   [[trillion|1,000,000,000,000]]  

Numbers less than [[0]] (such as [[−1]]) are not natural numbers.
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'''Natural numbers''', also called '''counting numbers''', are the [[number]]s used for counting things. Sometimes the special number [[zero]] is called a natural number. Sometimes [[one]] is called the smallest natural number. Natural numbers are always whole numbers ([[integers]]) and never less than zero.

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 "for ever". There is no [[infinity|infinite]] natural number. Any natural number can be reached by adding 1 enough times to the smallest natural number.

==Non-natural numbers==
The following types of number are not natural numbers:
* Numbers less than 0 ([[negative number]]s), for example, −2 −1
* [[Fraction (mathematics)|Fractions]], for example, ½ 3¼
* [[Decimal]]s, for example, 7.675
* [[Irrational number]]s, for example, <math>{\sqrt{2}}</math>, <math>\pi</math> ([[Pi (mathematical constant)|pi]])
* [[Imaginary number]]s, for example, <math>{\sqrt{-1}}</math> ('''i''')
* [[infinity]], for example, <math>\infty</math> <math> \aleph_0</math>

==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 and 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
# ''P'' is true of 0
# if ''P'' of ''n'' then ''P'' of ''n''+1

==Special natural numbers==

*[[Even number]]s: 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]]s: 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]]s: 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]]s: 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. 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:

{| border="0" cellpadding="2" cellspacing="1" style="float:left; margin-right:1em; background:#e3e3e3;"
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!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
|}
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==Related pages==
* [[Integer]]

[[Category:Number theory]]

{{Math-stub}}

TCS Wiki - User contributions [en]

Natural number