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numbers examples 12345678910 11121314151617181920 21222324252627282930 31323334353637383940 41424344454647484950 51525354555657585960 61626364656667686970 71727374757677787980 81828384858687888990 919293949596979899 100200300400500 600700800900 10002000300040005000 6000700080009000 10,000100,0001,000,000 1,000,000,0001,000,000,000,000 Numbers less than or equal to0(such as−1) are not natural numbers (ratherIntegers).

Natural numbers,also calledcounting numbers,are thenumbersused for counting things. Natural numbers are the numbers small children learn about when they first started to count. Natural numbers are alwayswhole numbers(integersexcludingnegative numbers) and often exclude zero, in which caseoneis the smallest natural number. The set of natural numbers can be represented by the symbol$\mathbb {N}$.

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 isinfinitein 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 numbers), for example, −2 and −1
• Fractions,for example, ½ and 3¼
• Decimals,for example, 7.675
• Irrational numbers,for example,${\sqrt {2}}$and$\pi$(pi)
• Imaginary numbers,for example,${\sqrt {-1}}$(i)
• infinity,for example,$\infty$and$\aleph _{0}$
• 0 (if not include as a natural number)

Basic operations

• Addition;The sum of two natural numbers is a natural number.$l+m=n$
  • $l+n=n+l$
  • $n+0=n$
  • $(l+m)+n=l+(m+n)$
• Multiplication:The product of two natural numbers is a natural number.$l\times m=n$
  • $l\times m=m\times n$
  • $n\times 0=0$
  • $n\times 1=n$
  • $(l\times m)\times n=l\times (m\times n)$
  • $l\times (m+n)=(l\times m)+(l\times n)$

• 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": Ifnis smaller thanmthenmminusnis a natural number. If n < m then m - n = p.
  • if l - m = n then l = n + m
  • ifnis greater thanm,thenmminusnis not a natural number
  • if l = m - n and p < n then l > m - p

• Division:If$l\times m=n$then$n/m=l$

• Mathematical induction:If these two things are true of any propertyPof natural numbers, thenPis true of every natural number
  • ifPis true of 1
  • and ifPofnthenPofn+1
  • thenPis true of all natural numbers

Special natural numbers

• Even numbers:Ifn=mx 2, thennis an even number
  • The even numbers are 0, 2, 4, 6, and so on. Zero is the smallest (or first) even number.
• Odd numbers:Ifn=mx 2 +1, thennis 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:Ifn=mxl,andmandlare not 0 or 1, thennis a composite number.
  • The composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21 and so on.
• 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 numbers:Ifn=mxm,thennis a square.nis the square ofm.
  • The squares are 0, 1, 4, 9, 16, 25, 36, 49 and so on.

How to write it

$\mathbf {N}$or$\mathbb {N}$is the way to write thesetof 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:

Symbol	Meaning
$\mathbb {N} ^{+}$	Positive numbers, without zero
$\mathbb {N} ^{*}$	Positive numbers without zero
$\mathbb {N} _{0}$	Positive numbers, with zero
$\mathbb {N} _{>0}$	Positive numbers without zero
$\mathbb {N} \setminus \{0\}$	Positive numbers without zero

Related pages

• 1 (number)
• Integer
• Zero