Natural numbers, also called counting numbers, are the numbers used for counting things. Natural numbers are the numbers small children learn about when they first start to count. Natural numbers are always whole numbers (integers) and exclude zero, so one is the smallest natural number. The set of natural numbers can be represented by the symbol ${\displaystyle \mathbb {N} }$.^[1][2]

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 infinite in size. Any natural number can be reached by adding 1 enough times to the smallest natural number.

The following types of numbers are not natural numbers:

• 0
• Numbers less than 0 (negative numbers), for example, −2 and −1
• Fractions, for example, ${\displaystyle {\frac {1}{2))}$ and ${\displaystyle {\frac {31}{4))}$
• Fractional numbers, for example, 7.675
• Irrational numbers, for example, ${\displaystyle {\sqrt {2))}$ and ${\displaystyle \pi }$ (pi)
• Imaginary numbers, for example, ${\displaystyle {\sqrt {-1))}$ (i)
• Complex numbers, for example, ${\displaystyle 1+2i}$
• infinities, for example, ${\displaystyle \infty }$ and ${\displaystyle \aleph _{0))$

• Addition: The sum of two natural numbers is a natural number. ${\displaystyle l+m=n}$
• Subtraction: The difference of two natural numbers is a whole number
  • if ${\displaystyle m}$ is greater than ${\displaystyle n}$, then ${\displaystyle m}$ minus ${\displaystyle n}$ is a natural number
• Multiplication: The product of two natural numbers is a natural number. ${\displaystyle l\times m=n}$
• Division: The quotient of two natural numbers is a rational number

• Ordering: Of two natural numbers, if they are not the same, then one is bigger than the other, and the other is smaller. ${\displaystyle m=n}$, ${\displaystyle m>n}$, or ${\displaystyle m<n}$
  • if ${\displaystyle l>m}$ then ${\displaystyle l+n>m+n}$
  • if ${\displaystyle l>m}$ and ${\displaystyle l>0}$ then ${\displaystyle lxn>lxm}$, which is the same as ${\displaystyle n>m}$.
  • One is the smallest natural number: ${\displaystyle 1=n}$ or ${\displaystyle 1<n}$
  • There is no largest natural number ${\displaystyle n<n+1}$

• Mathematical induction: If these two things are true of any property ${\displaystyle P}$ of natural numbers, then ${\displaystyle P}$ is true of every natural number
  • if ${\displaystyle P}$ is true of 1
  • and if ${\displaystyle P}$ of ${\displaystyle n}$ then ${\displaystyle P}$ of ${\displaystyle n+1}$
  • then ${\displaystyle P}$ is true of all natural numbers

• Even numbers: If ${\displaystyle n=m\times 2}$, then ${\displaystyle n}$ is an even number
  • The first even natural numbers are 2, 4, 6, 8, and so on.
• Odd numbers: If ${\displaystyle n=m\times 2+1}$, then ${\displaystyle n}$ is an odd number
  • A number is either even or odd but not both.
  • The first odd natural numbers are 1, 3, 5, 7, and so on.
• Composite numbers: If ${\displaystyle n=m\times l}$, and ${\displaystyle m}$ and ${\displaystyle l}$ are not 0 or 1, then ${\displaystyle n}$ is a composite number.
  • The first composite (non-prime) natural numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 21 and so on.
• Prime numbers: If a number is not 1, and not a composite number, then it is a prime number.
  • The first prime (non-composite) natural numbers are 2, 3, 5, 7, 11, 13, 17 and so on. Two is the only even prime number.
  • There is an infinite number of prime numbers.
• Square numbers: If ${\displaystyle n=m\times m}$, then ${\displaystyle n}$ is a square. ${\displaystyle n}$ is the square of ${\displaystyle m}$.
  • The first natural squares are 1, 4, 9, 16, 25, 36, 49 and so on.

${\displaystyle \mathbb {N} }$ is the way to write the set of all natural numbers.^[1] While 0 is not a natural number, it is possible to create a set that includes both the set of natural numbers and the number zero. This set is written ${\displaystyle \mathbb {N} \cup \{0\))$.^[2]

• Integer

1. ↑ ^{1.0 1.1} "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-11.
2. ↑ ^{2.0 2.1} Weisstein, Eric W. "Natural Number". mathworld.wolfram.com. Retrieved 2020-08-11.

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