Natural number

Natural numbers, also called counting numbers, are the numbers 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 infinite natural number. Any natural number can be reached by adding 1 enough times to the smallest natural number.
Contents
Nonnatural numbers
The following types of number are not natural numbers:
 Numbers less than 0 (negative numbers), for example, −2 −1
 Fractions, for example, ½ 3¼
 Decimals, for example, 7.675
 Irrational numbers, for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle {\sqrt{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \pi} (pi)
 Imaginary numbers, for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle {\sqrt{1}}} (i)
 infinity, for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \infty} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \aleph_0}
Basic operations
 Addition; The sum of two natural numbers is a natural number. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle l + m = n}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle l +n = n + l}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle n + 0 = n}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle (l + m) + n = l + (m + n)}
 Multiplication": The product of two natural numbers is a natural number. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle l \times m = n}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle l \times m = m \times n}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle n \times 0 = 0}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle n \times 1 = n}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle (l \times m) \times n = l \times (m \times n)}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle 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 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle l \times m = n} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle n / m = l}
 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 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 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 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: 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
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathbf{N}} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathbb{N}} 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:
Symbol  Meaning 

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathbb{N}^+}  Positive numbers, without zero 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathbb{N}^*}  Positive numbers without zero 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathbb{N}_0}  Positive numbers, with zero 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathbb{N}_{>0}}  Positive numbers without zero 
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \mathbb{N} \setminus \{0\}}  Positive numbers without zero 