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# Natural number

 numbers examples  1   2   3   4   5   6   7   8   9   10   11 12 13 14 15 16 17 18 19   20   21 22 23 24 25 26 27 28 29   30   31 32 33 34 35 36 37 38 39   40   41 42 43 44 45 46 47 48 49   50   51 52 53 54 55 56 57 58 59   60   61 62 63 64 65 66 67 68 69   70   71 72 73 74 75 76 77 78 79   80   81 82 83 84 85 86 87 88 89   90   91 92 93 94 95 96 97 98 99   100   200   300   400   500   600   700   800   900   1000   2000   3000   4000   5000   6000   7000   8000   9000   10,000   100,000   1,000,000   1,000,000,000   1,000,000,000,000   Numbers less than or equal to 0 (such as −1) are not natural numbers (rather Integers).

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 started to count. Natural numbers are always whole numbers (integers excluding 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 $\mathbb{N}$.

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 is 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:

## 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": 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 $l \times m = n$ then $n / m = l$
• Mathematical induction: If these two things are true of any property P of natural numbers, then P is true of every natural number
• 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 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

$\mathbf{N}$ or $\mathbb{N}$ 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
$\mathbb{N}^+$ Positive numbers, without zero
$\mathbb{N}^*$ Positive numbers without zero
$\mathbb{N}_0$ Positive numbers, with zero
$\mathbb{N}_{\gt 0}$ Positive numbers without zero
$\mathbb{N} \setminus \{0\}$ Positive numbers without zero