A natural number is a positive integer, a positive whole number. The natural numbers are also called counting numbers. The symbol for the set of natural numbers is N. The definition of natural numbers in set notation is N = {1, 2, 3, ...}.
Natural numbers were the first numbers used by people. They were used to count whole objects such as cattle and sheep. The concept of zero as a number was not introduced until much later, and negative numbers much later than that.
The set of natural numbers is used to define the size of infinite sets. The size, or cardinality of the natural numbers is called countable and is denoted aleph zero (ℵ_{0}). Any infinite set with a one to one correspondence with the natural numbers is said to also be countable.
Property | Description |
---|---|
Associativity | The set of natural numbers is associative with respect to addition, subtraction, multiplication and division. |
Commutativity | The set of natural numbers is commutative with respect to addition and multiplication. The set of natural numbers is not commutative with respect to subtraction or division. |
Additive identity | The additive identity for natural numbers is 0. |
Multiplicative identity | The multiplicative identity for natural numbers is 1. |
Closure | The set of natural numbers is closed with respect to addition and multiplication. The set of natural numbers is not closed with respect to subtraction and division. |
Discrete | The set of natural numbers is a discrete (not continuous) set. |
Cardinality | The cardinality of the set of natural numbers is defined to be ℵ_{0}. |
Table 1: Properties of the natural numbers. |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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