The natural numbers start off as follows: 1, 2, 3, 4, 5, ... . The "..." means that the list goes on forever. We give this set the name N. Giuseppe Peano gave five properties of this set. If a number is in N, then its successor is also in N. Thus, there is no greatest number, because we can always add one to get a larger one. N is an infinite set. Since it is infinite, N can never be exhausted by removing its members one at a time.
The set of natural numbers is closed with respect to addition and multiplication, which means that if you add (or multiply) two natural numbers together, you get another natural number. This isn't true with respect to subtraction or division, however. You can subtract one natural number from another and not get a natural number, or you can divide one natural number by another and not get a natural number. For example, 3 − 5 = −2, and 5 ÷ 2 = 2.5. −2 and 2.5 are not natural numbers. The set of rational numbers is, however, closed with respect to addition, multiplication, subtraction and division. Such a set is called a field.
See also: Number Systems.