The Natural Numbers

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 a natural number. This isn't true with respect to subtraction or division, however. 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.