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.