 In the late 1800s, the mathematician Giuseppe Piano produced a set of axioms or postulates that can be used to develop number properties. In simple terms, starting with the counting numbers they are as follows. One is a number. Every number n has one and only one successor number, n plus one. And no two different numbers have the same successor number. In other words, if n plus one equals m plus one, then n has to equal m. From these postulates and our basic operator definitions, a number of properties exist that we can use to manipulate numbers and solve equations. Here are the properties for the natural or counting numbers. Although we tend to take them for granted, mathematicians have to prove each and every one. You'll find a proof that a plus b equals b plus a for natural numbers in the appendix. Natural numbers are closed for addition and multiplication. By closed we mean that these operations on numbers in the set produce numbers that are also in the set. When we include zero and negative numbers, we get the integer number line. The set of integers adds closure for subtraction.