## ABSTRACT

The algebraic structure of Q consists of addition and multiplication together with those familiar properties pertaining to addition and multiplication in Q. The algebraic structure of Q is called a field. More generally, a field is any set having the properties listed in Theorem 20.2, so not only does Q form a field, but the system E1 forms a field as well. The system of real numbers is discussed in the next section. If in addition to the algebraic properties of a field there is the presence of an order relation which has the properties listed in Theorem 20.5, the resulting structure is called an ordered field. Therefore, the system Q of rational numbers is an ordered field. Here is the working definition of an "ordered" field.