**[below]** *field properties* because of the mathematical convention calling a *field* any set, together with two functions, + and **·**, satisfying these properties. They express the fact that the real numbers are a field.

PROPERTY I. (COMMUTATIVITY).

For every *a,b* in ** R**, we have

and

PROPERTY II. (ASSOCIATIVITY).

For every *a,b,c* in ** R**, we have

(

and

(

PROPERTY III. (DISTRIBUTIVITY).

For every *a,b,c* in ** R**, we have

PROPERTY IV. (EXISTENCE OF NEUTRAL ELEMENTS).

There are distinct elements 0 and 1 of ** R** such that for all

and

PROPERTY V. (EXISTENCE OF ADDITIVE AND MULTIPLICATIVE INVERSES).

For any *a* in ** R** there is an element of

and for any nonzero

such that

See ita-2-1-3-pg16.