The Field Properties

( of the real number system )

The five properties listed [below] are called the 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
a + b = b + a
and
a b = b a

PROPERTY II. (ASSOCIATIVITY).
For every a,b,c in R, we have
( a + b ) + c = a + ( b + c )
and
( a b ) c = a ( b c )

PROPERTY III. (DISTRIBUTIVITY).
For every a,b,c in R, we have
a ( b + c ) = a b + a c

PROPERTY IV. (EXISTENCE OF NEUTRAL ELEMENTS).
There are distinct elements 0 and 1 of R such that for all a in R we have
a + 0 = a
and
a 1 = a

PROPERTY V. (EXISTENCE OF ADDITIVE AND MULTIPLICATIVE INVERSES).
For any a in R there is an element of R, denoted -a, such that
a + ( -a ) = 0,

and for any nonzero a in R there is an element of R, denoted

such that

See ita-2-1-3-pg16.