THE FIELD PROPERTIES
The symbol R will be used to represent the set of real numbers (to review sets, go to SETS)
CLOSURE: An operation is closed on a given set if the result obtained by performing the operation yields a number from the given set.
COMMUTATIVE: For all a and b in R,
a + b = b + a and ab = ba
ASSOCIATIVE: For all a and b in R,
(a + b) + c = a + (b + c) and (ab)c = a(bc)
IDENTITY: A. There exists a unique real number (0) such that for every n in R,
n + 0 = 0 + n = n (0 is called the additive identity)
B. There exists a unique real number (1) such that for every n in R,
n*1 = 1*n = n (1 is called the multiplicative identity).
INVERSE: A. For every n in R there exists a unique real number (denoted by -n) such that
n + (-n) = (-n) + n = 0 (-n is called the additive inverse of n)
B. For every n in R (except 0) there exists a unique real number (1/n) such that
n(1/n) = (1/n)n = 1 (1/n is called the multiplicative inverse, or reciprocal of n)
DISTRIBUTIVE: For all a, b and c in R, a(b + c) = ab + ac.