 MR. TOWNSLEY'S
MATH DEPARTMENT

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.

E-mail Mr. Townsley at ktownsley@central-clinton.k12.ia.us

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