FOR THE STUDENT OF MATHEMATICS
PROPERTIES LIST 1.

Assume that p is any real number, q is any real number, r is any real number and t is any real number.

NUMBER OPERATIONS:

CLOSURE:                                   I.     p+q is a real number                   II.     pq is a real number.

COMMUTATIVE:                           III.     p+q = q+p                                 IV.     pq = qp

ASSOCIATIVE:                             V.     p+(q+r) = (p+q)+r                       VI.     p(qr) = (pq)r

IDENTITY:                                   VII.     p+0 = p = 0+p                           VIII.     p1 = p = 1p

INVERSE (0 is not equal to 1):       IX.     p+(-p) = 0                                 X.     p(1/p) = 1     for p not = 0

DISTRIBUTIVE:                                               XI.               p(q+r) = pq+pr

Definition 1. Subtraction: p - q = p + (-q).     Definition 2. Division for q not = 0, p/q = p (1/q)

NUMBER RELATIONS:

PROPERTY     EQUALITY p = q               INEQUALITY p is less than q means p + t = q for positive t.

REFLEXIVE:    XII.  p = p                                   XIII.    p is not less than p

SYMMETRIC:   XIV.   If p = q then q = p             XV.     If p is less than q, then q is not less than p.

TRANSITIVE:  XVI. If p = q and q = r then p = r   XVII. If p is less than q and q is less than r, then p is less than r.

SUBSTITUTION:  XVIII.  Any number, letter or algebra combination of numbers or letters may be
substituted for p, q, or r in the properties listed above unless stated otherwise.
Substitution: If a = b, then b may be put in place of a in any statement.

2 QUANTITIES = THE SAME:                             XIX.     If p = q and r = q, then p = r.

EQUALITY:                   XX.     If p = q, then p + r = q + r.             XXI.     If p = q, then p r = q r.

EQUALITY:                 XXII.     If p = q, then p - r = q - r.               XXIII.     If p = q, then p/r = q/r.   r not equal to 0.

EQUALITY:     XXIV.   If p = q and r = t, then p + r = q + t.           XXV.   If p = q and r = t, then p/r = q/t.   r , t not 0.

INVERSES FOR p:     XXVI.     p's additive inverse is unique.       XXVII.     p's multiplicative inverse is unique.