FOR THE STUDENT OF MATHEMATICS
PROPERTIES 1.


Know and apply these algebra properties and new ones to everything.

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

NUMBER OPERATIONS:

PROPERTY                                   ADDITION                                           MULTIPLICATION

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

NUMBER RELATIONS:

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

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.

NUMBERS AND GEOMETRY

The numbers p and q may locate points on one line so: p and q locate the same point when p = q. p and q locate different points when not equal to each other. Definition 1. Subtraction: p - q = p + (-q).
Definition 2. Absolute Value is written | p - q | and is the positive one of ( p - q ) or ( q - p ). If p and q locate points on a horizontal line then the absolute value of ( p - q ) gives the distance between p and q.
Also, if p and q locate points on a horizontal line and p is less than q, then we generally consider p on the left of q. In fact, p is less than 0 is another way to say p is negative.
Geometry: Definition 3. distance between p and q corresponds to this absolute value, | p - q |, in algebra.

Copyright © 2009 with all rights reserved by William V. Thayer