FOR THE STUDENT OF MATHEMATICS
PROPERTIES LIST 2.

Starting with Property XIX on Properties List 1, challenge yourself to show how each property may result from earlier properties.

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

Definition 3. Square   p2 = pp       Definition 4. Cube   p3 = ppp

NUMBER OPERATIONS:

XXVIII.     -(-p) = p                                   XXIX.     1/(1/p) = p

XXX.     -(-(-p)) = -p                             XXXI.     0p = 0   Zero Product Property

XXXII.     0-p = -p                                   XXXIII.     (-1)p = -p

XXXIV.     p - (-q) = p + q                         XXXV.     p/(1/q) = p q

XXXVI.     p - q = -q + p                         XXXVII.     p/q) = (1/q)p

DISTRIBUTIVE:                                                   XXXVIII.     p( q - r ) = pq - pr

DISTRIBUTIVE:                                                   XXXIX.     p( q + r + t ) = pq + pr + pt

XXXX.     (-p)q = -(pq) = p(-q)

XXXXI.     (-p)(-q) = pq

XXXXII.     -( p + q) = -p - q

XXXXIII.     -( p - q ) = -p + q = q - p

XXXXIV.     ( p + q ) ( r + t )= pr + qr + pt + qt

XXXXV.     ( p + q )2 =( p + q ) ( p + q )= p2 + 2pq + q2

XXXXVI.     ( p - q )2 =( p - q ) ( p - q )= p2 - 2pq + q2

XXXXVII.     ( p + q ) ( p - q )= p2 - q2

XXXXVIII.     ( p + q )3 = p3 + 3p2q + 3pq2+ q3