FOR THE STUDENT OF MATHEMATICS
PROPERTIES LIST 3.

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, t is any real number and u is any real number.

Definition 5.   p4 = pppp

NUMBER OPERATIONS:

PROPERTY
IL.     ( p + q )4 = p4 + 4p3q + 6p2q2 + 4pq3+ q4

L.     ( p + q ) ( r + t + u ) = ( p + q )r + ( p + q )t + ( p + q )u = pr + qr + pt + qt + pu + qu

LI.     If pq = 0, then p = 0 or q = 0

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, p ≠ q .

Definition 6. 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. | p - q | means the distance between p and q.

Definition 7. In geometry distance between p and q corresponds to the absolute value   | p - q |   in algebra.

PROPERTY
LII.     | p | is a positive number or zero

LIII.     | p q | = | p | | q |

LIV.     | p/q | = | p |/| q |

LV.     || p || = | p |