FOR THE STUDENT OF MATHEMATICS
PROPERTIES LIST 5.

Starting with Property XIX on Properties List 1, show how each property results 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.

PROPERTY
LXIV.         | p - q |   =   | q - p |

LXV.         | p + q |   ≤   | p | + |q |       Triangle Inequality

LXVI.         | p | - | q |   ≤   | p - q |

LXVII.         | p - q |   ≤   | p - r | + | q - r |

Definition 12. Square Root of p ≥ 0,     √ ( p )   =   t     ⇔   there is a t ≥ 0 such that   t2 = p.

LXVIII.         √ ( p2 )   =   | p |     for any real number p

LXIX.         If   p = q,   then   p2   =   q2.

LXX.         If   0   <   p   <   q,   then   p2   ≤   q2.

LXXI.         If   | p |   =   | q |,   then   p   =   q   or   p   =   -q .

RATIONAL OPERATIONS [ Note: No denominator can be zero. 0/p = 0 and p/0 is undefined. ]

Definition 13. Addition ( Subtraction )   p/q + r/t   =   ( pt + rq )/( qt )

Definition 14. Multiplication   ( p/q )( r/t )   =   ( pr )/( qt )

Definition 15. Division   ( p/q )  ÷   ( r/t )   =   ( p/q )/( r/t )   =   ( pt )/( qr )

PROPERTIES