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                  
        LXXII.         Amy's Interval Theorem Link                               LXXIII.         Fraction Link    

        LXXIV.         Exponents and Radicals Link                             LXXV.    Logarithm Theorems Link

        LXXVI.         Basic Summation Link                                       LXXVII.     Power Summation Link

        LXXVIII.       General Summation Link                                     LXXIX.     Trigonometry Link    

        LXXX.         Hyperbolic Functions Link                                   LXXXI.     Intervals Link    



Properties:     List #1 Link     List #2 Link     List #3 Link     List #4 Link   and   Demonstrations

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