Page 12 - Schaum's Outline of Theory and Problems of Advanced Calculus
P. 12

CHAP. 1]                             NUMBERS                                      3


                        8.  For any a there is a number x in R such that x þ a ¼ 0.
                               x is called the inverse of a with respect to addition and is denoted by  a.
                        9.  For any a 6¼ 0 there is a number x in R such that ax ¼ 1.
                                                                                             1
                               x is called the inverse of a with respect to multiplication and is denoted by a  or 1=a.
                        Convention: For convenience, operations called subtraction and division are defined by
                                             1
                                       a
                     a   b ¼ a þð bÞ and ¼ ab , respectively.
                                       b
                        These enable us to operate according to the usual rules of algebra. In general any set, such as R,
                     whose members satisfy the above is called a field.

                     INEQUALITIES
                        If a   b is a nonnegative number, we say that a is greater than or equal to b or b is less than or equal to
                     a, and write, respectively, a A b or b % a.If there is no possibility that a ¼ b,we write a > b or b < a.
                     Geometrically, a > b if the point on the real axis corresponding to a lies to the right of the point
                     corresponding to b.


                     EXAMPLES.  3 < 5or5 > 3;  2 <  1or  1 >  2; x @ 3means that x is a real number which may be 3 or less
                     than 3.
                        If a, b; and c are any given real numbers, then:

                        1. Either a > b, a ¼ b or a < b    Law of trichotomy
                        2. If a > b and b > c, then a > c  Law of transitivity
                        3. If a > b, then a þ c > b þ c
                        4. If a > b and c > 0, then ac > bc
                        5. If a > b and c < 0, then ac < bc




                     ABSOLUTE VALUE OF REAL NUMBERS
                        The absolute value of a real number a, denoted by jaj,isdefined as a if a > 0,  a if a < 0, and 0 if
                     a ¼ 0.

                     EXAMPLES.                     3  3   p ffiffiffi  p ffiffiffi 2, j0j¼ 0.
                                                   4  4     2j¼
                                j  5j¼ 5, jþ 2j¼ 2, j  j¼ , j
                        1. jabj¼ jajjbj                    or jabc ... mj¼ jajjbjjcj ... jmj
                        2. ja þ bj @ jajþ jbj              or ja þ b þ c þ     þ mj @ jajþjbjþ jcjþ    jmj
                        3. ja   bj A jaj  jbj
                        The distance between any two points (real numbers) a and b on the real axis is ja   bj¼jb   aj.



                     EXPONENTS AND ROOTS
                                                                                     p
                        The product a   a ... a of a real number a by itself p times is denoted by a , where p is called the
                     exponent and a is called the base.  The following rules hold:
                             p  q   pþq                         p r  pr
                        1. a   a ¼ a                       3. ða Þ ¼ a
                            a p  p q                                a p
                                                               a p
                        2.    ¼ a                          4.     ¼
                            a q                                b    b  p
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