We With this analogy, we will define the real

We have been working with real numbers since
elementary school, however they are not an easy subject to define. In a similar
way, when a biologist finds a new bacteria, it will try to classify it
according to its features (morphology, intra/extra cellular structures,
metabolism, growth, reproduction, etc.. ). With this analogy, we will define
the real numbers, , as the set of points (numbers) that satisfy the following five
properties (also called field properties) plus what is called axiom of
order and least upper bound property: 

    1.  Commutativity: For any , we have that

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    2.  Associativity: For every , we have that



    3.  Distributivity: For every  we have



    4.  Existence of neutral elements: There are
distinct elements 0 and 1 of  such that for all  we have  and .

    5.  Existence of multiplicative and additive
inverses: For any , there is an element in  denoted by  such that , and for any nonzero  there is an element of  denoted by , such that .

    6.  Axiom of order: There is a subset  of  such that 

        (a) If , then  

        (b) For
any , one and only one of the following options is true




    7.  The least upper bound property: A non empty
set of real numbers that is bounded from above as a least upper bound.


There are three objects involved in this definition,
the set of points  (denoted by ), and the two operations denoted by . Properties 1-5 may sound familiar (trivial), but remember that in
principle  can be any set, and  may not represent the usual sum and
multiplication. The last two properties of the real numbers differentiate them
from other sets (fields) in the following way: 

    1.  The axiom of order is what makes us think
that real numbers are aligned up in a row, one number following the other. In
other words, given two numbers you can always decide if they are equal or in
other case, which one is bigger.

    2.  The least upper bound property basically says
that there are no holes or gaps in our model of real numbers as a straight

 Subsets of the
real numbers are denoted with square brackets and parenthesis as follows: Given
two real numbers,  

    1.   (these are called open) is
the set of values of x with  

    2.   (thess are called closed) is
the set of values of x with  

    3.   is the set of values of x with  

    4.   is the set of values of x with  

    5.   is the set of values of x with  

    6.   is the set of values of x with  

    7.   is the set of values of x with  

 It is possible
to use the same algebraic properties of numbers when solving inequalities of
real numbers. One of the most useful inequlities is called  

Theorem 1.1 For any  real numbers