We With this analogy, we will define the real

We have been working with real numbers sinceelementary school, however they are not an easy subject to define.

In a similarway, when a biologist finds a new bacteria, it will try to classify itaccording to its features (morphology, intra/extra cellular structures,metabolism, growth, reproduction, etc.. ). With this analogy, we will definethe real numbers, , as the set of points (numbers) that satisfy the following fiveproperties (also called field properties) plus what is called axiom oforder and least upper bound property:      1.  Commutativity: For any , we have that                                               2.

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  Associativity: For every , we have that                          3.  Distributivity: For every  we have                                                  4.  Existence of neutral elements: There aredistinct elements 0 and 1 of  such that for all  we have  and .     5.  Existence of multiplicative and additiveinverses: 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) Forany , one and only one of the following options is true                                                    7.  The least upper bound property: A non emptyset 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 inprinciple  can be any set, and  may not represent the usual sum andmultiplication.

The last two properties of the real numbers differentiate themfrom other sets (fields) in the following way:     1.  The axiom of order is what makes us thinkthat real numbers are aligned up in a row, one number following the other. Inother words, given two numbers you can always decide if they are equal or inother case, which one is bigger.     2.  The least upper bound property basically saysthat there are no holes or gaps in our model of real numbers as a straightline.  Subsets of thereal numbers are denoted with square brackets and parenthesis as follows: Giventwo real numbers,       1.   (these are called open) isthe set of values of x with      2.   (thess are called closed) isthe 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 possibleto use the same algebraic properties of numbers when solving inequalities ofreal numbers. One of the most useful inequlities is called  Theorem 1.1 For any  real numbers