| 1.1.1 The Real Numbers : |
Real Numbers are the elements of a set, denoted by  , with the following properties: |
| 1) Algebraic properties of real numbers: |
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There are two binary operations defined on , one called addition, denoted by
 , and the other called multiplication, denoted by
 , with the usual algebric properties: for all
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There exist two distinct elements in , denoted by 0 and 1, with following properties: |
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0 +  = for all  ; 1 = for all 0  .
The elements 0, read as zero, is called the additive identity and 1, read as one, is called the multiplicative identity. |
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For every , there exists unique element  such that + (- ) = 0 ; for 0 in , there exists unique element  such that  = 1. |
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| 2) Order properties of real numbers: |
There exists an order, denoted by <, between the elements of  with the following properties: |
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For   , one and only one of the following relations hold :  . |
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There are two more properties that real numbers have which we shall describe later : |
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Archimedean property |
