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 ![](../../../images/Math%20Images/Lecture-1_clip_image002_0016.gif) .
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 ![](../../../images/Math%20Images/Lecture-1_clip_image002_0013.gif) such that ![](../../../images/Math%20Images/Lecture-1_clip_image002_0016.gif) ![](../../../images/Math%20Images/Lecture-1_clip_image002_0016.gif) = 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 ![](../../../images/Math%20Images/Math_clip_image002_0233.gif) , 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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3) |
Archimedean property |