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rational_numbers [2016/04/23 13:49]
nikolaj
rational_numbers [2016/04/23 13:54] (current)
nikolaj
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 ===== Rational numbers ===== ===== Rational numbers =====
 ==== Framework ==== ==== Framework ====
-$\dots,\,-\tfrac{4}{3},​\,​0,​\,​\tfrac{1}{17},​\,​1,​\,​7.528,​\,​9001,​\dots$+$\dots,\,-\frac{4}{3},​\,​0,​\,​\frac{1}{17},​\,​1,​\,​7.528,​\,​9001,​\dots$
  
-The rational numbers ${\mathbb{Q}}$ can be defined as the field of characteristic 0 which has no proper ​subfield. In less primitive notions, it's the field of fractions for the integral domain of [[natural ​number]]s. The second order theory of rationals (see the note below) describes a countable collection.+The rational numbers ${\mathbb{Q}}$ can be defined as the field of characteristic 0 which has no proper ​sub-field. In less primitive notions, it's the field of fractions for the integral domain of [[natural ​numbers]]. The second order theory of rationals (see the note below) describes a countable collection.
  
 The rationals can also be set up straight forwardly from tuples of natural numbers. ​ The rationals can also be set up straight forwardly from tuples of natural numbers. ​
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 For all $m$ For all $m$
  
-^ $\sum_{k=0}^m x^k = \dfrac{1}{1-x}(1-x^{m+1})$ ^ +^ $\dfrac{1}{1-x}=\dfrac{1}{1-x\cdot x^{m}}\sum_{k=0}^m ​x^k$ ^ 
-^ $\sum_{k=0}^m ​(1-y)^k \dfrac{1}{y}-\dfrac{1}{y}(1-y)^{m+1}$ ^+^ $\dfrac{1}{y}=\dfrac{1}{1-(1-y)\cdot(1-y)^{m}}\sum_{k=0}^m (1-y)^k$ ^
  
 == Logic == == Logic ==
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