# Differences

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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 == |