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rational_numbers [2014/12/27 19:32] nikolaj |
rational_numbers [2016/04/23 13:54] 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. | ||
| ----- | ----- | ||
| - | |||
| === Discussion === | === Discussion === | ||
| + | == Theorems == | ||
| + | For all $m$ | ||
| + | |||
| + | ^ $\dfrac{1}{1-x}=\dfrac{1}{1-x\cdot x^{m}}\sum_{k=0}^m x^k$ ^ | ||
| + | ^ $\dfrac{1}{y}=\dfrac{1}{1-(1-y)\cdot(1-y)^{m}}\sum_{k=0}^m (1-y)^k$ ^ | ||
| + | |||
| + | == Logic == | ||
| In first order logic, being of characteristic zero ("$\forall n.\,(1+1+\dots+1)_{n\ \text{times}}\neq 0$") requires an axiom schema. But even the induction axiom of the Peano axioms requires a schema. | In first order logic, being of characteristic zero ("$\forall n.\,(1+1+\dots+1)_{n\ \text{times}}\neq 0$") requires an axiom schema. But even the induction axiom of the Peano axioms requires a schema. | ||
