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order_structure_of_real_numbers [2013/09/08 15:00] nikolaj |
order_structure_of_real_numbers [2015/06/30 16:14] nikolaj |
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| ===== Order structure of real numbers ===== | ===== Order structure of real numbers ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#FFBB00: $\langle \mathbb R,\le \rangle$ | | + | | @#FFBB00: definiendum | @#FFBB00: $\langle \mathbb R,\le \rangle$ | |
| We define the total order over the real numbers (in the Dedekind cut model) via $r<s \equiv r\subset s$, i.e. | We define the total order over the real numbers (in the Dedekind cut model) via $r<s \equiv r\subset s$, i.e. | ||
| - | | @#55EE55: $s \subseteq r \Leftrightarrow s\ge r$ | | + | | @#55EE55: postulate | @#55EE55: $s \subseteq r \Leftrightarrow s\ge r$ | |
| + | |||
| + | ----- | ||
| + | === Theorems === | ||
| + | Inequality of arithmetic and geometric means (AM-GM inequality): | ||
| + | |||
| + | ^ $\frac{1}{n}\sum_{k=1}^n x_k \ge \left(\prod_{k=1}^n x_k\right)^\frac{1}{n}$ ^ | ||
| - | ==== Discussion ==== | ||
| === Reference === | === Reference === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Real_number|Real number]], [[http://en.wikipedia.org/wiki/Construction_of_the_real_numbers|Construction of the real numbers]], [[http://en.wikipedia.org/wiki/Dedekind_cut|Dedekind cut]] | + | Wikipedia: |
| - | ==== Parents ==== | + | [[http://en.wikipedia.org/wiki/Real_number|Real number]], |
| - | === Requirements === | + | [[http://en.wikipedia.org/wiki/Construction_of_the_real_numbers|Construction of the real numbers]], |
| + | [[http://en.wikipedia.org/wiki/Dedekind_cut|Dedekind cut]], | ||
| + | [[https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means|Inequality of arithmetic and geometric means]] | ||
| + | |||
| + | ----- | ||
| + | === Context === | ||
| [[Real number]] | [[Real number]] | ||
| === Element of === | === Element of === | ||
| [[Total order]] | [[Total order]] | ||
