98 lines
4.3 KiB
Markdown
98 lines
4.3 KiB
Markdown
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---
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title: Archimedean Property
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TARGET DECK: Obsidian::STEM
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FILE TAGS: algebra::archimedean
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tags:
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- algebra
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---
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## Overview
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If $x, y \in \mathbb{R}^+$, then there exists a positive integer $n$ such that $nx > y$. This fundamental property usually follows from the [[bounds#Completeness Axiom|completeness axiom]].
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%%ANKI
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Basic
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What does the Archimedean property of the reals state?
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Back: If $x, y \in \mathbb{R}^+$, then there exists a positive integer $n$ such that $nx > y$.
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Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
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<!--ID: 1724155425457-->
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END%%
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%%ANKI
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Basic
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How is the Archimedean property of the reals geometrically interpreted?
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Back: Any finite-length line segment can be covered by a finite number of line segments of some positive length.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724155425465-->
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END%%
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%%ANKI
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Basic
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The Archimedean property of the reals posits the existence of what mathematical entity?
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Back: A positive integer.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724155425470-->
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END%%
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%%ANKI
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Basic
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Given positive reals $x$ and $y$, what does the Archimedean property conclude?
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Back: There exists a positive integer $n$ such that $nx > y$.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724155425481-->
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END%%
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%%ANKI
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Basic
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Given reals $x$ and $y$, what does the Archimedean property conclude?
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Back: Indeterminate. We expect $x$ and $y$ to be positive reals.
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724155425487-->
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END%%
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%%ANKI
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Basic
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Which property is roughly described as "the reals have no infinitely large element?"
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Back: The Archimedean property of the reals.
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Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
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<!--ID: 1724155425492-->
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END%%
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%%ANKI
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Basic
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Which property of the reals is depicted in the following?
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![[archimedean-property.png]]
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Back: The Archimedean property.
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Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
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<!--ID: 1724155425498-->
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END%%
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%%ANKI
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Basic
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How is the following diagram stated analytically?
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![[archimedean-property.png]]
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Back: For any $A, B \in \mathbb{R}^+$, there exists a positive integer $n$ such that $nA > B$.
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Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
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<!--ID: 1724155425503-->
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END%%
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%%ANKI
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Basic
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What mathematical entities are assumed to exist in the formulate of the Archimedean property of the reals?
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Back: Two positive real numbers.
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Reference: “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
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<!--ID: 1724155425509-->
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END%%
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%%ANKI
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Basic
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What axiom of the real-number system is used to prove its Archimedean property?
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Back: The least upper bound axiom (i.e. the completeness axiom).
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Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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<!--ID: 1724155536942-->
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END%%
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## Bibliography
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* “Archimedean Property,” in _Wikipedia_, June 23, 2024, [https://en.wikipedia.org/w/index.php?title=Archimedean_property](https://en.wikipedia.org/w/index.php?title=Archimedean_property&oldid=1230567137).
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* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
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