---
title: Archimedean Property
TARGET DECK: Obsidian::STEM
FILE TAGS: algebra::archimedean
tags:
  - algebra
---

## Overview

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]].

%%ANKI
Basic
What does the Archimedean property of the reals state?
Back: If $x, y \in \mathbb{R}^+$, then there exists a positive integer $n$ such that $nx > y$.
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).
<!--ID: 1724155425457-->
END%%

%%ANKI
Basic
How is the Archimedean property of the reals geometrically interpreted?
Back: Any finite-length line segment can be covered by a finite number of line segments of some positive length.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1724155425465-->
END%%

%%ANKI
Basic
The Archimedean property of the reals posits the existence of what mathematical entity?
Back: A positive integer.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1724155425470-->
END%%

%%ANKI
Basic
Given positive reals $x$ and $y$, what does the Archimedean property conclude?
Back: There exists a positive integer $n$ such that $nx > y$.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1724155425481-->
END%%

%%ANKI
Basic
Given reals $x$ and $y$, what does the Archimedean property conclude?
Back: Indeterminate. We expect $x$ and $y$ to be positive reals.
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1724155425487-->
END%%

%%ANKI
Basic
Which property is roughly described as "the reals have no infinitely large element?"
Back: The Archimedean property of the reals.
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).
<!--ID: 1724155425492-->
END%%

%%ANKI
Basic
Which property of the reals is depicted in the following?
![[archimedean-property.png]]
Back: The Archimedean property.
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).
<!--ID: 1724155425498-->
END%%

%%ANKI
Basic
How is the following diagram stated analytically?
![[archimedean-property.png]]
Back: For any $A, B \in \mathbb{R}^+$, there exists a positive integer $n$ such that $nA > B$.
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).
<!--ID: 1724155425503-->
END%%

%%ANKI
Basic
What mathematical entities are presumed in the Archimedean property of the reals?
Back: Two positive real numbers.
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).
<!--ID: 1724155425509-->
END%%

%%ANKI
Basic
What axiom of the real-number system is used to prove its Archimedean property?
Back: The least upper bound axiom (i.e. the completeness axiom).
Reference: Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).
<!--ID: 1724155536942-->
END%%

## Bibliography

* “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).
* Tom M. Apostol, _Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra_, 2nd ed. (New York: Wiley, 1980).