4.2 KiB
| title | TARGET DECK | FILE TAGS | tags | |
|---|---|---|---|---|
| Archimedean Property | Obsidian::STEM | algebra::archimedean |
|
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.
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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.
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%%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).
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%%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).
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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).
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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).
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%%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.
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Basic
Which property of the reals is depicted in the following?
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Back: The Archimedean property.
Reference: “Archimedean Property,” in Wikipedia, June 23, 2024, https://en.wikipedia.org/w/index.php?title=Archimedean_property.
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Basic
How is the following diagram stated analytically?
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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.
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%%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.
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%%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).
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Bibliography
- “Archimedean Property,” in Wikipedia, June 23, 2024, https://en.wikipedia.org/w/index.php?title=Archimedean_property.
- Tom M. Apostol, Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra, 2nd ed. (New York: Wiley, 1980).