Approximation of Irrationals

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

If is irrational and is any positive integer, there is a fraction with and for which .

[more]

Proof. Let be a positive integer. Then by the pigeonhole principle, among the points (where denotes the fractional part of ), there are at least two numbers such that . Then for some integer . The statement is proved if we put .

[less]

Contributed by: Izidor Hafner (March 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Reference

[1] D. Benko, "A New Approach to Hilbert's Third Problem," American Mathematical Monthly114(8), 2007 pp. 665–676.



Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send