Are all rational numbers normal?

Click For Summary

Discussion Overview

The discussion revolves around the concept of normal numbers in relation to rational numbers, exploring whether all rational numbers can be classified as normal. The scope includes theoretical definitions and properties of normal numbers and rational numbers.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant suggests a specific rational number as an example of a normal number, referencing a link to a decimal representation.
  • Another participant argues that no rational numbers are normal, explaining that normal numbers must have all possible strings of digits occurring equally, while rational numbers are eventually repeating.
  • A third participant reiterates that the example given is a terminating decimal and emphasizes that it does not contain strings of length 21 or higher.
  • A later reply acknowledges a misunderstanding regarding the definition of normal numbers and questions whether the example could be classified as a simply normal number.

Areas of Agreement / Disagreement

Participants generally disagree on whether rational numbers can be considered normal, with some asserting that they cannot, while others explore the implications of the definitions involved.

Contextual Notes

There are limitations in the discussion regarding the definitions of normal numbers and the implications of terminating versus repeating decimals, which remain unresolved.

gummz
Messages
32
Reaction score
2
Physics news on Phys.org
On the contrary, NO rational numbers are normal! I think you have a wrong idea of what a "normal number" is. A "normal number" is one such that all possible strings of digits, n digits long, occur an equal number of times. A rational number is always eventually repeating (thinking of a terminating decimal as "repeating" 0s).

The number you give is a terminating decimal with 20 digits. It contains NO strings of length 21 or higher!
 
HallsofIvy said:
The number you give is a terminating decimal with 20 digits. It contains NO strings of length 21 or higher!

He means ##0.\overline{0123456789}##, not a terminating decimal but still not normal.
 
Ah, I looked at the definition more closely and now I see my misunderstanding. It must be any string of any length, not simply a digit. But this is a simply normal number, is that correct?
 

Similar threads

  • · Replies 14 ·
Replies
14
Views
1K
  • · Replies 4 ·
Replies
4
Views
2K
  • · Replies 7 ·
Replies
7
Views
2K
  • · Replies 2 ·
Replies
2
Views
2K
  • · Replies 3 ·
Replies
3
Views
2K
  • · Replies 43 ·
2
Replies
43
Views
6K
Replies
4
Views
3K
  • · Replies 8 ·
Replies
8
Views
2K
Replies
11
Views
3K
  • · Replies 5 ·
Replies
5
Views
2K