Countable vs Finite Rationals in (0,1)

  • Thread starter Thread starter lmedin02
  • Start date Start date
  • Tags Tags
    Finite
Click For Summary
SUMMARY

There are countably many rational numbers in the interval (0,1), meaning they can be put into a one-to-one correspondence with the positive integers. This does not imply that they can be listed in a finite manner, as there is no last rational number in this set. The concept of countability distinguishes between finite and infinite sets, with countably infinite sets allowing for an infinite listing process without a terminal element.

PREREQUISITES
  • Understanding of countable vs. uncountable sets
  • Familiarity with rational numbers and their properties
  • Basic knowledge of set theory
  • Concept of one-to-one correspondence
NEXT STEPS
  • Study the concept of uncountable sets, particularly the real numbers
  • Learn about Cantor's diagonal argument
  • Explore the implications of countability in different mathematical contexts
  • Review the properties of rational numbers in various intervals
USEFUL FOR

Mathematics students, educators, and anyone interested in set theory and the properties of rational numbers.

lmedin02
Messages
52
Reaction score
0

Homework Statement


Are there countably many rational numbers in the interval (0,1) or are there finitely many?


Homework Equations





The Attempt at a Solution


I am confused. There are countably many rational numbers in the interval (0,1). Does this mean I can list them all in such a way that I can theoretically stop at the very last element and therefore say that there are finitely many rational numbers in (0,1).
 
Physics news on Phys.org
lmedin02 said:

Homework Statement


Are there countably many rational numbers in the interval (0,1) or are there finitely many?


Homework Equations





The Attempt at a Solution


I am confused. There are countably many rational numbers in the interval (0,1). Does this mean I can list them all in such a way that I can theoretically stop at the very last element and therefore say that there are finitely many rational numbers in (0,1).

No. Countably many does not mean you can count them like you would a bag of marbles 1,2,3,...25, for example, and be done with it. That would be finitely many. It means there is a 1-1 correspondence between the rationals on (0,1) and the positive integers, and, as you know, there isn't a last one of those.
 
thank you
 

Similar threads

Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K
Closed set with rationals question
  • · Replies 2 ·
Replies
2
Views
2K
Does an Open Cover of Rationals in [0,1] Necessarily Encompass All Irrationals?
  • · Replies 10 ·
Replies
10
Views
2K
Proving Uncountability of (0,1): A Puzzling Challenge
  • · Replies 2 ·
Replies
2
Views
2K
Showing that a pathological function is only continuous at 0
  • · Replies 7 ·
Replies
7
Views
2K
Proof that algebraic numbers are countable
  • · Replies 6 ·
Replies
6
Views
3K
High School Is this a finite argument of the countable infiniteness of the rationals?
  • · Replies 21 ·
Replies
21
Views
4K
Undergrad I don't understand Dedekind cuts
  • · Replies 14 ·
Replies
14
Views
1K
Accumulation point of rational numbers in (0,1)
  • · Replies 5 ·
Replies
5
Views
7K
Finite and Countable union of countable sets
  • · Replies 3 ·
Replies
3
Views
3K