Can Rationals Be Factored with Non-Natural Exponents?

  • Context: Undergrad 
  • Thread starter Thread starter DaTario
  • Start date Start date
Click For Summary

Discussion Overview

The discussion revolves around the concept of extending the uniqueness of factorization principle to rational numbers by allowing prime number exponents to be integers rather than restricted to natural numbers. Participants explore whether this extension is meaningful and how it relates to the factorization of rational and irrational numbers.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants propose that rational numbers can be expressed with prime factors raised to integer exponents, citing examples like ##3.5 = 2^{-1} \times 7##.
  • Others argue that rational numbers already have a unique representation as fractions of coprime integers, which inherently leads to unique factorization.
  • A participant questions the application of this factorization method to irrational numbers, suggesting it does not work.
  • There is a request for references on the topic of factorization of rationals, indicating a desire for further study.
  • Some participants reference historical knowledge of rational and irrational numbers, suggesting that the concepts are well-established.

Areas of Agreement / Disagreement

Participants express differing views on the validity and implications of extending factorization principles to rational numbers. While some agree on the uniqueness of factorization for rationals, others challenge the applicability of these ideas to irrational numbers, leaving the discussion unresolved.

Contextual Notes

Limitations include the lack of consensus on the significance of extending factorization to rationals and the unclear implications for irrational numbers. The discussion also highlights a reliance on definitions and historical context without resolving the mathematical intricacies involved.

DaTario
Messages
1,097
Reaction score
46
Hi All,

Is there any sense in proposing an extension of the principle of the uniqueness of fatorization to rationals by allowing the exponents of the prime numbers to be integers numbers (except 0) and not only natural (except 0)?

For example, ## 3.5 = 2^{-1} \times 7 ##

Best wishes,

DaTario
 
Mathematics news on Phys.org
What is new about it? Rational numbers (apart from 0) have a unique way to be expressed as fraction of two coprime numbers, that is a well-known result. Those two coprime numbers have a unique factorization each, together they lead to the form you posted.
 
DaTario said:
Hi All,

Is there any sense in proposing an extension of the principle of the uniqueness of fatorization to rationals by allowing the exponents of the prime numbers to be integers numbers (except 0) and not only natural (except 0)?

For example, ## 3.5 = 2^{-1} \times 7 ##
##3.5 = \frac 7 2 = 7 \times 2^{-1}##
Every rational number ##\frac m n## can be rewritten as ##m \times n^{-1}##. However, when people speak of factoring numbers, all numbers involved are usually integers.
 
Thank you, mfb, you have put it in a very clear form.

Thank you, Mark44, as well.

So is it correct to say that rational number admit unique fatorization based on prime number raised at integer exponents?
 
What happens when one attempts to apply this fatorization method to some irrational number?

Do you have, by the way, some reference to indicate? I would like to start a study on this subject.

Thank you again,
 
DaTario said:
What happens when one attempts to apply this fatorization method to some irrational number?
It does not work.

DaTario said:
Do you have, by the way, some reference to indicate?
We are discussing the definition of rational and irrational numbers. References? Even the old Greeks knew the concepts of rational and irrational numbers. Check the wikipedia articles, they have tons of references.
 
mfb said:
It does not work.

We are discussing the definition of rational and irrational numbers. References? Even the old Greeks knew the concepts of rational and irrational numbers. Check the wikipedia articles, they have tons of references.
Sorry, mfb, I was kindly asking you to indicate references containing issues concerning the factorization of rationals. I have never seen anything like this in the books of math I have read.
 
And I was pointing to a website where you can find those references.
 
  • #10
mfb said:
Even the old Greeks knew the concepts of rational and irrational numbers. Check the wikipedia articles, they have tons of references.

Thank you, mfb, sincerely. I hope you are not being forced by anyone or by any reason to answer questions you don´t want to answer.

Best Regards,

DaTario
 

Similar threads

High School Extending the Fundamental Theorem of Arithmetic to the rationals
  • · Replies 35 ·
2
Replies
35
Views
5K
Undergrad Why is the p-adic order of zero considered infinite?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad The Fundamental Theorem of Arithmetic and Rational Numbers
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Prime factorization and real exponents
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Prove: Rational Number Squared Has Prime Factors w/ Even Exponent
  • · Replies 14 ·
Replies
14
Views
8K
Undergrad How to Understand Non-Rational Exponents Intuitively?
  • · Replies 7 ·
Replies
7
Views
4K
Undergrad Average of prime factors growth
  • · Replies 12 ·
Replies
12
Views
2K
High School Are Inverse Problems More Complex Than Direct Problems in Mathematics?
  • · Replies 13 ·
Replies
13
Views
3K
Graduate Is fractal nature dependent on the number of steps?
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Two integers and thus their squares have no common factors
  • · Replies 43 ·
2
Replies
43
Views
7K