Can Rationals Be Factored with Non-Natural Exponents?

[DaTario]

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

 

[mfb]

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.

 

[Mark44]

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.

 

[DaTario]

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?

 

[mfb]

Sure.

 

[DaTario]

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,

 

[mfb]

What happens when one attempts to apply this fatorization method to some irrational number?

It does not work.

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.

 

[DaTario]

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.

 

[mfb]

And I was pointing to a website where you can find those references.

 

[DaTario]

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