The Fundamental Theorem of Arithmetic and Rational Numbers

[e2m2a]

TL;DR

Is there a similar theorem to the fundamental theorem of arithmetic that applies to all rational numbers?

The fundamental theorem of arithmetic applies to prime factorizations of whole numbers. Can this theorem also correctly be invoked for all rational numbers? For example, if we take the number 3.25, it can be expressed as 13/4. This can be expressed as 13/2 x 1/2. This cannot be broken down further into smaller rational numbers, so these two rational factors are unique. So, I guess my question is, as long as the number we are factoring is a rational number, then there is only one unique factorization of rational numbers that it can be broken down into? So in a way the fundamental theorem of arithmetic applies to rational numbers too?

 

[fresh_42]

There are rings in which this theorem holds, so called unique factorization domains. The theorem doesn't make sense over the rationals, as every number different from zero is a unit, so there is no unique representation possible. Of course any rational number $q$ itself is already factored, i.e. the rationals considered as a ring fulfill the theorem trivially by $q=q.$

If you forget what rational numbers are, namely a field, and consider them as canceled fractions, then you can of course factorize numerator and denominator separately. But this is strictly speaking not the rational numbers anymore, but a quotient ring of the Cartesian product of the integers with themselves.

 

[e2m2a]

I need to study the meaning of quotient rings of Cartesian products. Thanks for the reply.

 

[fresh_42]

It means that you forget the fact that you can divide and consider all pairs $(a,b)$ of integers instead, where two pairs $(a,b)$ and $(c,d)$ are considered equivalent, if $ab=bc$. The pairs are the Cartesian product $\mathbb{Z} \times \mathbb{Z}$ and the identification rule is the quotient ring, since certain pairs are put into the same equivalence class and we only consider the classes, not the pairs anymore.

E.g. $\dfrac{1}{2}=\dfrac{2}{4}=\dfrac{-17}{-34}=\ldots$ and we only consider the canceled representant $1/2$, i.e. its equivalence class.

 

[e2m2a]

I guess what I am saying is this. Any rational number can be expressed as the quotient a/b, where a and b are whole numbers. Each of these whole numbers, both the numerator and the denominator, have a unique factorization by the fundamental theorem. Thus every rational number has a unique factorization of smaller rational numbers. For example, 1/2 can not be broken down any further. 2/4 can be factored by 2/2 x 1/2 and -17/-34 can be factored by 17/1 x 1/2 x 1/17.

 

[fresh_42]

Yes, but it's meaningless. The theorem says: "... unique up to units." and the rational numbers are all units, except zero. What you did is, you considered a certain ring of pairs instead of rational numbers and each component is a UFD.

But why do you want to do this? You haven't factored a rational number, you only defined what a canceled fraction is. The rational numbers don't have any primes.

 

[e2m2a]

I need to study this out more. Thanks.