Why the proof of given two rational numbers their sum is rational involves the sum of both numbers. Wouldn't this be a contradiction?

Question

In this question why doesn't proof of sum of two rational number is rational not proving the irreducibility of fraction $\frac{ad+bc}{bd}$?, the author gives a proof of why given two rationals the sum is also a rational. In some part of the proof he sums the two rationals. Why this statement doesn't lead to a contradiction since we are trying to prove exactly that?. My guess is that probably since he doesn't state that their sum is a rational number the proof is correct

Answer 1

The manipulations that show $\frac ab+\frac cd=\frac{ad+bc}{bd}$ are valid for any real numbers (more generally, any elements of a field); they don't rely on $a,b,c,d$ being integers. After that, we observe that if $a,b,c,d$ are integers, then $ad+bc$ and $bd$ are also both integers, implying that $\frac{ad+bc}{bd}$ is rational.

Answer 2

At this point we assume one has already gone through the effort of defining $\Bbb N,\Bbb Z$ and addition and multiplication for each of these sets as well.

We continue on to define $\Bbb Q$ as $(\Bbb Z\times \Bbb Z^*)/\simeq$ where $\simeq$ is the equivalence relation $(a,b)\simeq (c,d)\iff ad = bc$ and we define addition in $\Bbb Q$ to be $(a,b)+(c,d) = (ad+bc,bc)$ and multiplication in $\Bbb Q$ to be $(a,b)\cdot(c,d)=(ac,bd)$ where the operations that occur inside of the tuples are the additions and multiplications already defined for integers.

For conveniences sake we introduce the notation $\frac{a}{b}$ to be used in place of writing as $(a,b)$.

Now, there is some work to be done with these definitions still to show that the proposed addition and multiplications are in fact well-defined, which might be what your question is technically asking for, but this follows immediately from the definitions used for integers and our equivalence relation. The sum of rationals is rational by the very definitions we just introduced and does not need to be proven.

Concerns about irrational numbers and real numbers should not be present at the moment as generally to define the reals one uses $\Bbb Q$ as a stepping stone and questions about the rationality of the sum of rationals is far more fundamental.