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
Why the proof of given two rational numbers their sum is rational involves the sum of both numbers. Wouldn’t this be a contradiction
irrational-numbersproof-explanationrational numberssummation
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Best Answer
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.