[Math] Is this a valid proof of “a, b are rational, b ≠ 0, r is irrational, then a + br is irrational”

discrete mathematicsproof-verification

Theorem: If a and b are rational numbers, b ≠ 0, and r is an irrational number, then a + br is irrational.

Proof: Assume that if a and b are rational numbers, b ≠ 0, and r is an irrational number, then a + br is rational.

By the definition of rational, we can substitute a and b with fractions where p, q, m, n are particular but arbitrary integers.

    a = p/q   b = m/n

    a + br = p/q + (m/n)r/1

           = p/q + mr/n

           = (pn + qmr) / qn

Since r is irrational, we know that both the numerator and the denominator cannot be rational numbers, which implies a + br is irrational, which contradicts the fact that a + br is rational. This contradiction shows the supposition is false, therefore the theorem is true.

Best Answer

" $$ a + br =...= (pn + qmr) / qn $$ ... Since r is irrational, we know that both the numerator and the denominator cannot be rational numbers.

"

I think your conclusion is illogical / not deductive.

Your assumption was:

"Assume that if a and b are rational numbers, b ≠ 0, and r is an irrational number, then a + br is rational."

The numerator, $pn + qmr, $ is in the form $c+dr$ where $c$ and $d$ are rational and $r$ is irrational. How did you deduce that the numerator, $pn + qmr, $ is irrational, without assuming the thing you're trying to prove?

In fact, your assumption implies that the numerator $is$ rational.

You were on the right track in your proof until this part:

$a + br = p/q + (m/n)r/1$

I think you got "caught up in the maths" and forgot about the logical reasoning of the proof.

I would slightly modify the first line of your proof, which I assume you intended to be a proof by contradiction:

Proof: Assume that if a and b are rational numbers, b ≠ 0, and r is an irrational number, $and \ that \ $ a + br is rational. (Now your goal is to prove that some sort of contradiction will arise.)

By the definition of rational, we can substitute a and b with fractions where p, q, m, n are particular but arbitrary integers. (this bit is fine)

Your assumption assumed $a+br \ $ is rational, so now you should write:

$a + br = p/q$

and take it from there. Remember your goal now is to get a contradiction based on the fact that r is irrational and the rest of the numbers are rational.

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