Does this proof I made make sense?
Proof//
$\mathbf a$ is the rational number, $\mathbf b$ is the irrational number.
Assume that $\mathbf {a * b}$ is rational due to proof by contradiction. Therefore, $\mathbf {a * b = P}$ for some rational number $\mathbf P$.
In another word, $\mathbf {rational * irrational = rational}$.
If you divide the rational number, from both side, we get $\mathbf {b = \frac P a}$ which also translates to $\mathbf {irrational = \frac {rational} {rational}}$ which then becomes
$\mathbf {irrational = rational}$.
However this reaches a contradiction because $\mathbf {irrational \neq rational}$.
Therefore, proving that the product between an irrational number and rational number is equal to an irrational number.
$\Box$
$\mathbf {EDIT:}$ I would like to apologize for the tough wording of the proof, I just completed this on a test, and I tried to write it word for word as I did from the test to see if I was on the right track.
Best Answer
The premise is almost correct. The wording is pretty tough to read. But more important than the wording, is a missed an edge case.
You tried to use the fact (implicitly) that the quotient of two rationals is rational. But that's not exactly true. The denominator cannot be $0$.
You need to cite that the original rational $a$ is non-zero, then cite explicitly that the quotient of two non-zero rationals is rational. That'll at least make the proof correct.