How to prove that $X$ is not rational $X = Z - \pi \times Y$ where $Y$ is rational , $Z$ is an integer, $\times$ means multiplication

Question

Excuse me for a silly question like this. I am 60 years old retired engineer and want to learn some basic math I di did not learn earlier.

I know an example where $(x + y\pi)$ can be an integer, where $y$ is rational , * means multiplication But how to prove the general case if $x$ is always rational , always non-rational or both depending upon the case.

Thanks

Answer 1

Suppose for a contradiction $x$ is rational. Then $x=a/b$ where $a,b$ are coprime integers and $b\neq0$. Also assume $y=c/d$ for coprime integers $c$ and $d$, and furthermore assume $c$ is not zero (so $y$ is not zero). Then, $$ x = z - \pi y $$ if and only if $$ \frac{a}{b} = z - \pi \frac{c}{d} $$ Multiply by $bd$: $$ ad = zbd - \pi cb $$ Then, $$ \pi cb = zbd - ad $$ Divide by $cb$ (this is possible since both are nonzero): $$ \pi = \frac{zbd-ad}{cb} $$ This implies that $\pi$ is rational which is a contradiction.