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

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.
