Since $a$ and $b$ are both integers, there is only a countable amount of numbers $a+b\pi$. Thus not every real number can be expressed as such.
But is there a way to determine if $x$ can be expressed as $a+b\pi$, like a fundamental property of $x$? What about if $a$ and $b$ are rational? Or if instead of $\pi$ you take another irrational number?
Best Answer
The set $ \left\{ a+b\pi, (a,b)\in \mathbb{Z}^2 \right\}$ is dense in $\mathbb{R}$, meaning that given any real number, you can construct a sequence of elements of this set that converges to this number . This is due to $\pi$ being irrational, which can be proved at a college level in a guided exercise. The more general fact that $a\mathbb{Z}+b\mathbb{Z}$ is dense in $\mathbb{R}$ iff $\frac{a}{b} \notin \mathbb{Q}$ can be proved from the density of the additive subgroups of $\mathbb{R}$, usually using the greatest lower bound property.
To answer your question about the fundamental property, I guess more detail is needed on the assumptions you can make on $x$. For other irrational numbers such as square roots, there's a whole theory known as algebraic field extensions.