Can all irrational numbers be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational? I am curious to know whether all irrational numbers can be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational.  

(Almost similar to how all complex numbers can be written as $x + iy$, with $x$ and $y$ real.)
Let $\sqrt3=u+v\sqrt2$. Then by squaring, $3=u^2+2\sqrt2uv+2v^2$ and $\sqrt2=\dfrac{3-u^2-2v^2}{2uv}$ is a rational number !?
No. It is not possible.  For examples,

$$\pi, e, e^\pi, \pi^e$$