Can all irrational numbers be written in the form $u + v\sqrt{2}$, with $u$ and $v$ rational?

Question

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.)

Answer 1

Another reason that not all irrationals can be written as $u+v\sqrt 2$ with $u,v\in \Bbb Q$: there are only countably many reals of that form, but the irrationals are uncountable.

Answer 2

No.

Counterexample: $$\forall u,v \in \mathbb{Q}, \, \sqrt{3} \neq u + v\sqrt{2}.$$

Answer 3

No. It is not possible. For examples,

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

Answer 4

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 !?