# Yes/No :$\mathbb{R}$ is isomorphic to $\mathbb{R}\oplus \mathbb{R}$ as vector spaces over $\mathbb{Q}$

Yes/No :$$\mathbb{R}$$ is isomorphic to $$\mathbb{R}\oplus \mathbb{R}$$ as vector spaces over $$\mathbb{Q}$$

# My attempt : yes

i think $$\mathbb{R}\cong \mathbb{R}\oplus \mathbb{R} \cong2 \mathbb{R}$$ both have same dimension that is dim$$( \mathbb{R} )= 1$$

Is its true ?

• You say $\dim(\mathbb{R})=1$. Dimension over what? Sep 9 '19 at 9:19
• @jasmine If the dimension of $\Bbb R$ as a vector field over $\Bbb Q$ is $1$, then all elements are in the $\Bbb Q$-span of a single non-zero real number, say $1\in \Bbb R$. Can you express $\sqrt 2$ as a linear combination of $1$ with rational coefficients? (It seems strange to write "linear combination" with only one vector, but still.) Sep 9 '19 at 9:22
• math.stackexchange.com/questions/26781 Sep 9 '19 at 9:24
• thanks u @ Fred
– Fred
Sep 9 '19 at 9:41
• As clarified over here, the problem seems to lie at $\Bbb R\oplus\Bbb R\cong 2\Bbb R$. Sep 10 '19 at 9:41

$$\mathbb R$$ as a vector space over $$\mathbb Q$$ is not finite-dimensional !