# Meaning of tangent space to $\mathbb{R}$

If the tangent space $$T_{x_0}\mathbb{R}^n$$ is the tangent space to $$\mathbb{R}^n$$ at the point $$x_0$$, as the set of all vectors applied to $$x_0$$ and so it is isomorphic to $$\mathbb{R}^n$$, that is the set of all vectors applied in $$0$$: $$\textbf{what is the meaning of T_{x_0}\mathbb{R}, and why it is isomorphic to \mathbb{R}?}$$

I mean in $$\mathbb{R}$$ there is no a notion of vector so I can't imagine what is the tangent space in this case. Can you help me?

If you define tangent spaces for any differentiable manifold, you'll notice that when your manifold is a (finite-dimensional) real vector space $$V$$ you get to identify $$T_p V$$ with $$V$$ at any point $$p$$ (to see why, see for example this question).
The case of $$\Bbb R$$ is simply the one-dimensional case of this phenomenon, since $$\Bbb R$$ is (the) one-dimensional real vector space.
(I don't really get why you would say "there's no notion of vector in $$\Bbb R$$", but you can easily "visualize" what a vector in $$\Bbb R$$ is as you can do in any other $$\Bbb R^n$$: as an arrow going from the origin, i.e. $$0$$, to any element of the space. That is, you're looking at every real number $$t$$ as a vector of length $$|t|$$, going left or right depending on the sign of $$t$$, respectively negative or positive sign.)