Is $\mathbb{R}$ a subset of $\mathbb{R}^2$?

Is it correct to say that $\mathbb{R}$ is a subset of $\mathbb{R}^2$? Or, put more generally, given $n,m\in\mathbb{N}$, $n<m$, is $\mathbb{R}^n$ a subset of $\mathbb{R}^m$?

Also, strictly related to that: what is then the “relationship” between the set $\{(x,0)\in\mathbb{R}^2,x\in\mathbb{R}\}\subset\mathbb{R}^2$ and $\mathbb{R}$? Do they coincide (I would say no)? As vector spaces, do they have the same dimension (I would say yes)?

If you could give me a reference book for this kind of stuff, I would really appreciate it.

Thank you very much in advance.

(Please correct the tags if they are not appropriate)

Solutions Collecting From Web of "Is $\mathbb{R}$ a subset of $\mathbb{R}^2$?"

It’s not really true that $\mathbb R^n$ is a subset of $\mathbb R^m$ when $n<m$. It is true that there is a subspace of $\mathbb R^m$ that is isomorphic to $\mathbb R^n$, but unfortunately, there are way too many of them, and there is no real way to pick one of them usefully as “the obvious embedding.”

A simple example when $n=1$ and $m=2$ is that the $x-$ and $y-$axes are both embeddings of $\mathbb R^1$ into $\mathbb R^2$, and there is no “natural” way to choose between these enbeddings (or any other embeddings of the real link into $\mathbb R^2$.)

I wouldn’t say so, even though every onedimensional subspace of $\mathbb{R}^n$ is isomorphic to $\mathbb{R}$, but there is no natural embedding.

But a more or less funny is, that even thought nearly everyone say that $\mathbb{R}\not\subset\mathbb{R}^2$ many mathematicans say that $\mathbb{R}\subset\mathbb{C}$
even though there is a canocial transformation from $\mathbb{C}$ to $\mathbb{R}^2$.

I guess sometime one stops to distinguish often between things which are isomorphic but not the same.

Strictly speaking, $\mathbb R$ is not a subset of $\mathbb R^2$.

However, depending on how you construct your number systems, the natural number $1$ is different from the rational number $1$, and both are distinct from the real number $1$. And don’t get me started on complex numbers or quaternions!

This type of imprecision allows us to actually talk about the objects in question without worrying too much about auxiliary details. $\mathbb{R}$ can clearly be embedding into $\mathbb{R}^2$ preserving virtually any property of $\mathbb{R}$ in the process. Once this is realised linguistic license is not too detrimental to rigour.