Is $\mathbb{R}$ a subset of $\mathbb{R}^n, \ n>1$? I am very confused right now... I think it is not, since it is like comparing apples and black holes... but I am not sure anymore... woahoah $\overset{\times \times}{\sim}$
So, is $\mathbb{R}$ a subset of $\mathbb{R}^n, \ n>1$?
 A: No, because you're comparing real numbers to ordered $n$-tuples of real numbers, and real numbers are not ordered $n$-tuples.
However, there is a way to naturally identify a subset of $\mathbb R^n$ with $\mathbb R$, for example by identifying $x\in\mathbb R$ with $(x,0,\dots,0)\in\mathbb R^n$ (it is trivial to verify that this map is a bijection). Indeed it is also an injective homomorphism when you think of it as e.g. a map between vector spaces, so it is a very natural way to identify $\mathbb R$ with a subset of $\mathbb R^n$. But no, $\mathbb R$ itself is not a subset.
A: I suppose that you define $\mathbb R^n$ as$$\overbrace{\mathbb R\times\mathbb R\times\cdots\times\mathbb R}^{n\text{ times}}.$$So, no, $\mathbb R$ is not a subset of $\mathbb R^n$, since the elements of $\mathbb R^n$ are $n$-tuples of real numbers, none of which is a real number itself.
A: This is simply an add on. 
What can be confusing is that R² is , in a way and loosely speaking , "made out" real numbers. For real numbers are the elements of the couples that belong to R². 
Rk : More precisely they are the elements of the elements of couples, for the couple ( a, b) is standardly defined as a set of sets : ( a, b) = { {a} , {a,b}} See : https://en.wikipedia.org/wiki/Ordered_pair#Informal_and_formal_definitions
But, in set theory, it is NOT TRUE that : " the elements of the elements of a set S are also elements of S". ( In other words, the membership relation is NOT transitive). 
So, from " real numbers are the elements of the couples that are the elements of R²" one CANNOT infer that " real numbers are elements of R²"
In this respect sets do  not behave  in the same way as other objects. 
Rk : saying that the membership relation is not transitive does not mean that it is intransitive. It can happen that the member of the member of a set S is also a member of S. For example, in the construction of natural numbers, the number 1 is a member of the number 2 = {O,1} , the number 2 is a member of the number 3 , and 1 is still a member of 3 for 3 is defined as equal to the set { 0,1,2} . 
