I know $\mathbb{R}$ is the real number line. What really is $\mathbb{R}^n$?

Question

I know $\mathbb{R}$ is the real number line. What really is $\mathbb{R}^n$?

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EDIT (based on comments below): What actually is the result of a cartesian product? That is something I failed to grasp too. What do you get when you multiply the number line $\Bbb{R}$ with itself? I mean the final result? Thanks.

Answer 1

I'm not sure what you mean by "really". First and foremost, $\mathbb{R}^n$ is the set of $n$-tuples with real entries. A set has no specific structure. Then again, most mathematicians use $\mathbb{R}^n$ as shortcut for the vector space triplet $(\mathbb{R}^n,+,\,\cdot\,)$, that is, $\mathbb{R}^n$ with an addition and scalar multiplication. You can learn more about that here. Others again use it as shortcut for $(\mathbb{R}^n,+,\,\cdot\,,\langle\cdot,\cdot\rangle)$, the aforementioned vector space together with a scalar product. Etc.

But I guess maybe you just think about the dimension indicated by $n$. $n$ stands for any positive natural number, i.e. one arbitrary of $1,2,3,4,\ldots$ For $n=1$ we usally write $\mathbb{R}$ instead of $\mathbb{R}^1$. As you mentioned, this can be visualized as the real number line. $\mathbb{R}^2$ would be the plane, two-dimensional, but infinitely thin and the plane extending forever. $\mathbb{R}^3$ is the three-dimensional space, infinite in all directions you could look. $\mathbb{R}^4$ gets difficult, because, as we are mere three-dimensional beings, we are having serious trouble going on beyond. $\mathbb{R}^5$ is even more difficult and it doesn't get better with bigger numbers.

However, lack of visualization has hardly ever hindered any true mathematician to make some calculations. We mathematicians cannot draw these high-dimensional spaces, but we can think about them, in an abstract way. In other words, in comparison to the reality we can embrace, $\mathbb{R}^n$ is nothing but a construction of mind shared across many people.

Answer 2

It is Euclidean space, which can be thought of as ordered $n$-tuples of real numbers.

For example, $\mathbb{R}^3$ is the set of all ordered triples $(a,b,c)$, where $a,b,c\in\mathbb{R}$.