What are $\mathbb{R}^2$ and $\mathbb{R}^3$?

Question

What are $\mathbb{R}^2$ and $\mathbb{R}^3$?

I've seen them referred to, I'm guessing they refer to a plane an a space that can be defined in Cartesian coordinates in 2 and 3 dimensions respectively, and the coordinates are in the set of real numbers.

Answer 1

Generally, $\mathbb{R}^n$ is the set of all $n$-tuples of real numbers. For example $(1,0,-2)\in\mathbb{R}^3$.

There is a Wikipedia article on real coordinate space.

Answer 2

$\mathbb{R}$ is the set of real numbers.

$\mathbb{R}^2 = \{(x,y) \,\,|\,\, x\in \mathbb{R} ,\, y\in \mathbb{R}\}$

$\mathbb{R}^3 = \{(x,y,z) \,\,|\,\, x\in \mathbb{R} ,\, y\in \mathbb{R},\, z\in \mathbb{R}\}$

The geometric properties depend on the extra structures.