Can you imagine that? (I)

In $\mathbb R^{2}$ , there are $2^2=4$ different quadrants .

In $\mathbb R^{3}$ , there are $2^3=8$ different octants .

In $\mathbb R^{4}$ , there are $2^4=16$ different orthants.

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In $\mathbb R^{10}$ , there are $\boxed{2^{10}=1024}$ different orthants.

Generalization:

Let $\mathbb R^{n}$ be the $n-dimensional$ space. Then there are $2^n$ different orthants .

You can distill the solution down to the basics. We know that a quadrant, octant, etc... Can be defined as a permutation of the values +or - assigned to the number of dimensions n in $\mathbb{R}^n.$ Therefore, there exist $2^n$ orthants in $\mathbb{R}^n.$

I think you should change the link to wikipedia because the answer is there