Why is a sphere in an $n $-dimensional space called $(n-1) $-sphere? Isn't it natural to call a sphere in 3D a 3-sphere, a sphere in 2D (i.e. a circle) a 2-sphere, etc?
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$\begingroup$tl;dr: You are describing the sphere itself, not a particular space in which it is represented.
It should first be made clear the distinction between a sphere and a ball: a sphere consists of the surface only, whereas a ball consists of both the surface and the contained volume.
Now that that's out of the way, then there's the question of how you define a sphere. A sphere in n-dimensional space with radius $r$ centered at $(c_1,c_2,\cdots,c_n)$ is very easily represented by $(x_1-c_1)^2+(x_2-c_2)^2+\cdots+(x_n-c_n)^2=r^2$. However, this is an implicit representation, not an explicit representation. An explicit representation entails a mapping of some other space, and in the case of a sphere in $n$-dimensional space, it happens to be the case that the lowest possible dimension of this other space is $n-1$. As such, spheres in $n$-dimensional space are said to be $n-1$-spheres.
The reason for this nomenclature is that this creates a description specific to the sphere itself and not to a specific environment of the sphere in which it is convenient to describe.
Consider a circle, that is, a $1$-sphere, for instance. Certainly, the equation $x_1^2+x_2^2=r^2$ may come to mind, and there are only two variables in this equation, $x$ and $y$, which might inspire you to call this a $2$-sphere. But a circle can also be represented in $\mathbb{R}^3$, $\mathbb{R}^4$, and so on (although of course, you would need to fix $x_3, x_4$ and so on as well). Why, then, should this circle not also be called a $3$-sphere or $4$-sphere? For that matter, is a $3$-sphere a circle or sphere?
$\endgroup$ $\begingroup$To expand a little - a small piece of a circle is like a line, a small piece of the surface of a sphere is like a plane.
The only way that a line can live in one dimension is if it is straight - a curve can live in many dimensions and still look like a line when you look at a short piece.
Likewise a surface can only live in two dimensions if it is flat. Surfaces exist in many dimensions, but they look like a plane when you take a small enough piece.
If we are creating a coherent useful idea of dimension, it turns out that we want a curve to be one-dimensional and a surface to be two dimensional - but what do we mean by a curve or a surface if we have many-dimensional space?
The idea of a manifold captures this - if our curve is locally like a line, and the local pieces can be coherently glued together to make the whole, it is a one-dimensional manifold. Likewise a surface is a two dimensional manifold if it looks locally like a plane wherever you look and you can glue the local pieces together in a coherent way.
There are some technicalities about precise definitions, to make these intuitive ideas rigorous. But that is why the dimensions come out as they do.
$\endgroup$ $\begingroup$Because sphere in $n$-dimensional space is $(n-1)$-dimensional object.
$\endgroup$ 3 $\begingroup$This is because the "$n-1$" sphere is the surface of the $n$ dimensional solid contained in the $n-1$ sphere, I.e it is one dimension lower.
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