I read about half space and hyper plane and keep getting confused about which is which and how people are using it. I would really appreciate if somebody can give me an example in simple language over the math one written on wikipedia.Half Space
I made a mistake of writing half planes instead of hyper plane. Corrected it.
A hyperplane is a subset of a Euclidean space of one less dimension than the whole space. As such, it is defined by one linear equation. In $\mathbb R^3$, the plane created by the $x$ and $y$ axes is one such, represented by $z=0$. The plane $x+y+z=0$ is another, tilted and going through the origin. Each side of such a plane is a half space. The same happens in higher dimensions. If the coordinates are $x_1, x_2, \ldots ,x_n$, there is a hyperplane $x_1=5$ which divides the space into two half spaces: one with all points that satisfy $x_1 \gt 5$ and one with the ones that have $x_1 \lt 5$. Similarly, there is another with $x_1+4x_2+3x_3 = 9$ which is inclined in those axes.
Intuitive answer for non-mathematicians:
A hyperplane divides a higher dimensional space into two half-spaces. A half-plane is one of the halves after the hyperplane has been split in two.
For example, if we cut a piece of cheese in half, the place where we cut is the hyperplane, while the two pieces are the half-spaces.
If we imagine that the knife forms a hyperplane, than if we split the blade into two, we will get half-planes.
Usually, half-plane is just another name to a half-space in $\mathbb{R}^{2}$. The general definition is half-spaces requires some understanding of linear algebra, and the wikipedia article covers it pretty well.
Mathworld article on Half-space relates half-space and hyperplane clearly as(quoting from the article):
A half-space is that portion of an n-dimensional space obtained by removing that part lying on one side of an (n-1)-dimensional hyperplane.
