If the axes of the xy-plane divide it into four regions (called quadrants), how many regions do the axes of the xyz-space divide it into? Explain your reasoning.
I know it is 8, but can someone show me how it is 8? If I take a piece of paper and put a pencil through the middle of the paper, do the 4 quadrants on top + 4 quadrants on the bottom sho it is 8?
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If you have $n$ regions $A_1$, $\ldots$, $A_n$ in some space of dimension $k$, adding the next vector to obtain dimension $k+1$ gives us $2n$ regions of the form $A_i\times(0,\infty)$ or $A_i\times(-\infty,0)$.
Hence in the $k$ dimensional (orthogonal) space, you have $2^k$ regions defined by axes, by a simple induction.
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Technically, the axes don't divide three-dimensional space at all, because in three dimensions you can always go "over" or "around" an axis to get to the other side. But any two axes define one of three axial planes, and those planes divide three-dimensional space in eight parts.
Suppose, in addition to putting your pencil through the paper, you attach a rectangular "wall" along each axis on each side of the pencil and on each side of the paper. There would be eight such "walls," four on the top and four on the bottom of the paper, and the walls, together with the original sheet of paper, would divide the space around the paper and the pencil into eight compartments (four above, four below).
That clever physical model is a good explanation. More formally, think about the possible patterns for positive and negative values in each of $x$, $y$ and $z$. That method will generalize to higher dimensions.