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Real number is often used to represent a point in a 1-dimensional number line. Real numbers are written as $a$ where $a \in \mathbb{R}$.

Complex number is often used to represent a point in a 2-dimensional plane. Complex numbers are written as $a + bi$ where $a,b \in \mathbb{R}$ and $i^2 = -1$.

What about higher dimensional space?

We can extend the concept of complex number, and write a point in a 3-dimensional space as $a + bi + cj$ where $a,b,c \in \mathbb{R}$, but how do we define i and j?

Certainly, this also applies to 4-dimensional space. We can write a point in 4D as $a + bi + cj + dk$, how do we define $i,j,k$?

How about even higher n-dimensional space? Is there a system to define i,j,k,... that will extend to arbitrary dimensions? Is there a name for it?

  • 0
    Very related: [Is there a third dimension of numbers?](http://math.stackexchange.com/questions/32100/is-there-a-third-dimension-of-numbers)2011-09-12
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    This is a very important question! You should read the wikipedia page on quaternions, which give the 4D solution. From there you will find links to discussion of other dimensions. The answers are intricate and rather surprising!2011-09-12
  • 0
    What kind of structure do you want on these numbers?2011-09-12

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