# Relationship between complex number and vectors

What is the relation between complex numbers and vectors?

I have read in some places "a complex number a 2-dimensional vector".

One can easily observe that $i\cdot i=-1$ in complex multiplication like the cross product of vectors.

Is vector only the generalisation of complex number to n dimensions?

Is there any fundamental difference between those two?

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If $z$ is a complex number, then $(Re(z),Im(z))$ is a vector in $\mathbf{R}^2$. – Harry Dec 23 '12 at 10:17

The set of complex numbers $\mathbb{C}$ is like the set of two dimensional real vectors $\mathbb{R}^2$ in the sense that you have a one-to-one mapping from the one into the other, except it has special rules for addition and multiplication, making it into a very special field.
Addition is the same but multiplication is different. One way to look at complex multiplication is that it is like a rotation in $\mathbb{R}^2$.