Why is it enough to have extended the numbers to include only one orthogonal imaginary axis? I am wondering in the context of roots of polynomials. I know that the orthogonality of imaginary axis w.r.t. real axis is not the only property of imaginary numbers but there is also the relation $i^2=-1$. But still I am not able to think why is the solution space of polynomials complete with expanding the uni-dimensional real axis to 2-dimensional Complex numbers?
2026-03-26 10:57:24.1774522644
Complex Numbers : Why stop at 2 dimensions?
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There are several other useful extensions of $\mathbb{R}$. You can google quaternions (real dimension 4) and octonions (real dimension 8). Each time the dimension increases, we loose some property of $\mathbb{R}$. For example multiplication of quaternions is not commutative, and for octonions it is not associative. But there is a theorem of Frobenius which say that in some natural sense these are all possible extensions of $\mathbb{R}$.