enter image description here This is an argand Diagram to show 2d Imaginary-real plane solutions to a said root of a number jut there to illustrate the normal complex solutions.
2026-03-31 10:15:36.1774952136
Can numbers as well as having roots in the Imaginary-Real plane also be said to have roots as quaternions or higher complex forms? If so how?
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Sure. The situation's no different from how a real number can have more complex roots than it has real roots. E.g. the fourth roots of $1$ in the reals are $1$ and $-1$, but in the complex numbers we get $i$ and $-i$ as well.
By embedding an algebraic structure $\mathcal{A}$ inside a larger algebraic structure $\mathcal{B}$ we can often get more variety at the cost of having fewer nice properties. In this case, our additional variety is "having more roots," and the nice property we lose is that the proof that an $n$th-degree polynomial over $\mathbb{C}$ has at most $n$ complex roots breaks down when we try to replace $\mathbb{C}$ with $\mathbb{H}$ (= the quaternions), and so there's no reason to not expect more roots to pop up. (As an exercise you might try to find out how many square roots $-1$ has in $\mathbb{C}$.)