is there is a way to visualise what it actually means that a polynomial has complex number roots?

if we take an example of finding a root for the equation $x^2+1 = 0$ in complex field, is there a way to visualise or think about the roots?

can you also please explain in an intuitive way the difference in geometry between the complex field and a real field.

thank you

  • $\begingroup$ Just think of it on the complex plane with complex number multiplication properties: rotating and dialating $\endgroup$ – Anas A. Ibrahim Jun 6 '20 at 8:07

Yes, you can visualize it. Given a polynomial $p(x)$, consider the graph of $\bigl|p(x)\bigr|$. The zeros of $p(x)$ are the points at which this surface touches tha plane $z=0$.

And you can solve the equation $x^2+1=0$ in $\Bbb C$ using the fact that $i^2=(-i)^2=-1$. So, $i$ and $-i$ are roots of $x^2+1$ and, since no number can have more that two square roots, there are no more roots. If you want to apply the method of the previous paragraph, just consider this surface:

enter image description here

In geometrical terms, the difference between $\Bbb R$ and $\Bbb C$ is that $\Bbb R$ is a line, whereas $\Bbb C$ is a plane.

  • $\begingroup$ can you please explain how you draw it on GeoGebra 3D? $\endgroup$ – user14732 Jun 6 '20 at 8:17
  • $\begingroup$ f(x,y)=sqrt((x^2 - y^2 - 1)^2 + 4x^2 y^2) $\endgroup$ – José Carlos Santos Jun 6 '20 at 8:24

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