# roots of polynomial in the complex numbers field

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

• Just think of it on the complex plane with complex number multiplication properties: rotating and dialating – 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: In geometrical terms, the difference between $$\Bbb R$$ and $$\Bbb C$$ is that $$\Bbb R$$ is a line, whereas $$\Bbb C$$ is a plane.
• f(x,y)=sqrt((x^2 - y^2 - 1)^2 + 4x^2 y^2) – José Carlos Santos Jun 6 '20 at 8:24