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A visual explanation of why complex roots of polynomials occur in conjugate pairs:

Show extension to 2+i:
Show extension to 2−i:

The black parabola is the function

z=(x−2)2+1

It clearly does not intersect the real x-axis, but has the pair of conjugate roots 2±i , shown as red spheres. This can be understood by extending the real parabola over the complex plane using the pair of absolute value functions

(x−2)2+(y∓1)2

where y is the imaginary part of the independent variable. Since the extension can happen in either direction, the solutions necessarily come in complex conjugate pairs.

Complete code for this example:



 


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