Complex Numbers (Extra): Visualizing Complex Roots

Hello! In the main article about complex numbers (which you can read here), we looked at: 

• what roots of an equation are, their importance, how to visualize them 
• how complex numbers were invented to find roots of special equations which didn't have real roots
• basic operations on complex numbers
• powers of i
• graphical representation and argument
• Euler's formula and de Moivre's theorem
• polar forms
• rotation by multiplication

In this article, I'll show you a way to visualize complex roots.

Let's say you have the following function: 

 z3 + z + 1 = f(z). 

I want to find the zeroes, so I can write:


 z3 + z + 1 = 0. 


Substitute z to be a complex number x + yi (a complex root of the equation):

(x + yi)3  + x + yi + 1 = 0. 

= x3  + y3i3  + 3x2yi + 3xy2 i2 + x + yi + 1 = 0.

= x3 + x + 3x2yi + yi - y3i - 3xy2 + 1 = 0.

=  (x3  + x - 3xy2 + 1) + i(y  - y3 + 3x2y) = 0.

Real part of f(z): (x3  + x - 3xy2  + 1)
Imaginary part of f(z): (y  - y3 + 3x2y) 

For f(z) to be zero, the real and imaginary parts have to be zero. So,


x3 + x - 3xy2  + 1 = 0.

y - y3 + 3x2y = 0.


Now, plot the solution sets for the two equations. You get two curves in the XY-plane we're all familiar with:



f(z) is zero only when Re(f(z)) and Im(f(z)) are 0 (equal). So, the roots of f(z) are the points of intersection of the curves! 


Hope you understand, and as always, thanks for reading!

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