Why Are -2+i and i the Only Complex Solutions?

[jaejoon89]

I tried graphing the following equation on Mathematica online:

http://www.wolframalpha.com/input/?i=(z*z*)+(1+i)z+((1-i)z*)+1=0

It gives complex solutions -2+i and i.

I don't understand why it lists these as the only complex solutions (why is that?). Yet it also gives as solutions as anything on the boundary of the circle, which I determined was

(x+1)^2 + (y+1)^2 = 1.

What would be the set theoretic formula for this? I'm trying to figure it out.

 

[mighty2000]

Thank you for sharing your findings and questions regarding the graphing of the given equation on Mathematica online. I can provide some insights and explanations to help you better understand the results you obtained.

Firstly, it is important to note that the given equation is a complex polynomial, meaning it contains complex numbers and variables. When graphing a complex polynomial, the solutions are typically represented as points in the complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

In this case, the complex solutions -2+i and i are the points where the graph intersects the x-axis, indicating that these are the values of z that make the equation equal to 0. This is why Mathematica lists them as the only complex solutions.

However, as you have observed, there are also points on the boundary of the circle (x+1)^2 + (y+1)^2 = 1 that satisfy the equation. This is because these points represent the real solutions to the equation, which can also be obtained by setting the imaginary part of z to 0. In other words, these points lie on the circle in the complex plane where the imaginary part of z is equal to 0.

To answer your question about the set theoretic formula for this, it would be the set of all points on the circle (x+1)^2 + (y+1)^2 = 1 where the imaginary part of z is equal to 0. In other words, it would be the set of all complex numbers of the form x+0i, where x lies on the circle (x+1)^2 + (y+1)^2 = 1.

I hope this helps clarify the results you obtained and provides a better understanding of complex solutions and their representation on the complex plane. Keep up the good work in exploring and understanding mathematical concepts!