How to Find Complex Numbers Where (z^2 bar{z})

Let's explore the problem of finding a complex number z such that z2 (bar{z}), the conjugate of z. The journey to solving this will involve expressing z in terms of its real and imaginary parts, setting up equations, and solving these step-by-step.

Expressing (z) and (bar{z})

Consider the complex number z expressed as:

z x yi, where x and y are real numbers, and i is the imaginary unit.

The conjugate of z is given by:

(bar{z}) x - yi

Substituting and Equating Parts

Substitute z into the equation z2 (bar{z}):

(x yi)2 x - yi

Expanding the left-hand side:

x2 2xyi (yi)2 x - yi

Simplifying using (i^2 -1):

x2 - y2 2xyi x - yi

This equation can now be split into its real and imaginary parts:

Real Part

x2 - y2 x

Imaginary Part

2xy -y

Solving the Imaginary Part

From the imaginary part, we can factor out y:

y(2x - 1) 0

This gives us two possible solutions:

Case 1: (y 0)

If (y 0), then z is a real number. The real part equation simplifies to:

x2 x

This can be factored as:

x(x - 1) 0

Thus, x can be either 0 or 1, leading to the solutions:

z 0 z 1

Case 2: (2x - 1 0)

This equation simplifies to:

x -1/2

Substitute x -1/2 into the real part equation:

(-1/2)2 - y2 -1/2

This simplifies to:

1/4 - y2 -1/2

Rearrange to find:

1/4 1/2 y2

Convert 1/2 to quarters:

1/4 2/4 y2 implies 3/4 y2

Take the square root:

y ±√3/2

This leads to two additional solutions:

z -1/2 √3/2i z -1/2 - √3/2i

Summary of Solutions

The complex numbers z that satisfy z2 (bar{z}) are:

z 0 z 1 z -1/2 √3/2i z -1/2 - √3/2i