Determining the Real Part of a Complex Number Given the Condition z^2 (z - 1)^2

In the realm of complex numbers, the equations involving the square of a complex number can often provide insightful results. This article delves into the algebraic and geometric interpretations of the given equation z^2 (z - 1)^2, focusing on the determination of the real part of the complex number z a bi.

Algebraic Solution

Let's consider a complex number z x yi, where x and y are real numbers, and i is the imaginary unit.

Step 1: Expand Both Sides of the Equation

First, we expand z^2 and (z - 1)^2 using the properties of complex numbers and polynomials.

Calculating z^2:

z^2 (x yi)^2 x^2 2xyi y^2i^2

Simplifying using the fact that i^2 -1, we get:

z^2 x^2 2xyi - y^2 (x^2 - y^2) 2xyi

Calculating (z - 1)^2:

(z - 1)^2 (x - 1 yi)^2 (x - 1)^2 2(x - 1)yi y^2i^2

Simplifying using the fact that i^2 -1, we get:

(z - 1)^2 (x - 1)^2 2(x - 1)yi - y^2 (x^2 - 2x 1 - y^2) 2(x - 1)yi

Step 2: Set the Two Expressions Equal and Simplify

Setting the two expressions equal to each other, we have:

(x^2 - y^2) 2xyi (x^2 - 2x 1 - y^2) 2(x - 1)yi

This equates to having:

x^2 - y^2 x^2 - 2x 1 - y^2 and 2xy 2(x - 1)y

From the real part, simplifying further, we find:

x^2 - y^2 x^2 - 2x 1 - y^2

2x - 1 0

x -frac{1}{2}

Geometric Interpretation

Consider the equation z^2 (z - 1)^2 geometrically. By examining the distance in the complex plane, we can see that it describes the set of points equidistant from -1 and the origin.

Method 2: Geometric Solution

Using the geometric interpretation, we know that z - 1 and z - 0 represent the distances from a point z to the points 1 and 0 (the origin).

The condition z - 1 ±(z - 0) suggests that the locus of points z is the perpendicular bisector of the line segment joining -1 and the origin. This results in a straight line parallel to the imaginary axis, with the real part being equal to -frac{1}{2}.

Conclusion

In conclusion, the real part of the complex number z a bi given the condition z^2 (z - 1)^2 is text{Re}z -frac{1}{2}. This value is derived both algebraically and geometrically, showcasing the power of both approaches in understanding complex number equations.