Optimizing the Complex Expression (z(z - 1)) Using Geometric Insights and Calculus

In this article, we will delve into the process of optimizing the complex expression (z(z - 1)) by leveraging geometric interpretations and calculus techniques. This exploration will involve a detailed analysis of the expression, its geometric significance, and how to determine its minimum value.

Geometric Interpretation

Let (z x iy) where (x) and (y) are real numbers. The complex number (z) can be thought of as a point in the complex plane, and the expression (z(z - 1)) can be interpreted as the sum of two distances. Specifically:

The distance from the origin (0 0i) (point ((0, 0))) to the point (z x iy). The distance from the point (z x iy) to the point (1 0i) (point ((1, 0))).

The expression (z(z - 1)) represents the sum of these two distances. To find the minimum value of (z(z - 1)), we need to minimize the total distance from the origin to the point (z) and from (z) to the point (1 0i).

According to the triangle inequality, the shortest path between two points in a plane is the straight line segment connecting them. Therefore, the minimum value of the sum of these distances occurs when (z) lies on the line segment connecting the origin ((0, 0)) and the point ((1, 0)).

Geometric Optimization

Let's denote the complex number (z) as follows:

When (z 0 0i), the distance to (1 0i) is 1 unit. When (z 1 0i), the distance to (0 0i) is 1 unit. For any intermediate point (z) on the line segment, the sum of the distances is always 1.

Mathematically, if (z t) where (t) ranges from 0 to 1 (on the line segment), then:

[z t]

[z - 1 t - 1 1 - t]

Thus, the expression becomes:

[z(z - 1) t(1 - t) 1 - t^2]

As (t) varies from 0 to 1, the minimum value of (1 - t^2) is (1), which occurs at (t 0) or (t 1).

Algebraic Optimization

Another approach involves expressing (z) in polar form as (z re^{iphi} r cos phi ir sin phi). Then, (z - 1 r cos phi - 1 ir sin phi), and the modulus (distance) is:

[|z - 1| sqrt{(r cos phi - 1)^2 (r sin phi)^2} sqrt{r^2 cos^2 phi - 2r cos phi 1 r^2 sin^2 phi} sqrt{r^2 - 2r cos phi 1}]

Therefore, the expression (z(z - 1)) becomes:

[zz - 1 r sqrt{r^2 - 2r cos phi 1}]

If (r) is fixed, the minimum value is achieved when the cosine term is maximized. The maximum value of (cos phi) is 1, which occurs when (phi 0, 2pi, 4pi, dots). Substituting (cos phi 1) gives:

[zz - 1 r sqrt{r^2 - 2r 1} r sqrt{(r - 1)^2} r|1 - r| r(1 - r) 1 - r^2]

When (r 1), the expression simplifies to:

[1 - 1^2 1 - 1 0]

However, this is not the minimum for the original expression (z(z - 1)), which simplifies to 2 when (r 1).

The minimum value of (z(z - 1)) is still achieved at (z t) on the line segment from (0 0i) to (1 0i), and this minimum value is (boxed{1}).