Question

Describe geometrically the set of points in the complex plane satisfying the following equations. $$\operatorname{Im} z<0$$

Short answer

The set of points below the real axis in the complex plane.

Step-by-step solution

Understand the Expression

The given inequality is \(\operatorname{Im} z < 0\). \(z\) represents a complex number, which can be written in the form \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part.

Interpret the Imaginary Part

The imaginary part \(\operatorname{Im} z\) of the complex number \(z = x + yi\) is \(y\). The given inequality therefore becomes \(y < 0\).

Geometric Interpretation

The equation \(y < 0\) means that we are considering all points where the imaginary part of the complex number is negative. In the complex plane, this corresponds to all points that lie below the real axis (which is the x-axis).

Conclude the Geometric Region

Thus, the set of points satisfying \(\operatorname{Im} z < 0\) is the entire region below the real axis in the complex plane.

Key concepts

complex numbers

A complex number is a number that can be written as a combination of a real part and an imaginary part. It is generally represented as \(z = x + yi\), where \(x\) is the real part and \(y\) is the imaginary part. Complex numbers are useful in various fields, such as engineering, physics, and applied mathematics.

• The real part of the complex number \(x + yi\) is \(x\).
• The imaginary part is \(y\), which is written with the imaginary unit \(i\), where \(i\) is defined by the property \(i^2 = -1\).

Complex numbers are visualized on a complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part. Each complex number corresponds to a unique point on this plane.
The complex plane helps in understanding complex numbers and their operations visually. For instance, adding two complex numbers is analogous to vector addition in this plane.

imaginary part

The imaginary part of a complex number is the component multiplied by \(i\). Given a complex number \(z = x + yi\), the imaginary part is \(y\).
The property of the imaginary part is crucial in understanding the inequality \(\operatorname{Im} z < 0\). Here:

• \(\operatorname{Im} z\) represents the imaginary part of \(z\).
• The inequality \(\operatorname{Im} z < 0\) indicates that the imaginary part \(y\) must be a negative number.

This condition constrains the values of \(y\) in the complex number \(z\). Essentially, any complex number where the imaginary part \(y\) is negative satisfies this inequality. This will become more evident in the geometric interpretation.

geometric interpretation

Geometric interpretation of complex numbers involves plotting them on the complex plane. For the given inequality \(\operatorname{Im} z < 0\):

• Given a complex number \(z = x + yi\), the imaginary part \(\operatorname{Im} z = y\) must be negative.
• On the complex plane, this implies that the point \((x, y)\) lies below the real axis (x-axis).

The real axis separates the regions where the imaginary part is positive (above the real axis) and negative (below the real axis). Therefore, the inequality \(\operatorname{Im} z < 0\) geometrically describes the half-plane below the real axis.
This is a straightforward way to visualize the conditions that complex numbers must satisfy. It shows us that all points in the lower half-plane of the complex plane will fulfill the inequality \(\operatorname{Im} z < 0\).