How to Draw the Region 1 ≤ |z| ≤ 2 in the Complex Plane

The complex plane, also known as the Argand plane, is a fundamental concept in complex analysis. Understanding how to draw regions involving the modulus (absolute value) of a complex number is crucial for many applications. This guide will walk you through the steps to visualize the region defined by the inequality 1 ≤ |z| ≤ 2 in the complex plane.

Understanding the Notation

In the complex plane, the symbol |z| represents the modulus or absolute value of the complex number z. The inequality 1 ≤ |z| ≤ 2 describes the region between two circles centered at the origin.

Identifying the Circles

Two circles are involved in this description: The circle defined by |z| 1 has a radius of 1. The circle defined by |z| 2 has a radius of 2.

Draw the Circles

Circle 1 (|z| 1): Draw a small circle with a radius of 1 centered at the origin (0,0). All points on this circle satisfy the equation |z| 1. Circle 2 (|z| 2): Draw a larger circle with a radius of 2 also centered at the origin. All points on this circle satisfy the equation |z| 2.

By drawing these two circles, you will have two concentric circles in the complex plane.

Shading the Region

The region you want to highlight is the area between the two circles, excluding the points on the circles themselves. This area forms an annular region (a doughnut-shaped region) where the modulus of z lies between 1 and 2.

Visualization

An illustration of the circles and the shaded region in the complex plane.

The inner circle, with radius 1, is represented by the smaller circle. The outer circle, with radius 2, is represented by the larger circle. The shaded area, which is not shown in this ASCII representation, is the annular region between the two circles.

Conclusion

The region 1 ≤ |z| ≤ 2 consists of all points in the complex plane that lie within the annular region between the two circles, excluding the boundaries of the circles themselves. This region is also referred to as an annulus in mathematical terms.