Geometric Interpretation of Complex Regions: Sketching the Region Defined by (z - 2i

Understanding and visualizing regions in the complex plane can often be a daunting task, especially when dealing with inequalities. In this article, we will explore a specific region in the complex plane defined by the inequality (|z - 2i|

Geometric Representation of Complex Numbers

Complex numbers can be represented as points in the complex plane, where the horizontal axis corresponds to the real part (re) and the vertical axis corresponds to the imaginary part (im). A complex number (z) can be written as (z x yi), where (x) and (y) are real numbers, and (i) is the imaginary unit. In vector form, this can be represented as (vec{z} langle x, y rangle).

Distance Between Complex Points

The distance between two complex points (z_1 x_1 y_1i) and (z_2 x_2 y_2i) is given by the modulus of their difference, which in vector form is (vec{z_1} - vec{z_2}). Mathematically, this is expressed as:

[ |z_1 - z_2| sqrt{(x_1 - x_2)^2 (y_1 - y_2)^2} ]

The Region Defined by (|z - 2i|

Consider the inequality (|z - 2i| [ |(x yi) - (0 2i)| This simplifies to:

[ sqrt{x^2 (y - 2)^2} Squaring both sides, we get:

[ x^2 (y - 2)^2 This inequality represents a circle in the complex plane with its center at (0 2i) (or (0, 2) in Cartesian coordinates) and a radius of 1. The region defined by this inequality includes all complex numbers (z x yi) that are within a distance of 1 from the point (2i).

Sketching the Region

To sketch the region, follow these steps:

1. **Identify the Center**: The center of the circle is at ((0, 2)) in the complex plane.2. **Determine the Radius**: The radius is 1.3. **Draw the Circle**: Draw a circle with the center at ((0, 2)) and a radius of 1. This circle includes all points that satisfy the inequality.4. **Shade the Interior**: Shade the interior of the circle to represent the region defined by (|z - 2i| By following these steps, you can accurately sketch the region in the complex plane described by the inequality (|z - 2i|

Conclusion

Understanding the geometric interpretation of regions in the complex plane is crucial for solving complex number problems and visualizing mathematical concepts. By leveraging the properties of complex numbers and their geometric representation, we can easily sketch and describe regions such as (|z - 2i|

Keywords

Complex Plane, Complex Numbers, Geometric Interpretation