Explainer: Regions in the Complex Plane

In this explainer, we will learn how to use loci to identify regions in the complex plane.

Before we work with regions in the complex plane, we will briefly recap some of the equations we use to define circles, lines, and half lines in the complex plane.

The types of regions we will consider in this explainer are the ones defined in terms of inequalities such as |π‘§βˆ’π‘§1|<π‘Ÿ and composite regions defined in terms of multiple inequalities. For example, let us consider the region defined by |𝑧+1+𝑖|<2. To represent this on an Argand diagram, we first consider the circle defined by |𝑧+1+𝑖|=2. We notice that this is a circle of radius 2 centered at (βˆ’1,βˆ’1). Since the inequality is a strict inequality, we should draw this circle using a dashed line to represent the fact that the shaded region does not include its boundary.

Once we have drawn the boundary lines, we need to shade the region. To do this correctly, we need to consider whether we are to shade the interior or the exterior of the circle. This will be dictated by the direction of the inequality. If we are considering an inequality in the form of |π‘§βˆ’π‘§1|<π‘Ÿ, we will be concerned with the interior of the circle, whereas if we are considering |π‘§βˆ’π‘§1|>π‘Ÿ, we will be concerned with the exterior of the circle. The figure shows the regions in the complex plane represented by |𝑧+1+𝑖|<2 and |𝑧+1+𝑖|>2.

We will now demonstrate how we can recognize a region of the complex plane given to us in terms of an inequality.

Example 1: Representing Regions in the Complex Plane

Which of the following represents the region of the complex plane defined by βˆ’πœ‹2≀arg(𝑧+3βˆ’2𝑖)<πœ‹4?

Answer

We begin by considering the boundaries of the region. There are two we need to consider: arg(𝑧+3βˆ’2𝑖)=πœ‹4 and arg(𝑧+3βˆ’2𝑖)=βˆ’πœ‹2. Starting with the first of these, we notice that it is a half line emanating from βˆ’3+2𝑖 which makes a positive angle of πœ‹4 with the positive horizontal. Similarly, the second boundary of the region is defined by arg(𝑧+3βˆ’2𝑖)=βˆ’πœ‹2, which is a half line radiating from βˆ’3+2𝑖, which makes a negative angle of πœ‹2 with the positive horizontal. All of the figures include both of these boundary lines.

The region is defined by βˆ’πœ‹2≀arg(𝑧+3βˆ’2𝑖)<πœ‹4; therefore, it will be the region between these two half lines. This rules out both (a) and (e) as possible answers.

We now consider whether the boundary points are included in the region or not. Since arg(𝑧+3βˆ’2𝑖)<πœ‹4 is a strict inequality, this boundary of the region should be represented with a dashed line. This rules out (c) as a possible answer.

We are left with either (b) or (d) as possible answers. The difference between these two figures is whether we have used a solid circle β€’ or a hollow circle ∘ to represent the end point βˆ’3+2𝑖. Recall that we use the solid circle to indicate that the line includes the end point, whereas the hollow circle represents the fact that the end point is not included. Furthermore, recall that the argument of a complex number is not defined for zero.

Therefore, the region defined by βˆ’πœ‹2≀arg(𝑧+3βˆ’2𝑖)<πœ‹4 will not include the end point and hence the correct representation of the region is (b).

Example 2: Describing Regions in the Complex Plane

The figure shows a region in the complex plane.

Write an algebraic description of the shaded region.

Answer

There is more than one way to describe a circle in the complex plane. For example, we could try to use the form |π‘§βˆ’π‘§1|=π‘˜|π‘§βˆ’π‘§2|. However, in most cases, it is a significantly more complicated process to try to find two points whose distance to the given circle are in constant ratio. Therefore, we tend to express regions involving circles using the form |π‘§βˆ’π‘§1|=π‘Ÿ. This is generally much easier to do since all we need to do is identify the center and radius.

In the figure, the center has been given to us. Therefore, we can simply state that 𝑧1=4+𝑖. However, we need to calculate the radius. Firstly, we notice that the circle intersects the imaginary axis at 7𝑖. Therefore, to find the radius, we can either use the Pythagorean theorem or simply evaluate the modulus of the difference of this complex number with 𝑧1.

Hence, π‘Ÿ=|7π‘–βˆ’(4+𝑖)|=|βˆ’4+6𝑖|.

Using the definition of the modulus, we have π‘Ÿ=√(βˆ’4)2+62=√52=2√13.

Notice that we are considering the region exterior to this circle. Therefore, we are considering the points which are further from the center than the radius. Furthermore, a solid line has been used in the figure which represents that the boundary points are included. Hence, the region is described by the following inequality: |π‘§βˆ’4βˆ’π‘–|β‰₯2√13.

To describe more interesting regions in the complex plane, we often want to talk about the regions that are subject to multiple constraints on the value of 𝑧. One of the common ways to do this is by using set notation and set operations.

Recall that the notation {π‘§βˆˆβ„‚βˆΆRe(𝑧)<0} means that we are talking about the set of all complex numbers which have a positive real part. This notation is referred to as set builder notation. Furthermore, when talking about sets, there are a number of common operations we perform.

Unions, Intersections, and Complements

For two sets 𝐴,π΅βŠ‚π‘ˆ, where π‘ˆ is the universal set which contains both 𝐴 and 𝐡, we define the union of 𝐴 and 𝐡 to be the set of all elements that are in 𝐴 or in 𝐡. Notice that when we say β€œor” we are using the inclusive or; that is, we mean that an element is in 𝐴 or 𝐡 or in both 𝐴 and 𝐡. We use the notation 𝐴βˆͺ𝐡 to represent the union. Using set builder notation, we can write 𝐴βˆͺ𝐡={π‘₯βˆˆπ‘ˆβˆΆπ‘₯∈𝐴orπ‘₯∈𝐡}.

The intersection of 𝐴 and 𝐡 is defined as the set of all elements that are in both 𝐴 and 𝐡. We write this as 𝐴∩𝐡. Using set builder notation, we can write 𝐴∩𝐡={π‘₯βˆˆπ‘ˆβˆΆπ‘₯∈𝐴andπ‘₯∈𝐡}.

Finally, we define the complement of a set 𝐴 to be all of the elements that are in the universal set but not in 𝐴. We use the notation 𝐴 to represent this. Using set builder notation, we can write this as 𝐴={π‘₯βˆˆπ‘ˆβˆΆπ‘₯βˆ‰π΄}.

We can represent each one of these using a Venn diagram as follows.

Example 3: Representing Composite Regions in the Complex Plane

We define the regions 𝐴, 𝐡, and 𝐢 in the complex plane as 𝐴={π‘§βˆˆβ„‚βˆΆRe(𝑧)<4},𝐡={π‘§βˆˆβ„‚βˆΆ|𝑧|≀|π‘§βˆ’8βˆ’12𝑖|},𝐢={π‘§βˆˆβ„‚βˆΆ|π‘§βˆ’6βˆ’5𝑖|<5}.

Which of the following figures could represent the region of the complex plane defined by ο€Ίπ΄βˆ©π΅ο†βˆͺ𝐢?

Answer

The best approach to solving this problem is first to consider the regions represented by 𝐴, 𝐡, and 𝐢 separately. Beginning with 𝐴, we will first consider the region represented by 𝐴={π‘§βˆˆβ„‚βˆΆRe(𝑧)<4} and then its complement. Notice that 𝐴 represents all the complex numbers with real parts less than 4. Hence, this will be the region to the left of the line π‘₯=4. Notice that since the inequality is strict, we will represent the boundary with a dashed line. The figure below shows the region 𝐴.

We, however, do not need to consider the region 𝐴; we need to consider 𝐴. This will be the region of all the points not in 𝐴. We can think about this visually, by inverting the regions we have shaded in the plane. Notice, when doing this, that we need to change the boundary from a dashed line to a solid line. Alternatively, we can think about this more abstractly as follows. Since 𝐴 represents all complex numbers with a real part less than 4, 𝐴 will represent all complex numbers with a real part greater than or equal to 4. Whichever approach we use, we can represent