In this explainer, we will learn how to identify complex numbers plotted on an Argand diagram and discover their geometric properties.
One of the most fascinating things about complex numbers is that they introduce a geometric interpretation to familiar arithmetic operations. When working with purely real numbers, we could express them on a one-dimensional number line. Thinking in this way gave us additional insight into their properties. Whereas, with the introduction of , we can add a second dimension and consider complex numbers as points in a plane, we will find that visualizing complex numbers in this way will give us additional insight into their properties.
Definition: Argand Diagram
The complex numbers can be represented geometrically on a two-dimensional plane with two perpendicular axes representing the real and imaginary parts of the number respectively. The complex number is represented by the point in Cartesian coordinates. This plane is referred to as the complex plane, the Argand plane, or the Argand diagram.
Let us begin with a simple example where we will determine the Cartesian coordinates of a complex number on an Argand diagram.
Example 1: Coordinates of Complex Numbers on an Argand Diagram
If the number is represented on an Argand diagram by the point , determine the Cartesian coordinates of that point.
From the definition of the Argand diagram, we know that the complex number will be represented by a point with Cartesian coordinates .
Hence, will be represented by the point .
In our next example, we will identify complex numbers and their conjugates from an Argand diagram.
Example 2: Representing Complex Numbers on an Argand Diagram
Seven complex numbers , , , , , , and are represented on the Argand diagram.
- Which of the complex numbers is ?
- What complex number is represented by ?
- Which complex number has equal real and imaginary parts?
- Which two complex numbers are a conjugate pair? What is their geometric relationship?
According to the definition of the Argand diagram, the complex number will be represented by the point . Reading these coordinates off the plane, we find that .
We begin by reading off the coordinates of from the Argand diagram as which according to the definition represent the complex number . Hence, .
A complex number with equal real and imaginary parts will lie on the line . Drawing this line on the Argand diagram, we find that only one of the numbers lies on this line: .
Recall that the complex conjugate of is . Therefore, we could plot at the point and we could plot at the point . Hence, the points representing a complex number and its conjugate both have the same -value but opposite -values. Looking at the diagram we have been given, we see there are only two pairs of points with the same -coordinates: and and and . Considering and , we find that the -coordinate of is 3 whereas the -coordinate of is . Hence, these two are not a complex conjugate pair, whereas, considering and , we find that the -coordinate of is 3 and the -coordinate of is . Therefore, they are a complex conjugate pair. Furthermore, we can see that, as a complex conjugate pair, the points and are related by reflection in the real axis (-axis).
Using Argand diagrams, we can interpret addition of complex numbers geometrically. For two complex numbers and , their sum can be expressed as . If we plot these numbers on the Argand diagram, we would plot the points , , and . Considering these points suggests some sort of equivalence between complex numbers and vectors. This is in fact true, and for a number of operations with complex numbers, considering them to be vectors in the Argand diagram is actually most informative. In particular, for addition and subtraction, we can consider the two complex numbers and to represented vectors with components , respectively. In this way, addition of complex numbers can be interpreted as vector addition. For example, adding the complex numbers and using the parallelogram rule can be represented as follows.
In the next example, we will calculate the addition of two complex numbers using the graphical approach shown above.
Example 3: Finding the Sum of Two Complex Numbers Represented on an Argand Diagram
Using the Argand diagram shown, find the value of .
One way to add complex numbers given in an Argand diagram is to read off the values and add them algebraically. We recall that the point on an Argand diagram represents the complex number . Thus, we find expressions for and by identifying the points.
We can see that is at , so , and is at , so .
Now we can add the numbers by adding their real and imaginary components respectively:
Thus, the answer is .
Additionally, we note that if we plot as the point on the diagram, it is one of the ends of the diagonal of a parallelogram whose opposite vertices are and .
Having just seen how the addition of complex numbers can be represented using the parallelogram law on an Argand diagram, we might wonder how other geometric constructions correspond to operations with complex numbers. For instance, what about the midpoint between two complex numbers? Let us explore this idea in the next example.
Example 4: Finding the Midpoint of Two Complex Numbers Drawn on an Argand Diagram
What complex number lies at the midpoint of and on the given complex plane?
To find the midpoint between two points, there are various methods available to us. One way is to use the formula for the midpoint of a line segment. Specifically, for given endpoints and , the midpoint is
From the Argand diagram, we can see that has coordinates and has coordinates . Therefore, using the formula, their midpoint is
Recall that the point on an Argand diagram represents the complex number . Thus, the complex number lying at the midpoint of and is .
Let us explore the significance of the answer to the last example. By reading the points off of the Argand diagram, we can see that and are located at coordinates and , respectively, which means that they are equal to
If we consider their sum, we have
Note that this is double , which was the midpoint we calculated. This is in fact a general property that applies to the midpoint of any two numbers on the complex plane.
Property: Midpoint of Complex Numbers
The midpoint of the line segment between two complex numbers and on an Argand diagram corresponds to the complex number given by
Let us further explore geometric relations between points on an Argand diagram and their corresponding complex counterparts in the following example.
Example 5: Finding the Multiplication of a Complex Number on an Argand Diagram by a Real Number
Using the Argand diagram below, find the value of .
We are being asked to find what is given on an Argand diagram, which we can determine by finding the complex number that refers to and multiplying it by .
Recall that a point with coordinates on an Argand diagram refers to the complex number . Since is at , this means that .
Now, we can multiply this number by to find the new number:
Although not required, we can plot the effect of the multiplication from the previous question on a diagram.
Interpreting this geometrically, we can see that the distance from the origin has been doubled, but in the opposite direction. This corresponds to a dilation of the point with a scale factor of centered at the origin. Alternatively, we can think of it as a rotation by radians about the origin followed by a dilation with a scale factor of 2. This is a concept we can generalize.
Property: Multiplication by Real Numbers on an Argand Diagram
If a complex number is multiplied by a real number , this corresponds to a dilation with scale factor centered at the origin on the Argand diagram.
We will now turn our attention to the geometric interpretation of multiplying by .
Example 6: Finding the Quadrant in Which a Complex Number Lies on an Argand Diagram
Consider the complex number . If is represented on an Argand diagram by the point , in which quadrant of the Argand plane does lie?
The most straightforward way to tackle this problem is to start by calculating directly:
Now, let us draw this on an Argand diagram (along with the original point for reference). Recall that the number corresponds to the point on an Argand diagram. Thus, we will plot the points and for and respectively.
Using the convention that the top-right quadrant is the first quadrant, and continuing counterclockwise, we can see that our point (i.e., the point representing ) lies in the second quadrant.
Let us geometrically interpret the transformation from the previous example. In the Argand diagram, we can see that the point remained at the same distance from the origin, but its angle with the real axis changed. Specifically, the original point was rotated around the origin by an angle of radians (positive since it is a counterclockwise rotation). This is a quality that holds for general cases.
Property: Multiplication by 𝑖 on an Argand Diagram
If a complex number is multiplied by , this corresponds to a (counterclockwise) rotation by about the origin on the Argand diagram.
Incidentally, we can see how this ties in with the previous property of multiplication by real numbers. If a complex number is multiplied by twice, this is the same as multiplying by . Using the definition from before, this would be a dilation with scale factor . Equivalently, using our new definition, this would be two rotations by (or one rotation by ), which ends up having the same effect.
Let us recap the main things we have learned in this explainer.
- Complex numbers can be interpreted as points or vectors in the Argand diagram.
- Many operations with complex numbers can be interpreted geometrically.
|Adding||Translation by vector|
|Conjugation||Reflection in the real axis|
|Multiplication by a real number||Dilation centered at the origin with scale factor|
|Multiplication by||Rotation counterclockwise by radian centered at the origin|