In this explainer, we will learn how to find the distance and midpoint of two complex numbers in the complex plane.
The geometry of the complex plane and the simplicity in manipulating complex numbers can be combined to create powerful tools for solving geometric problems. Complex numbers provide us with simple ways to think about translations, dilations, and even rotations. This can often lead to much cleaner and simpler derivations of geometric theorems and facts. In this explainer, we will focus on the concepts of distance and finding midpoints. In many ways, these concepts are just the tip of the iceberg. However, gaining a strong foundation in the basics of the geometry of the complex plane will serve us extremely well when dealing with more advanced concepts.
Broadly speaking, we can interpret complex numbers in two distinct geometric ways: as points in a plane or as vectors in the plane. As we will see, both of these interpretations will help us as we work with complex numbers geometrically.
The idea of a distance between two numbers is something we have come across before. When dealing with real numbers, we define the absolute value of a real number to be its distance from the origin. Once we have introduced this concept, we are able to define the distance between two real numbers as the absolute value of their difference. Hence, the distance between the real numbers and is given by .
We will find that we are able to do similar things when working with complex numbers. We will begin by looking at a simple example of the distance between two imaginary numbers.
Example 1: Distance between Two Imaginary Numbers
What is the distance between the numbers and in the complex plane?
Since we have two purely imaginary numbers, we can consider their difference in exactly the same way as we would consider the difference between two real numbers. Firstly, we find their difference: .
Since we consider the distance of from the origin to be the same as the distance of 1 from the origin, we can conclude that the distance between and is 10.
For purely imaginary numbers, we do not really need to introduce any new concepts. We simply use the fact that the distance of from the origin is the same as the distance of 1 from the origin. However, complex numbers in general have both a horizontal component (representing the real part) and a vertical one (representing the imaginary part). Therefore, to find the distance between two complex numbers, we will need to use the Pythagorean theorem (or, equivalently, the Cartesian distance formula). We will use the next example to demonstrate how we do this.
Example 2: Distance between Two Complex Numbers
Find the distance between the complex numbers and shown on the complex plane. Give your answer in an exact simplified form.
We begin by finding the horizontal distance between and . From the figure, we can see that the -coordinates of and are and 6 respectively. Therefore, the horizontal distance between them is given by . Similarly, we can find the vertical distance. Noting that the -coordinates of and are 7 and , respectively, we find that the vertical distance between them is . We can represent this on an Argand diagram as follows.
To find the distance between these two points, we can use the Pythagorean theorem which states that
Notice that the calculation in the last question is equivalent to finding the difference between the two complex numbers. Then, considering the result as a vector, we find its magnitude. The next example demonstrates how, by using the same method as the previous example, we can derive a general formula for the distance between two complex numbers.
Example 3: General Form for the Distance between Two Complex Numbers
What is the distance between the complex numbers and , where , , , and are real?
We begin by finding the horizontal distance between and . The horizontal distance is equivalent to the difference of their real parts. Hence, the horizontal distance is given by . Similarly, we find the vertical distance between them. This is equivalent to the difference of their imaginary parts. Therefore, the vertical distance between them is given by .
To find the distance between these two points, we can use the Pythagorean theorem which, within this context, states that
For any real number , . Hence, we can ignore the modulus signs. Then, finally, we take the positive square root to derive the following formula for the distance between two arbitrary complex numbers:
Notice that if we set one of the complex number to zero, we get the formula for the distance from to the origin to be
This is the generalization of the absolute value to complex numbers and we give it a special name; we call it the modulus.
Definition: Modulus of a Complex Number
The modulus of a complex number is defined as
Equivalently, this can be written as
It represents the distance of the complex number from the origin.
Notice that we can use the concept of the modulus to define distance between two complex numbers and as .
Distance between Two Complex Numbers
For two complex numbers and , we define the distance between them to be Equivalently, this can be written as
Notice that, in this way, we have defined distance between two complex numbers in a way that is completely analogous to the way we define the distance between two real numbers. We will now turn our attention to how we can apply this knowledge to geometric problems in the complex plane.
Example 4: Geometry in the Complex Plane
A complex number lies at a distance of from and at a distance of from . Is the triangle formed by the points , , and a right triangle?
We would like to know whether the triangle formed by the points , , and is a right triangle or not. We could try to calculate the angles in the triangle and see whether one of them is a right angle. However, an easier approach would be to use the reverse Pythagorean theorem to check whether the lengths of the sides , , and satisfy , where is the length of the longest side. We have already been given the lengths of two of the sides, so we only need to calculate the distance between the points , and . Recall that we can find the distance between two complex numbers by evaluating the modulus of their difference. Hence, the distance between , and is given by
Substituting in the values of and , we have
Using the definition of the modulus, we have
We now can check whether these three side lengths satisfy the Pythagorean theorem. To do this, we first need to identify the longest side, which is the side between and . Hence, we need to verify whether is equal to the sum of the squares of the other two sides:
Since the side lengths satisfy the Pythagorean theorem, we can conclude that the triangle is indeed a right triangle.
We will now turn our attention to the idea of the midpoint between two complex numbers. Once again, we will see how the definition for real numbers naturally generalizes to complex numbers.
Example 5: The Midpoint of Two Complex Numbers
Find the midpoint of and .
To find the midpoint of the two complex numbers. We consider its real and imaginary parts separately. The real part of will have a real part which is equal to the average of the real parts of and .
If this is not obvious, you can think of finding the distance between the real parts which we could write as , where and are the real parts of the two numbers . Then, we halve it to find the distance to the midpoint. This is equal to . Finally, we can add this to the smaller real part , which gives
Therefore, the real part of is equal to . Similarly, the imaginary part is the average of the imaginary parts of and . Hence, the imaginary part of is equal to . Hence, the complex number representing the midpoint of and is .
The previous example demonstrates that the formula for the midpoint of two complex numbers and can be simply written as
Notice that this is the exact analog to the formula for the midpoint of two real numbers (also referred to as the mean).
Example 6: Applications of the Midpoint
Let , , and be complex numbers such that lies at the midpoint of the line segment connecting to . Given that and find .
The midpoint of a pair of complex numbers is their average. Hence,
We would like to find the value of the complex number . We can therefore rearrange this formula to make the subject. First, by multiplying by 2, we get
Hence, Substituting the values of and into this formula gives
There is an alternative way to interpret the formula for the midpoint of two complex numbers by using vectors. If we consider the complex numbers and to represent the vectors and whose tails lie at the origin and heads at the points and respectively. We can consider the vector from to . We can write this as since we traverse vector in the negative direction and then traverse in the positive direction.
If we want to get to the midpoint of and , we can traverse vector in the positive direction and then add half of the vector that represents the displacement from to . We can write this in vector notation as
This corresponds to the equivalent calculation involving complex numbers: .
Using this interpretation, it is easy to generalize this formula to find any point on the line passing through and : instead of adding half of the vector that represents the displacement from to , we can add any arbitrary multiple as follows:
Once again, this corresponds to the equivalent calculation involving complex numbers: .
In the final example, we will apply this result to derive an interesting result related to the geometry of triangles.
Example 7: Centroid
A triangle has its vertices at points , , and in the complex plane. Find an expression for the centroid of the triangle in terms of , , and . You can use the fact that the centroid divides the median in a ratio of .
Recall that a median of a triangle is the line that passes through one of the vertices and the midpoint of the opposite edge. Hence, we should start by finding the midpoint of one of the sides. We will find the midpoint of the side connecting to . Using the formula for the midpoint of two complex numbers, we have
We can now apply the fact that the centroid divides the median in a ratio of . In particular, the centroid lies a third of the way from to .
The vector from to can be represented by the complex number . To get to point , we need to traverse the vector and then traverse a third of this vector from to . Hence,
Substituting our expression for in terms of and gives
The last example demonstrates how, by using complex numbers to represent points in the plane, we can derive beautiful and simple results with relative ease.
- The distance between two complex numbers and can be expressed in terms of the modulus of a complex number as which is equivalent to
- The midpoint of two complex numbers and lies at their arithmetic average; that is,
- The midpoint formula can be generalized to find a point any fraction of the way between two points or even on the line extending beyond them. A point that is some given fraction along the line segment from to is given by If , lies on the line that extends beyond , and if , lies on the line that extends beyond .