Explainer: Distance and Midpoints on the Complex Plane

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 𝑎 and 𝑏 is given by |𝑎𝑏|.

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 3𝑖 and 7𝑖 in the complex plane?

Answer

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: 7𝑖(3𝑖)=10𝑖.

Since we consider the distance of 𝑖 from the origin to be the same as the distance of 1 from the origin, we can conclude that the distance between 3𝑖 and 7𝑖 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 𝑖 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 𝑧 and 𝑧 shown on the complex plane. Give your answer in an exact simplified form.

Answer

We begin by finding the horizontal distance between 𝑧 and 𝑧. From the figure, we can see that the 𝑥-coordinates of 𝑧 and 𝑧 are 2 and 6 respectively. Therefore, the horizontal distance between them is given by |26|=8. Similarly, we can find the vertical distance. Noting that the 𝑦-coordinates of 𝑧 and 𝑧 are 7 and 3, respectively, we find that the vertical distance between them is |7(3)|=10. We can represent this on an Argand diagram as follows.

To find the distance 𝑑 between these two points, we can use the Pythagorean theorem which states that 𝑑=8+10=241.

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 𝑎+𝑏𝑖 and 𝑥+𝑦𝑖, where 𝑎, 𝑏, 𝑥, and 𝑦 are real?

Answer

We begin by finding the horizontal distance between 𝑎+𝑏𝑖 and 𝑥+𝑦𝑖. The horizontal distance is equivalent to the difference of their real parts. Hence, the horizontal distance is given by |𝑎𝑥|. 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 |𝑏𝑦|.

To find the distance 𝑑 between these two points, we can use the Pythagorean theorem which, within this context, states that 𝑑=|𝑎𝑥|+|𝑏𝑦|.

For any real number 𝑐, |𝑐|=𝑐. 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: 𝑑=(𝑎𝑥)+(𝑏𝑦).

Notice that if we set one of the complex number to zero, we get the formula for the distance from 𝑧=𝑎+𝑏𝑖 to the origin to be 𝑑=𝑎+𝑏.

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 𝑧=𝑎+𝑏𝑖 is defined as |𝑧|=𝑎+𝑏.

Equivalently, this can be written as |𝑧|=((𝑧))+((𝑧)).ReIm

It represents the distance of the complex number 𝑧 from the origin.

Notice that we can use the concept of the modulus to define distance between two complex numbers 𝑧 and 𝑧 as |𝑧𝑧|.

Distance between Two Complex Numbers

For two complex numbers 𝑧=𝑎+𝑏𝑖 and 𝑧=𝑎+𝑏𝑖, we define the 𝑑 distance between them to be 𝑑=|𝑧𝑧|. Equivalently, this can be written as 𝑑=(𝑎𝑎)+(𝑏𝑏).

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 𝑤 lies at a distance of 52 from 𝑧=3+5𝑖 and at a distance of 45 from 𝑧=62𝑖. Is the triangle formed by the points 𝑤, 𝑧, and 𝑧 a right triangle?

Answer

We would like to know whether the triangle formed by the points 𝑤, 𝑧, and 𝑧 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 𝑎, 𝑏, and 𝑐 satisfy 𝑎+𝑏=𝑐, where 𝑐 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 𝑧, and 𝑧. Recall that we can find the distance between two complex numbers by evaluating the modulus of their difference. Hence, the distance between 𝑧, and 𝑧 is given by 𝑑=|𝑧𝑧|.

Substituting in the values of 𝑧 and 𝑧, we have 𝑑=|3+5𝑖(62𝑖)|=|9+7𝑖|.

Using the definition of the modulus, we have 𝑑=9+7=130.

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 𝑧 and 𝑧. Hence, we need to verify whether 𝑑 is equal to the sum of the squares of the other two sides:52+45=50+80=130=𝑑.

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 3+5𝑖 and 713𝑖.

Answer

To find the midpoint 𝑚 of the two complex numbers. We consider its real and imaginary parts separately. The real part of 𝑚 will have a real part which is equal to the average of the real parts of 3+5𝑖 and 713𝑖.

If this is not obvious, you can think of finding the distance between the real parts which we could write as 𝑏𝑎, where 𝑎 and 𝑏 are the real parts of the two numbers 𝑏>𝑎. Then, we halve it to find the distance to the midpoint. This is equal to 12(𝑏𝑎). Finally, we can add this to the smaller real part 𝑎, which gives 𝑎+12(𝑏𝑎)=𝑎+𝑏2.

Therefore, the real part of 𝑚 is equal to 7+32=5. Similarly, the imaginary part is the average of the imaginary parts of 3+5𝑖 and 713𝑖. Hence, the imaginary part of 𝑚 is equal to 5132=4. Hence, the complex number representing the midpoint of 3+5𝑖 and 713𝑖 is 𝑚=54𝑖.

The previous example demonstrates that the formula for the midpoint 𝑚 of two complex numbers 𝑧 and 𝑧 can be simply written as 𝑚=𝑧+𝑧2.

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 𝑧, 𝑚, and 𝑧 be complex numbers such that 𝑚 lies at the midpoint of the line segment connecting 𝑧 to 𝑧. Given that 𝑧=4+5𝑖 and 𝑚=12+20𝑖 find 𝑧.

Answer

The midpoint of a pair of complex numbers is their average. Hence, 𝑚=𝑧+𝑧2.

We would like to find the value of the complex number 𝑧. We can therefore rearrange this formula to make 𝑧 the subject. First, by multiplying by 2, we get 2𝑚=𝑧+𝑧.

Hence, 𝑧=2𝑚𝑧. Substituting the values of 𝑧 and 𝑚 into this formula gives 𝑧=2(12+20𝑖)(4+5𝑖)=28+35𝑖.

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 𝑧 and 𝑧 to represent the vectors z and z whose tails lie at the origin and heads at the points 𝑧 and 𝑧 respectively. We can consider the vector from 𝑧 to 𝑧. We can write this as +zz since we traverse vector z in the negative direction and then traverse z in the positive direction.

If we want to get to the midpoint of 𝑧 and 𝑧, we can traverse vector z in the positive direction and then add half of the vector that represents the displacement from 𝑧 to 𝑧. We can write this in vector notation as zzzzz+2=+2.

This corresponds to the equivalent calculation involving complex numbers: 𝑧+𝑧2.

Using this interpretation, it is easy to generalize this formula to find any point on the line passing through 𝑧 and 𝑧: instead of adding half of the vector that represents the displacement from 𝑧 to 𝑧, we can add any arbitrary multiple as follows: zzzzz+𝑘()=(1𝑘)+𝑘.

Once again, this corresponds to the equivalent calculation involving complex numbers: (1𝑘)𝑧+𝑘𝑧.

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 𝑎, 𝑏, and 𝑐 in the complex plane. Find an expression for the centroid of the triangle in terms of 𝑎, 𝑏, and 𝑐. You can use the fact that the centroid divides the median in a ratio of 21.

Answer

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 𝑚 of the side connecting 𝑏 to 𝑐. Using the formula for the midpoint of two complex numbers, we have 𝑚=𝑏+𝑐2.

We can now apply the fact that the centroid divides the median in a ratio of 21. In particular, the centroid lies a third of the way from 𝑚 to 𝑎.

The vector from 𝑚 to 𝑎 can be represented by the complex number 𝑎𝑚. To get to point 𝑐, we need to traverse the vector 𝑚 and then traverse a third of this vector from 𝑚 to 𝑎. Hence, 𝑐=𝑚+𝑎𝑚3=𝑎+2𝑚3.

Substituting our expression for 𝑚 in terms of 𝑏 and 𝑐 gives 𝑐=𝑎+23=𝑎+𝑏+𝑐3.

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.

Key Points

  1. The distance 𝑑 between two complex numbers 𝑧=𝑥+𝑦𝑖 and 𝑧=𝑥+𝑦𝑖 can be expressed in terms of the modulus of a complex number as 𝑑=|𝑧𝑧|, which is equivalent to 𝑑=(𝑥𝑥)+(𝑦𝑦).
  2. The midpoint 𝑚 of two complex numbers 𝑧 and 𝑧 lies at their arithmetic average; that is, 𝑚=𝑧+𝑧2.
  3. 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 𝑚 that is some given fraction 0𝑘1 along the line segment from 𝑧 to 𝑧 is given by 𝑚=(1𝑘)𝑧+𝑘𝑧. If 𝑘>1, 𝑚 lies on the line that extends beyond 𝑧, and if 𝑘<0, 𝑚 lies on the line that extends beyond 𝑧.

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