01 / 20

Distance & Midpoints
in the Complex Plane

Learn to calculate the distance between complex numbers using the modulus of their difference, and find the midpoint of a segment by averaging endpoints.

Algebra 2

What Is the Complex Plane?

Every complex number \( z = a + bi \) can be plotted as a point in a two-dimensional plane. The horizontal axis represents the real part \( a \), and the vertical axis represents the imaginary part \( b \).

Re Im -2 -1 1 2 3
\(i\)
\(2i\)
\(-i\)
\(3+2i\)
\(-1-i\)

From Coordinates to Complex Numbers

You already know the distance formula for two points \( (x_1, y_1) \) and \( (x_2, y_2) \):

Since a complex number \( z = a + bi \) corresponds to the point \( (a, b) \), the same formula applies. In complex-number language, this distance equals the modulus (absolute value) of the difference \( z_2 - z_1 \).

The Modulus of a Complex Number

The modulus of \( z = a + bi \) measures its distance from the origin \( (0,0) \):

Re Im O
\(z = 3 + 4i\)
\(a = 3\)
\(b = 4\)
\(|z| = 5\)

Distance as Modulus of the Difference

Re Im
\( z_1 = 1 + i \)
\( z_2 = 4 + 5i \)
\( \Delta a = 3 \)
\( \Delta b = 4 \)
\( d = 5 \)

Midpoint as the Average

Re Im
\( z_1 = 2 + i \)
\( z_2 = 6 + 3i \)
\( M = 4 + 2i \)
 4
\(2i\)

Distance vs. Midpoint

Distance
Uses subtraction then modulus:
\( |z_2 - z_1| \)

Result: a non-negative real number (a length).
Midpoint
Uses addition then average:
\( \frac{z_1 + z_2}{2} \)

Result: a complex number (a point on the plane).

Step-by-Step Approach

1
Identify \( z_1 \) and \( z_2 \). Write each in the form \( a + bi \).
2
For distance: Compute \( z_2 - z_1 \), then find the modulus \( |z_2 - z_1| = \sqrt{(\Delta a)^2 + (\Delta b)^2} \).
3
For midpoint: Compute \( \frac{z_1 + z_2}{2} \) by averaging real and imaginary parts separately.
4
Simplify. Simplify radicals for distance; express the midpoint in \( a + bi \) form.

Example 1: Integer Coordinates

Find the distance and midpoint between \( z_1 = 2 + 3i \) and \( z_2 = 5 - i \).

Re Im
\( z_1 = 2+3i \)
\( z_2 = 5-i \)
\( M = 3.5+i \)
\( d = 5 \)

Example 2: Rational and Irrational Distances

Find the distance between \( z_1 = -3 + 2i \) and \( z_2 = 1 + 5i \).

1
Subtract: \( z_2 - z_1 = (1-(-3)) + (5-2)i = 4 + 3i \)
2
Modulus: \( |4+3i| = \sqrt{16 + 9} = \sqrt{25} = \) 5

Now try \( z_1 = 1 + i \) and \( z_2 = 3 + 4i \):

1
Subtract: \( z_2 - z_1 = 2 + 3i \)
2
Modulus: \( |2+3i| = \sqrt{4+9} = \) \( \sqrt{13} \)

Example 3: Midpoint with Negatives

Find the midpoint between \( z_1 = -4 + 6i \) and \( z_2 = 2 - 2i \).

1
Add: \( z_1 + z_2 = (-4+2) + (6+(-2))i = -2 + 4i \)
2
Divide by 2: \( M = \frac{-2+4i}{2} = \) \( -1 + 2i \)
Re Im
\( z_1 = -4+6i \)
\( z_2 = 2-2i \)
\( M = -1+2i \)

Example 4: Distance and Midpoint Together

Given \( z_1 = -1 - 3i \) and \( z_2 = 5 + i \), find both the distance and midpoint.

Why Does This Work?

1
Complex subtraction \( z_2 - z_1 \) gives a vector from \( z_1 \) to \( z_2 \), whose components are \( (\Delta a, \Delta b) \).
2
The modulus applies the Pythagorean theorem to that vector: \( \sqrt{(\Delta a)^2 + (\Delta b)^2} \).
3
Complex addition then halving averages each coordinate independently, placing the result at the geometric center of the segment.

Common Mistakes

✕ Forgetting to Square
Writing \( \sqrt{3 + 4} \) instead of \( \sqrt{3^2 + 4^2} \). Always square each component before adding.
✕ Sign Errors
Subtracting \( (-3) \) as \( -3 \) instead of \( +3 \). Distribute the negative carefully: \( a_2 - (-3) = a_2 + 3 \).
✕ Midpoint as Subtraction
Using \( \frac{z_2 - z_1}{2} \) for the midpoint. The midpoint formula uses addition: \( \frac{z_1 + z_2}{2} \).

Summary Table

Distance Midpoint
Formula \( |z_2 - z_1| \) \( \frac{z_1 + z_2}{2} \)
Operation Subtract, then modulus Add, then halve
Result type Real number (\( \geq 0 \)) Complex number
Geometric meaning Length of segment Center of segment
Equivalent to Coordinate distance formula Coordinate midpoint formula

Question 1 of 5

What is the distance between \( z_1 = 1 + 2i \) and \( z_2 = 4 + 6i \)?

Question 2 of 5

What is the midpoint of the segment from \( z_1 = -2 + 4i \) to \( z_2 = 6 + 2i \)?

Question 3 of 5

The distance between \( z_1 = -1 + i \) and \( z_2 = 2 - 3i \) is:

Question 4 of 5

If the midpoint of \( z_1 \) and \( z_2 = 8 + 2i \) is \( M = 3 + 5i \), what is \( z_1 \)?

Question 5 of 5

Which expression gives the distance between \( z_1 \) and \( z_2 \) in the complex plane?