This lesson shows how every complex number corresponds to a point on the complex plane, so the distance and midpoint formulas from coordinate geometry apply directly, letting you measure how far apart two complex numbers are and find the point exactly between them.
Complex Numbers as Points on a Plane
Every complex number \(z = a + bi\) can be matched to the ordered pair \((a, b)\) on the complex plane, where the horizontal axis measures the real part and the vertical axis measures the imaginary part. Once a complex number is a point, questions like "how far apart are these two complex numbers?" or "what complex number sits exactly between them?" turn into ordinary coordinate geometry problems.
That means the same distance formula and midpoint formula you already use for points \((x_1, y_1)\) and \((x_2, y_2)\) apply here too, just written in terms of real and imaginary parts.
Distance Between Two Complex Numbers
Let \(z_1 = a + bi\) and \(z_2 = c + di\). Plotting them as \((a, b)\) and \((c, d)\) and applying the Pythagorean theorem to the horizontal and vertical legs between them gives:
\( d(z_1, z_2) = \sqrt{(c - a)^2 + (d - b)^2} \)
This is exactly the same formula used for two points on a regular \(xy\)-plane. There is also a shortcut: the distance between \(z_1\) and \(z_2\) is the modulus (absolute value) of their difference, \( |z_1 - z_2| \). If you have already studied the absolute value of complex numbers, you can subtract the two numbers first and then take the modulus of the result instead of plugging into the coordinate formula directly.
The blue segment shows the distance between \(z_1\) and \(z_2\); the green point M is their midpoint.
Worked Example: Distance
Find the distance between \(z_1 = 5 + 2i\) and \(z_2 = 1 - 4i\).
Match the coordinates: \((a, b) = (5, 2)\) and \((c, d) = (1, -4)\). Substitute into the formula:
So \(z_1\) and \(z_2\) are \(2\sqrt{13}\) units apart on the complex plane.
Midpoint of Two Complex Numbers
The midpoint of \(z_1 = a + bi\) and \(z_2 = c + di\) is the complex number that averages the real parts and averages the imaginary parts separately:
\( M = \dfrac{a + c}{2} + \dfrac{b + d}{2}i \)
An equivalent way to remember this is that the midpoint is simply \(\dfrac{z_1 + z_2}{2}\), which lands exactly halfway along the segment joining the two points.
Worked Example: Midpoint
Find the midpoint of \(z_1 = 5 + 2i\) and \(z_2 = 1 - 4i\), the same pair used above.
Average the real parts: \( \dfrac{5 + 1}{2} = 3 \). Average the imaginary parts: \( \dfrac{2 + (-4)}{2} = -1 \).
So the midpoint is \( M = 3 - i \), matching the green point plotted in the diagram above, right between \(z_1\) and \(z_2\).
Putting It Together
Distance and midpoint problems with complex numbers almost always come down to two steps: identify the real and imaginary parts of each number, then substitute those values into the formula that matches what is being asked. Because a complex number is nothing more than a point on the plane, no new formulas need to be memorized beyond the ones already familiar from coordinate geometry, only the labels change from \(x\) and \(y\) to real and imaginary parts.