Pre-Calculus
Complex numbers (advanced)
Distance and midpoint in the complex plane
20 practice questions
0 video lessons
Theory + worked examples
Theory
Because the complex plane is just the coordinate plane with axes relabeled, distance and midpoint work exactly as in coordinate geometry:
- Distance between \(z_1\) and \(z_2\) is \(|z_1-z_2|\) — the modulus of their difference.
- Midpoint is \(\dfrac{z_1+z_2}{2}\) — the average.
\(|z-w|\) is a distance. So \(|z-w|=r\) describes a circle of radius \(r\) centered at \(w\).
The distance between two complex numbers is \(|z_1-z_2|\).
Distance and midpoint formulas.
Distance and midpoint:
\[d(z_1,z_2)=|z_1-z_2|=\sqrt{(a_1-a_2)^2+(b_1-b_2)^2},\qquad M=\dfrac{z_1+z_2}{2}\]
\(|z-w|=r\) is a circle; \(|z-w|<r\) its interior.
How to find distance and midpoint
- Distance: subtract the numbers, then take the modulus.
- Midpoint: add the numbers and divide by 2.
- Loci: read \(|z-w|=r\) as a circle centered at \(w\).
Example 1 — Distance between two points
Find the distance between \(z_1=1+i\) and \(z_2=4+5i\).
Solution
Distance is the modulus of the difference.
| \(z_2-z_1\) | \(=\) | \((4-1)+(5-1)i=3+4i\) |
| \(|z_2-z_1|\) | \(=\) | \(\sqrt{3^2+4^2}=5\) |
Example 2 — Midpoint
Find the midpoint of \(z_1=2+3i\) and \(z_2=6-i\).
Solution
Average the two numbers.
| \(\dfrac{z_1+z_2}{2}\) | \(=\) | \(\dfrac{(2+6)+(3-1)i}{2}\) |
| \(=\) | \(\dfrac{8+2i}{2}=4+i\) |
Example 3 — Distance with a negative part
Find the distance between \(z_1=-2+i\) and \(z_2=1-3i\).
Solution
Subtract, then take the modulus.
| \(z_2-z_1\) | \(=\) | \((1+2)+(-3-1)i=3-4i\) |
| \(|z_2-z_1|\) | \(=\) | \(\sqrt{9+16}=5\) |
Example 4 — A circle in the complex plane
Describe all \(z\) with \(|z-(2+i)|=3\).
Solution
This is the set of points a distance 3 from \(2+i\) — a circle.
| \(|z-(2+i)|\) | \(=\) | \(3\) |
A circle of radius 3 centered at \(2+i\).
Common pitfalls
Distance is the modulus of the difference, not the difference of moduli.
Subtract carefully with negative parts. \((-2)-1=-3\), etc.
Midpoint stays complex. Keep the real and imaginary parts separate when averaging.
Frequently asked questions
How do you find the distance between two complex numbers?
Take the modulus of their difference: \(|z_1-z_2|\).
How do you find the midpoint of two complex numbers?
Average them: \(\dfrac{z_1+z_2}{2}\), keeping real and imaginary parts separate.
What does |z - w| = r represent?
A circle of radius \(r\) centered at \(w\) — all points a fixed distance from \(w\).
Is distance in the complex plane the same as in coordinate geometry?
Yes. The complex plane is the coordinate plane, so the distance and midpoint formulas are identical.
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