Distance and Midpoint on the Complex Plane

Lesson 6 of 9: Complex Number System

In this lesson:

• Compute distance using the modulus of a difference
• Find midpoints using the average of two complex numbers
• Recognize these as the coordinate formulas in compact notation

What You Will Learn Today

1. Calculate distance between two complex numbers as
2. Find the midpoint of a segment as
3. Recognize both formulas as coordinate-geometry formulas in compact form
4. Solve geometric problems: circle equations and segment length

The Same Formula in Two Languages

Coordinate form: Distance from to :

Complex form: Distance from to :

Same formula. Different notation.

Distance Formula: |z − w|

For and :

Special case: Distance from to the origin:

The modulus is the distance from the origin — confirmed.

Distance Worked Example: Two Points

Distance from the Origin Confirms the Modulus

Distance from to the origin:

Distance from to :

Quick Check: Compute Two Distances

1. Distance from to the origin
2. Distance from to

For problem 2: compute first, then take the modulus.

Midpoint: Average of Two Complex Numbers

Coordinate midpoint of and :

Complex midpoint of and :

Same formula. The complex form is the coordinate form in one expression.

Midpoint Formula: Average the Two Numbers

Example: ,

Verify: ✓

Finding the Unknown Endpoint From Midpoint

Given: , one endpoint . Find .

Check: midpoint of and is ✓

Quick Check: Compute and Reverse Midpoints

1. Midpoint of and
2. Midpoint of and is , and . Find .

Circle Equation: Distance From a Center

Coordinate: — circle at , radius .

Complex: — the same circle.

Center , not the origin.

Circle Identification: Read Center and Radius

1. → center (on real axis), radius

2. → center , radius

Coordinate form of Problem 2:

Find the Two Common Errors

A student computes:

1. Distance from to :

2. Midpoint of and :

What is wrong in each? Compute the correct answers.

Three Computations: Distance, Midpoint, Circle

1. Distance between and
2. Midpoint of and
3. Identify center and radius of

Problem 3: rewrite as .

Watch for These Three Common Errors

1. Distance: — use modulus of the difference
2. Midpoint: — midpoint is the average (add, not subtract)
3. Circle center: is centered at , not at the origin

What You Now Know About Complex Geometry

✓ Distance: — modulus of the difference
✓ Midpoint: — average of the two numbers
✓ Circle: — center , radius

Distance: use , not
Midpoint: use addition, not subtraction

You Have Completed the Complex Number System

The complex number system is an arithmetic system and a geometric space.