Represent Operations Geometrically on the Complex Plane

Lesson 5 of 9: Complex Number System

In this lesson:

• Represent addition, subtraction, and conjugation as geometric moves
• Describe multiplication as scaling and rotation
• Use polar form to compute powers of complex numbers

What You Will Learn Today

1. Represent addition and subtraction as vector operations (parallelogram rule)
2. Represent conjugation as reflection across the real axis
3. Describe multiplication as scaling by modulus and rotation by argument
4. Use polar form to compute powers using the argument multiplication rule

Recall: Complex Plane and Polar Form

• Plotting: sits at point — real axis horizontal, imaginary vertical
• Modulus: — distance from origin
• Argument: angle measured counterclockwise from positive real axis

Can you locate and on the complex plane right now?

Addition Is Placing Arrows Tip-to-Tail

The Parallelogram Rule for Addition

• Draw and as arrows from the origin
• Complete the parallelogram
• Sum is the diagonal

Triangle inequality:

Subtraction: Reflect w and Add

: reflect through the origin (negate both parts).

: add to →

Conjugation: Reflection Across the Real Axis

reflects across the real axis (horizontal axis).

→

Same horizontal position, mirrored vertically.

Conjugate pairs are symmetric about the real axis.

Quick Check: Three Geometric Operations

For and :

1. Plot using the parallelogram rule. What is ?
2. Plot by reflecting and adding. What is ?
3. Plot . Describe its position.

Powers of i: A Rotation Discovery

Each multiplication by is a 90° counterclockwise rotation.

: four steps, full circle.

Multiplying by i Rotates 90° Counterclockwise

has modulus and argument .

Multiplying any by : preserves length, adds 90° to direction.

Example: at angle

at angle () ✓

General Rule: Moduli Multiply, Arguments Add

For and :

Moduli multiply. Arguments add. Not the other way around.

Multiplication in Polar Form: Example

using polar form:

has ,

Square: ,

Verify algebraically: ✓

Find the Error in Each Statement

A student writes:

1. for ,

2. for computing the argument of a product

What is wrong in each? State the correct rule.

Quick Check: Rotation and Scaling

For and :

1. What is the geometric effect of multiplying by ?
2. Compute in polar form. Verify algebraically.

Hint: has modulus 1 and argument (or ).

The Hard Way: Expand Algebraically

Step 1:

Step 2:

Eight products, three rounds of like-term collection. Every step is an error opportunity.

De Moivre's Theorem: Multiply the Argument

Rule: Raise modulus to . Multiply argument by .

Multiply by — do not raise to .

The Canonical Example:

Step 1:

Step 2: (second quadrant: )

Step 3: Cube: ,

Step 4:

Second Example:

: ,

Apply De Moivre: ,

Verify algebraically: , ✓

Three Computations: No Help Provided

1. Compute and describe the result geometrically.
2. Compute using polar form. Verify algebraically.
3. Compute using De Moivre's theorem.

Watch for These Two Common Errors

1. — the diagonal is shorter.

2. — arguments add:

De Moivre: , not .

What You Now Know: Geometry of Operations

✓ Addition: parallelogram rule;
✓ Subtraction: tip-to-tail with
✓ Conjugation: reflection across the real axis
✓ Multiplication: ,
✓ Powers: De Moivre — and

Next Lesson: Distance on the Complex Plane

In HSN.CN.B.6, you'll compute the distance between two complex numbers.

The subtraction you learned today gives the distance vector. The modulus gives its length.