Find Conjugates and Quotients of Complex Numbers

Lesson 3 of 9: Complex Number System

In this lesson:

• Find the conjugate of any complex number
• Compute the modulus
• Divide complex numbers using the conjugate technique

What You Will Learn Today

1. Find the conjugate of any complex number
2. Compute the modulus using
3. Divide complex numbers by multiplying by the conjugate
4. Express division results in standard form

The Conjugate Product You Already Know

From last lesson:

The result is real. Its square root is .

We call the modulus of , written .

The modulus is the square root of the conjugate product.

Modulus as a Distance in the Plane

is the hypotenuse length — Pythagorean theorem.

The Conjugate: Flip the Imaginary Sign

Conjugate of :

Only the imaginary sign flips.

Conjugate vs. Negation: Know the Difference

These are different operations:

Conjugate: mirror across the real axis. Negation: rotate 180°.

The Modulus Formula Uses Both Parts

— modulus extends absolute value to complex numbers.

Quick Check: Conjugate and Modulus

Compute for each:

1. : find and
2. : find and

For problem 1: both parts give a whole-number modulus.

Recall: Rationalizing a Radical Denominator

• Problem: — radical in denominator is not simplified form
• Move: Multiply by — value unchanged, form changed
• Result: — denominator is now rational

Can you describe why multiplying by a form of 1 clears the radical?

Division Uses the Same Tool as Rationalization

Real algebra: — multiply by a form of 1

Complex division: — same move

The conjugate plays the role of in the denominator clearing.

Division Strategy: Multiply by One

To compute , multiply by :

The denominator is always real and positive.

You are multiplying by 1 — the value doesn't change, only the form.

Four Steps for Complex Division

Step 1: Identify the conjugate of the denominator.

Step 2: Multiply both numerator and denominator by it.

Step 3: Expand the numerator (FOIL, replace ).

Step 4: Denominator . Split into form.

Each step uses skills you already have.

Division Example: Two Plus Three i

Division Example: Five Plus i

Step 1: Conjugate of is

Step 2:

Step 3: Numerator:

Step 4: Denominator: . Result:

Guided Practice: Complete the Division

Given:

Steps 1–2 done:

Step 3: Expand numerator:

Step 4: Denominator . Write result as .

Complete steps 3 and 4.

Spot the Errors in This Division

Two errors are hidden above. Find and correct both.

Three Computations With No Scaffolding

Write each result in form or simplest radical form.

1. Find and for
2. Compute
3. Use modulus to check: is ?

Modulus of a Product Is Multiplicative

Property:

Let , :

• , so ✓

Both routes give the same answer.

Watch for These Three Common Errors

1. Wrong conjugate: , not — only imaginary sign flips
2. Numerator skipped: multiply to both top and bottom, not denominator only
3. Denominator sign: , not —

What You Now Know About Complex Numbers

✓ — imaginary sign flips
✓ — distance, both parts
✓ Divide: use to clear denominator
✓

Conjugate ≠ negation
, not

Next Lesson: The Complex Plane

In HSN.CN.B.4, you'll plot complex numbers as points on the complex plane.

The modulus is the distance from the origin to the point .

The conjugate is the reflection across the real axis.