Add, Subtract, Multiply Complex Numbers

Lesson 2 of 9: Complex Number System

In this lesson:

• Add and subtract complex numbers by combining like parts
• Multiply using FOIL and
• Compute conjugate products

What You Will Learn Today

1. Add and subtract complex numbers by combining like parts
2. Multiply complex numbers using distributive property and
3. Write any result in standard form
4. Compute and recognize the result is real

Recall: Combining Like Terms with Variables

• Like terms share the same variable: ,
• Addition:
• Distributing minus: — minus applies to both terms

Can you combine like terms without stopping to think about it?

You Already Know How to Do This

Adding polynomials:

Now replace with :

The mechanism is identical — behaves like for addition.

Addition: Combine Like Parts Side by Side

Adding Complex Numbers: The Rule

Real parts add to real parts. Imaginary parts add to imaginary parts.

Example:

Subtraction: Distribute the Minus Sign

Distribute the minus to both parts, then collect:

, not

Quick Check: Add and Subtract

Compute each and write in form:

Problem 2: remember to distribute the minus to both parts.

Multiplication Works Just Like FOIL

Compare — both use the distributive property:

The only difference: replace every with at the end.

Three Steps Every Multiplication Follows

Step 1: Distribute (FOIL)
Step 2: Replace with
Step 3: Collect real and imaginary parts → form

Every complex multiplication uses these three steps in order.

Worked Example:

Step 1 — Distribute:

Step 2 — Replace :

Step 3 — Collect:

Squaring via FOIL:

Expand — never skip the middle term:

Replace :

— the middle term survives

Spot the Error in These Products

Which result is wrong? What is the correct answer?

Guided Practice:

Step 1 done:

Step 2 — Replace :

Step 3 — Collect: real = , imaginary =

Complete steps 2 and 3 before advancing.

Compute These Three — No Scaffolding

Write each result in form.

Conjugates: A Pattern Worth Naming

and are called complex conjugates.

Notice what happens when you multiply them by FOIL:

The imaginary parts cancel. The result is real.

The Conjugate Product Is Always Real

The result is always a real number — never imaginary.

Worked Example:

Using the conjugate product formula:

Or verify by FOIL: ✓

Pure Imaginary Products and

Pure imaginary product:

Multiplying two pure imaginary numbers gives a real number.

Square root of a negative:

Check: ✓

Quick Check: Conjugates and Roots

Compute each:

Watch for These Three Common Errors

1. Minus distribution: : wrong ; right
2. Missing middle term: : wrong ; right
3. Forgetting : : wrong ; right

What You Now Know About Complex Arithmetic

✓ Add/subtract: combine real to real, imaginary to imaginary
✓ Multiply: FOIL, replace , collect terms
✓ Conjugate product: (always real)

Subtract: distribute minus to both parts
Square: FOIL completely — middle term survives
, not

Next Lesson: Dividing Complex Numbers

In HSN.CN.A.3, you'll divide .

The conjugate product from today is the key tool.