Imaginary Unit and Complex Numbers

Lesson 1 of 9: Complex Number System

In this lesson:

• Explain why is defined and what means
• Write complex numbers in standard form
• Classify numbers and evaluate powers of

What You Will Learn Today

1. Explain why is defined by
2. Write any complex number in standard form
3. Classify numbers as pure real, pure imaginary, or mixed
4. Evaluate integer powers of using the four-cycle

Recall: What Squaring Real Numbers Produces

• Positive: , — result is positive
• Negative: , — result is still positive
• Zero: — the only square that equals zero

Can you name a real number whose square is negative?

Can You Solve ?

has solutions . What about ?

• : ✗ — : ✗
• : ✗ — : ✗

Every square ≥ 0 — has no real solution.

All Real Squares Land at Zero or Greater

The real number line has no landing spot for .

Each Number System Extended to Solve a Problem

• Counting numbers → couldn't solve → invented integers
• Integers → couldn't solve → invented rationals
• Rationals → couldn't solve → invented irrationals
• Real numbers → can't solve → define complex numbers

The Imaginary Unit: Defined by

We define as a number satisfying:

This is a definition — not a calculation from previous rules.

Two solutions: , so both and solve .

Name Both Solutions to

What are the two solutions to ?

Write your answer before the next slide.

Introducing Standard Form:

Every complex number is written where and are real numbers.

• is the real part; is the imaginary part (a real coefficient)
• Like packages two values into one point, packages two reals

Reading and from Any Complex Number

Every real number is a complex number — just with imaginary part .

The Imaginary Part Is , Not

Given :

• Imaginary part = ✓ (the real coefficient)
• Imaginary part = ✗ (that's the imaginary term)

Compare: in , the slope is — not "."

The coefficient is the part; the full term includes the variable.

Three Categories of Complex Numbers

• Pure real: → e.g.
• Pure imaginary: → e.g.
• Mixed: → e.g.
• Zero: →

Real Numbers Live Inside Complex Numbers

Complex Numbers Cannot Be Ordered Like Reals

Is greater or less than ?

• Real-part comparison: larger ()
• Imaginary-part comparison: smaller ()

No consistent ordering exists — complex numbers live on a plane, not a line.

Spot the Error in These Identifications

One row has a wrong imaginary part. Which one, and what should it be?

Your Turn: Identify and

For each complex number, write the real part and imaginary part:

Write $a = $ and $b = $ for each before advancing.

Write Form and Classify Each

For each: write in form, state and , classify.

No hints — complete all five.

Higher Powers Need No New Rules

Replace with :

— cycle repeats.

Deriving the Period-4 Cycle of Powers

From :

• ← cycle repeats

Use the Remainder to Evaluate

Divide by ; the remainder gives the answer:

• : — : — : — :

Applying the Remainder Rule:

Step 1: Divide the exponent by 4

Step 2: Remainder 3 →

Check: , so ✓

Applying the Remainder Rule:

Step 1: Divide the exponent by 4

Step 2: Remainder 0 →

Multiples of 4 always give .

Quick Check: and

Evaluate each using the remainder method:

Use and identify the remainder for each.

Practice: Powers, Parts, and Classification

Evaluate each:

2. Real part of
3. Imaginary part of
4. Classify and explain why

What You Now Know About Complex Numbers

✓ ; both and solve
✓ Form ; is the real coefficient, not
✓ Powers cycle period 4; divide exponent by 4

, not
Reals are complex:

Next Lesson: Arithmetic with Complex Numbers

In HSN.CN.A.2, you'll add, subtract, and multiply complex numbers in standard form.

Every computation uses and the form from today.