Complex Number Operations

Complex Number Operations: The Magic Mirror World 🪞

Imagine you found a magic mirror. When you look into it, you see yourself—but something’s different. Everything on your right appears on your left, and vice versa. This mirror flips your reflection across an invisible line.

Complex numbers have their own magic mirror. It’s called the complex conjugate, and understanding it unlocks the secrets to dividing, solving equations, and mapping numbers on a beautiful plane.

Let’s explore this mirror world together!

1. Complex Conjugates: Your Number’s Mirror Twin

What’s a Conjugate?

Every complex number has a twin—a mirror image. If your number is a + bi, its conjugate is a - bi.

Think of it like this:

• The real part a stays the same (your face doesn’t change)
• The imaginary part bi flips sign (left becomes right)

Simple Examples

The Magic Property

When you multiply a number by its conjugate, something amazing happens—the imaginary part vanishes!

(3 + 2i) × (3 - 2i)
= 9 - 6i + 6i - 4i²
= 9 - 4(-1)
= 9 + 4
= 13 ✨ (A real number!)

Why does this work? The +bi and -bi cancel each other out, and i² = -1 turns the rest into a positive real number.

Formula to Remember

For any a + bi:

(a + bi)(a - bi) = a² + b²

This always gives you a positive real number. No imaginary parts left!

2. Dividing Complex Numbers: The Conjugate Trick

The Problem

Dividing by a complex number is tricky because we can’t have i in the denominator. It’s like having a splinter—we need to remove it!

The Solution: Multiply by the Conjugate

The trick: Multiply top AND bottom by the conjugate of the bottom number.

Step-by-Step Example

Let’s divide: (4 + 3i) ÷ (2 + i)

Step 1: Write as a fraction

  4 + 3i
  ------
  2 + i

Step 2: Find the conjugate of the bottom

• Bottom is 2 + i
• Conjugate is 2 - i

Step 3: Multiply top and bottom by this conjugate

  (4 + 3i)(2 - i)
  ---------------
  (2 + i)(2 - i)

Step 4: Expand the top

(4 + 3i)(2 - i)
= 8 - 4i + 6i - 3i²
= 8 + 2i - 3(-1)
= 8 + 2i + 3
= 11 + 2i

Step 5: Expand the bottom (using our magic property!)

(2 + i)(2 - i) = 2² + 1² = 4 + 1 = 5

Step 6: Final answer

  11 + 2i     11     2
  ------- = ---- + ---i
     5        5     5

Answer: 2.2 + 0.4i

Why This Works

The conjugate removes the imaginary part from the denominator, giving us a clean real number to divide by. Simple division then gives us our answer!

3. Complex Quadratic Solutions: When Reality Isn’t Enough

The Classic Problem

Remember the quadratic formula?

x = (-b ± √(b² - 4ac)) / 2a

Sometimes b² - 4ac is negative. In the past, we’d say “no solution.” But now we know better!

The Discriminant Tells All

The expression b² - 4ac is called the discriminant:

• Positive: Two real solutions
• Zero: One real solution
• Negative: Two complex solutions ✨

Example: Solving x² + 4x + 5 = 0

Step 1: Identify a, b, c

• a = 1, b = 4, c = 5

Step 2: Calculate the discriminant

b² - 4ac = 16 - 20 = -4

It’s negative! Time for complex numbers.

Step 3: Apply the quadratic formula

x = (-4 ± √(-4)) / 2
x = (-4 ± 2i) / 2
x = -2 ± i

Solutions: x = -2 + i and x = -2 - i

Notice Something?

The two solutions are conjugates of each other! This always happens with complex solutions to quadratic equations with real coefficients.

graph TD
    A["Quadratic Equation"] --> B{Discriminant}
    B -->|Positive| C["2 Real Solutions"]
    B -->|Zero| D["1 Real Solution"]
    B -->|Negative| E["2 Complex Conjugate Solutions"]

4. The Complex Number Plane: A New Kind of Map

Beyond the Number Line

Real numbers live on a line. But complex numbers need more space—they need a plane!

The Argand Diagram

Named after mathematician Jean-Robert Argand, this is a map where:

• Horizontal axis (x): Real part
• Vertical axis (y): Imaginary part

Every complex number a + bi becomes a point at (a, b).

Examples on the Plane

Visualizing Conjugates

On the plane, a conjugate is a reflection across the real axis (the horizontal line).

• 3 + 2i is at (3, 2)
• 3 - 2i is at (3, -2)
• They’re mirror images!

graph TD
    subgraph Complex Plane
    A["3 + 2i<br/>#40;3, 2#41;"]
    B["Real Axis"]
    C["3 - 2i<br/>#40;3, -2#41;"]
    end
    A -.->|Mirror| C

Distance from Origin: The Modulus

The distance from any complex number to the origin (0, 0) is called its modulus (or absolute value).

For a + bi:

|a + bi| = √(a² + b²)

Example: |3 + 4i| = √(9 + 16) = √25 = 5

This is just the Pythagorean theorem in action!

Fun Fact

When you multiply a number by its conjugate:

(a + bi)(a - bi) = a² + b²

This equals the modulus squared: |z|² = a² + b²

Putting It All Together: A Complete Example

Problem: Solve z² + 2z + 5 = 0 and plot the solutions.

Step 1: Find the solutions

• a = 1, b = 2, c = 5
• Discriminant: 4 - 20 = -16 (negative!)
• Solutions: z = (-2 ± √(-16)) / 2 = (-2 ± 4i) / 2 = -1 ± 2i

Step 2: Identify the conjugates

• z₁ = -1 + 2i
• z₂ = -1 - 2i (the conjugate!)

Step 3: Plot on the complex plane

• z₁ is at point (-1, 2)
• z₂ is at point (-1, -2)
• They’re reflections across the real axis!

Step 4: Find the modulus

|z₁| = √(1 + 4) = √5 ≈ 2.24

Both solutions have the same distance from the origin.

Key Takeaways

You Did It!

You’ve just explored the mirror world of complex numbers. You learned that:

1. Conjugates are like mirror twins—same real part, opposite imaginary part
2. Dividing complex numbers uses the conjugate trick to eliminate i from the bottom
3. Quadratic equations with negative discriminants give us beautiful complex conjugate pairs
4. The complex plane lets us see and understand complex numbers as points in 2D space

Complex numbers aren’t scary—they’re just numbers that live in a bigger, more interesting world. And now you have the tools to explore it!

Next time someone says “that’s impossible,” remember: complex numbers prove that with the right perspective, nothing is impossible! ✨