Complex Numbers

The Imaginary Unit

Problem: No real number satisfies x² = -1

Solution: Define imaginary unit i

Definition: i² = -1, or i = √(-1)

Powers of i:

• i¹ = i
• i² = -1
• i³ = i² · i = -1 · i = -i
• i⁴ = i² · i² = (-1)(-1) = 1
• i⁵ = i⁴ · i = 1 · i = i (pattern repeats!)

Pattern repeats every 4 powers

Example: Simplify Powers of i

Simplify: i²⁷

Divide exponent by 4:

27 = 4(6) + 3

So i²⁷ = i³ = -i

Answer: -i

Imaginary Numbers

Imaginary number: bi where b is real, i is imaginary unit

Examples:

• 3i
• -5i
• (√2)i

Simplifying square roots of negatives:

√(-a) = √(a) · √(-1) = i√a

Example: Simplify Square Roots

Simplify: √(-16)

Rewrite:

√(-16) = √(16) · √(-1)
       = 4i

Answer: 4i

Simplify: √(-12)

√(-12) = √(12) · i
       = √(4·3) · i
       = 2i√3

Answer: 2i√3

Complex Numbers

Complex number: a + bi

Where:

• a = real part
• b = coefficient of imaginary part (bi)
• If a = 0: purely imaginary
• If b = 0: real number

Examples:

• 3 + 4i
• -2 - 5i
• 7 (equivalent to 7 + 0i)
• 4i (equivalent to 0 + 4i)

Standard form: a + bi

Adding Complex Numbers

Add real parts and imaginary parts separately

(a + bi) + (c + di) = (a + c) + (b + d)i

Example 1: Add

(3 + 2i) + (5 + 4i)

Add parts:

= (3 + 5) + (2 + 4)i
= 8 + 6i

Answer: 8 + 6i

Example 2: With Negatives

(6 - 3i) + (-2 + 7i)

Add:

= (6 - 2) + (-3 + 7)i
= 4 + 4i

Answer: 4 + 4i

Subtracting Complex Numbers

Distribute negative, then add

(a + bi) - (c + di) = (a - c) + (b - d)i

Example: Subtract

(8 + 5i) - (3 + 2i)

Subtract parts:

= (8 - 3) + (5 - 2)i
= 5 + 3i

Answer: 5 + 3i

Multiplying Complex Numbers

Use distributive property (FOIL)

Remember: i² = -1

Example 1: Multiply

(2 + 3i)(4 + i)

FOIL:

= 2(4) + 2(i) + 3i(4) + 3i(i)
= 8 + 2i + 12i + 3i²
= 8 + 14i + 3(-1)
= 8 + 14i - 3
= 5 + 14i

Answer: 5 + 14i

Example 2: With Negatives

(1 - 2i)(3 + 4i)

FOIL:

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

Answer: 11 - 2i

Example 3: Square Complex Number

(3 + 2i)²

Expand:

= (3 + 2i)(3 + 2i)
= 9 + 6i + 6i + 4i²
= 9 + 12i + 4(-1)
= 9 + 12i - 4
= 5 + 12i

Answer: 5 + 12i

Complex Conjugates

Complex conjugate: Change sign of imaginary part

If z = a + bi, then z̄ = a - bi (read "z bar")

Examples:

• Conjugate of 3 + 4i is 3 - 4i
• Conjugate of 5 - 2i is 5 + 2i
• Conjugate of -i is i

Key property: z · z̄ is always real!

Example: Multiply by Conjugate

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

Difference of squares pattern:

= 2² - (3i)²
= 4 - 9i²
= 4 - 9(-1)
= 4 + 9
= 13

Answer: 13 (real number!)

Dividing Complex Numbers

Multiply numerator and denominator by conjugate of denominator

This "rationalizes" the denominator

Example 1: Simple Division

(6 + 8i) / (1 + i)

Multiply by conjugate (1 - i):

= (6 + 8i)(1 - i) / [(1 + i)(1 - i)]
= (6 - 6i + 8i - 8i²) / (1 - i²)
= (6 + 2i + 8) / (1 + 1)
= (14 + 2i) / 2
= 7 + i

Answer: 7 + i

Example 2: More Complex

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

Multiply by conjugate (4 + 3i):

= (3 + 2i)(4 + 3i) / [(4 - 3i)(4 + 3i)]
= (12 + 9i + 8i + 6i²) / (16 + 9)
= (12 + 17i - 6) / 25
= (6 + 17i) / 25
= 6/25 + (17/25)i

Answer: 6/25 + (17/25)i

Absolute Value (Modulus)

Absolute value (modulus) of a + bi:

|a + bi| = √(a² + b²)

Geometric meaning: Distance from origin in complex plane

Example: Find Modulus

Find: |3 + 4i|

Calculate:

|3 + 4i| = √(3² + 4²)
         = √(9 + 16)
         = √25
         = 5

Answer: 5

The Complex Plane

Complex plane (Argand diagram):

• Horizontal axis: real part
• Vertical axis: imaginary part
• Point (a, b) represents a + bi

Example: 3 + 2i is plotted at (3, 2)

Geometric operations:

• Addition: vector addition
• Modulus: distance from origin
• Conjugate: reflection over real axis

Solving Quadratic Equations with Complex Solutions

When discriminant b² - 4ac < 0, solutions are complex

Example: Solve Quadratic

Solve: x² + 2x + 5 = 0

Use quadratic formula:

x = [-2 ± √(4 - 20)] / 2
x = [-2 ± √(-16)] / 2
x = [-2 ± 4i] / 2
x = -1 ± 2i

Answer: x = -1 + 2i or x = -1 - 2i

Note: Complex solutions come in conjugate pairs!

Properties of Complex Numbers

Commutative: z₁ + z₂ = z₂ + z₁

Associative: (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)

Distributive: z₁(z₂ + z₃) = z₁z₂ + z₁z₃

Conjugate properties:

• (z₁ + z₂)̄ = z̄₁ + z̄₂
• (z₁ · z₂)̄ = z̄₁ · z̄₂
• z · z̄ = |z|²

Real-World Applications

Electrical engineering: AC circuit analysis (impedance)

Quantum mechanics: Wave functions

Signal processing: Fourier transforms

Fluid dynamics: Potential flow

Control theory: Stability analysis

Practice