Complex Numbers
Complete Notes & Formulae for Eleventh Grade (Algebra 2)
1. Introduction to Complex Numbers
The Imaginary Unit (i):
\[ i = \sqrt{-1} \]
\[ i^2 = -1 \]
The imaginary unit i is defined as the square root of -1. This allows us to work with square roots of negative numbers.
Standard Form of a Complex Number:
\[ z = a + bi \]
where:
• a = real part (Re(z))
• b = imaginary part (Im(z))
• i = imaginary unit
Both a and b are real numbers
Examples:
Example 1: \( 3 + 4i \)
Real part = 3, Imaginary part = 4
Example 2: \( -2 - 5i \)
Real part = -2, Imaginary part = -5
Example 3: \( 7 \) (purely real)
Real part = 7, Imaginary part = 0 → \( 7 + 0i \)
Example 4: \( 6i \) (purely imaginary)
Real part = 0, Imaginary part = 6 → \( 0 + 6i \)
Simplifying Square Roots of Negative Numbers:
Formula:
\[ \sqrt{-n} = i\sqrt{n} \quad \text{where } n > 0 \]
Examples:
• \( \sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i \)
• \( \sqrt{-25} = \sqrt{25} \cdot i = 5i \)
• \( \sqrt{-7} = i\sqrt{7} \)
2. Add and Subtract Complex Numbers
Addition Formula:
\[ (a + bi) + (c + di) = (a + c) + (b + d)i \]
Rule: Add the real parts together, add the imaginary parts together
Subtraction Formula:
\[ (a + bi) - (c + di) = (a - c) + (b - d)i \]
Rule: Subtract the real parts, subtract the imaginary parts
Examples:
Example 1: Addition
\( (5 + 3i) + (2 + 7i) \)
Step 1: Add real parts: \( 5 + 2 = 7 \)
Step 2: Add imaginary parts: \( 3i + 7i = 10i \)
Answer: \( 7 + 10i \)
Example 2: Subtraction
\( (8 - 4i) - (3 + 6i) \)
Step 1: Subtract real parts: \( 8 - 3 = 5 \)
Step 2: Subtract imaginary parts: \( -4i - 6i = -10i \)
Answer: \( 5 - 10i \)
Example 3: Mixed Operations
\( (6 + 2i) - (4 - 3i) + (1 + i) \)
Step 1: Distribute negative: \( 6 + 2i - 4 + 3i + 1 + i \)
Step 2: Combine real: \( 6 - 4 + 1 = 3 \)
Step 3: Combine imaginary: \( 2i + 3i + i = 6i \)
Answer: \( 3 + 6i \)
3. Complex Conjugates
Definition:
The complex conjugate of a complex number is obtained by changing the sign of the imaginary part.
\[ \text{If } z = a + bi, \text{ then } \overline{z} = a - bi \]
Notation: \( \overline{z} \) or \( z^* \)
Examples:
• \( z = 3 + 4i \) → \( \overline{z} = 3 - 4i \)
• \( z = -5 - 2i \) → \( \overline{z} = -5 + 2i \)
• \( z = 7 \) → \( \overline{z} = 7 \) (real number's conjugate is itself)
• \( z = 6i \) → \( \overline{z} = -6i \)
Important Property:
Product of Complex Conjugates:
\[ (a + bi)(a - bi) = a^2 + b^2 \]
The product is ALWAYS a real number!
Example:
\( (3 + 4i)(3 - 4i) = 3^2 - (4i)^2 = 9 - 16i^2 = 9 - 16(-1) = 9 + 16 = 25 \)
4. Multiply Complex Numbers
Multiplication Formula (FOIL Method):
\[ (a + bi)(c + di) = (ac - bd) + (ad + bc)i \]
Steps:
1. First: \( a \times c = ac \)
2. Outer: \( a \times di = adi \)
3. Inner: \( bi \times c = bci \)
4. Last: \( bi \times di = bdi^2 = -bd \)
5. Combine: \( ac + adi + bci - bd = (ac - bd) + (ad + bc)i \)
⚠️ Remember: \( i^2 = -1 \)
Detailed Examples:
Example 1:
\( (3 + 2i)(4 + 5i) \)
F: \( 3 \times 4 = 12 \)
O: \( 3 \times 5i = 15i \)
I: \( 2i \times 4 = 8i \)
L: \( 2i \times 5i = 10i^2 = 10(-1) = -10 \)
Combine: \( 12 + 15i + 8i - 10 = 2 + 23i \)
Answer: \( 2 + 23i \)
Example 2:
\( (2 - 3i)(1 + 4i) \)
F: \( 2 \times 1 = 2 \)
O: \( 2 \times 4i = 8i \)
I: \( -3i \times 1 = -3i \)
L: \( -3i \times 4i = -12i^2 = -12(-1) = 12 \)
Combine: \( 2 + 8i - 3i + 12 = 14 + 5i \)
Answer: \( 14 + 5i \)
5. Divide Complex Numbers
Division Method:
To divide complex numbers, multiply both numerator and denominator by the complex conjugate of the denominator.
\[ \frac{a + bi}{c + di} = \frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} = \frac{(a + bi)(c - di)}{c^2 + d^2} \]
Step-by-Step Process:
Step 1: Identify the complex conjugate of the denominator
If denominator is \( c + di \), conjugate is \( c - di \)
Step 2: Multiply numerator and denominator by the conjugate
Step 3: Multiply out the numerator using FOIL
Step 4: Simplify the denominator (will be a real number)
Use formula: \( (c + di)(c - di) = c^2 + d^2 \)
Step 5: Write in standard form \( a + bi \)
Detailed Example:
Divide: \( \frac{3 + 2i}{1 - 4i} \)
Step 1: Complex conjugate of \( 1 - 4i \) is \( 1 + 4i \)
Step 2: Multiply:
\( \frac{3 + 2i}{1 - 4i} \cdot \frac{1 + 4i}{1 + 4i} \)
Step 3: Numerator (FOIL):
\( (3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i^2 \)
\( = 3 + 14i + 8(-1) = 3 + 14i - 8 = -5 + 14i \)
Step 4: Denominator:
\( (1 - 4i)(1 + 4i) = 1^2 + 4^2 = 1 + 16 = 17 \)
Step 5: Final answer:
\( \frac{-5 + 14i}{17} = -\frac{5}{17} + \frac{14}{17}i \)
Answer: \( -\frac{5}{17} + \frac{14}{17}i \)
6. Absolute Values of Complex Numbers (Modulus)
Definition:
The absolute value (or modulus) of a complex number is its distance from the origin in the complex plane.
\[ |a + bi| = \sqrt{a^2 + b^2} \]
Notation: \( |z| \) or \( r \) (radius)
The result is ALWAYS a non-negative real number
Examples:
Example 1:
Find \( |3 + 4i| \)
\( |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
Answer: 5
Example 2:
Find \( |-5 + 12i| \)
\( |-5 + 12i| = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \)
Answer: 13
Example 3:
Find \( |6i| \)
\( |0 + 6i| = \sqrt{0^2 + 6^2} = \sqrt{36} = 6 \)
Answer: 6
Important Properties:
• \( |z| \geq 0 \) (always non-negative)
• \( |z| = 0 \) if and only if \( z = 0 \)
• \( |z \cdot w| = |z| \cdot |w| \)
• \( \left|\frac{z}{w}\right| = \frac{|z|}{|w|} \) (when \( w \neq 0 \))
• \( |z| = |\overline{z}| \) (complex number and its conjugate have same modulus)
7. Powers of i
The Repeating Pattern:
Powers of i follow a cyclical pattern that repeats every 4 powers:
| Power | Value | Calculation |
|---|---|---|
| \( i^0 \) | 1 | Any number to power 0 equals 1 |
| \( i^1 \) | \( i \) | By definition |
| \( i^2 \) | -1 | By definition |
| \( i^3 \) | \( -i \) | \( i^2 \cdot i = -1 \cdot i = -i \) |
| \( i^4 \) | 1 | \( i^2 \cdot i^2 = (-1)(-1) = 1 \) |
| \( i^5 \) | \( i \) | \( i^4 \cdot i = 1 \cdot i = i \) |
| \( i^6 \) | -1 | \( i^4 \cdot i^2 = 1 \cdot (-1) = -1 \) |
| \( i^7 \) | \( -i \) | \( i^4 \cdot i^3 = 1 \cdot (-i) = -i \) |
| \( i^8 \) | 1 | \( i^4 \cdot i^4 = 1 \cdot 1 = 1 \) |
Cycle Pattern:
\( 1, \; i, \; -1, \; -i, \; 1, \; i, \; -1, \; -i, \; ... \)
Quick Method to Find Any Power of i:
Formula:
Divide the exponent by 4 and find the remainder:
| Remainder when n ÷ 4 | \( i^n = \) |
|---|---|
| 0 | 1 |
| 1 | \( i \) |
| 2 | -1 |
| 3 | \( -i \) |
Examples:
Example 1: Find \( i^{23} \)
23 ÷ 4 = 5 remainder 3
Remainder = 3 → \( i^{23} = i^3 = -i \)
Answer: \( -i \)
Example 2: Find \( i^{100} \)
100 ÷ 4 = 25 remainder 0
Remainder = 0 → \( i^{100} = i^0 = 1 \)
Answer: 1
Example 3: Find \( i^{46} \)
46 ÷ 4 = 11 remainder 2
Remainder = 2 → \( i^{46} = i^2 = -1 \)
Answer: -1
8. Quick Reference Summary
Key Formulas:
\( i = \sqrt{-1}, \quad i^2 = -1 \)
Standard Form: \( z = a + bi \)
Addition: \( (a + bi) + (c + di) = (a + c) + (b + d)i \)
Subtraction: \( (a + bi) - (c + di) = (a - c) + (b - d)i \)
Multiplication: \( (a + bi)(c + di) = (ac - bd) + (ad + bc)i \)
Complex Conjugate: \( \overline{a + bi} = a - bi \)
Conjugate Product: \( (a + bi)(a - bi) = a^2 + b^2 \)
Absolute Value: \( |a + bi| = \sqrt{a^2 + b^2} \)
Division: \( \frac{a + bi}{c + di} = \frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} \)
Powers of i: Divide exponent by 4, use remainder
Remainder 0 → 1, Remainder 1 → i, Remainder 2 → -1, Remainder 3 → -i
📚 Study Tips
✓ Always remember \( i^2 = -1 \) when multiplying
✓ Combine like terms (real with real, imaginary with imaginary)
✓ To divide, multiply by conjugate of denominator
✓ Use remainder method for powers of i (divide exponent by 4)
✓ Practice FOIL method for multiplication