Complex Numbers
Complete Notes & Formulae for Twelfth Grade (Precalculus)
1. Complex Number Basics
Standard Form:
\[ z = a + bi \]
• \( a \) = real part
• \( b \) = imaginary part
• \( i \) = imaginary unit where \( i^2 = -1 \)
• \( i = \sqrt{-1} \)
2. Add and Subtract Complex Numbers
Rules:
Combine real parts separately and imaginary parts separately
Addition:
\[ (a + bi) + (c + di) = (a + c) + (b + d)i \]
Subtraction:
\[ (a + bi) - (c + di) = (a - c) + (b - d)i \]
Examples:
Add: \( (3 + 2i) + (5 - 4i) \)
Combine real parts: \( 3 + 5 = 8 \)
Combine imaginary parts: \( 2 + (-4) = -2 \)
Result: \( 8 - 2i \)
Subtract: \( (7 + 3i) - (2 + 5i) \)
Subtract real parts: \( 7 - 2 = 5 \)
Subtract imaginary parts: \( 3 - 5 = -2 \)
Result: \( 5 - 2i \)
3. Complex Conjugates
Definition:
The complex conjugate of \( z = a + bi \) is obtained by changing the sign of the imaginary part
\[ \text{If } z = a + bi, \text{ then } \overline{z} = a - bi \]
Properties:
1. \( \overline{\overline{z}} = z \) (conjugate of conjugate is original)
2. \( z + \overline{z} = 2a \) (sum is real)
3. \( z - \overline{z} = 2bi \) (difference is imaginary)
4. \( z \cdot \overline{z} = a^2 + b^2 \) (product is real and non-negative)
Example:
Find conjugate of \( z = 4 - 7i \)
\( \overline{z} = 4 + 7i \)
4. Multiply Complex Numbers
Formula:
Use the distributive property (FOIL) and remember that \( i^2 = -1 \)
\[ (a + bi)(c + di) = (ac - bd) + (ad + bc)i \]
Steps:
1. Multiply using FOIL method
2. Replace \( i^2 \) with -1
3. Combine like terms (real with real, imaginary with imaginary)
Example:
Multiply: \( (3 + 2i)(4 - 5i) \)
F: \( 3 \cdot 4 = 12 \)
O: \( 3 \cdot (-5i) = -15i \)
I: \( 2i \cdot 4 = 8i \)
L: \( 2i \cdot (-5i) = -10i^2 = -10(-1) = 10 \)
Combine: \( 12 + 10 + (-15i + 8i) \)
Result: \( 22 - 7i \)
5. Divide Complex Numbers
Method:
Multiply numerator and denominator by the conjugate of the denominator
\[ \frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} \]
Steps:
1. Find the conjugate of the denominator
2. Multiply both numerator and denominator by this conjugate
3. Simplify (denominator becomes real: \( c^2 + d^2 \))
4. Write in standard form \( a + bi \)
Example:
Divide: \( \frac{3 + 2i}{1 - 4i} \)
Conjugate of denominator: \( 1 + 4i \)
Multiply: \( \frac{(3 + 2i)(1 + 4i)}{(1 - 4i)(1 + 4i)} \)
Numerator: \( 3 + 12i + 2i + 8i^2 = 3 + 14i - 8 = -5 + 14i \)
Denominator: \( 1 + 16 = 17 \)
Result: \( \frac{-5 + 14i}{17} = -\frac{5}{17} + \frac{14}{17}i \)
6. Absolute Value (Modulus) of Complex Numbers
Definition:
The absolute value (or modulus) represents the distance from the origin in the complex plane
\[ |z| = |a + bi| = \sqrt{a^2 + b^2} \]
\[ |z|^2 = z \cdot \overline{z} = a^2 + b^2 \]
Properties:
• \( |z| \geq 0 \) (always non-negative)
• \( |z| = 0 \) if and only if \( z = 0 \)
• \( |z_1 \cdot z_2| = |z_1| \cdot |z_2| \)
• \( \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \) (if \( z_2 \neq 0 \))
Examples:
Find \( |3 + 4i| \)
\( |3 + 4i| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} \)
Result: \( |3 + 4i| = 5 \)
Find \( |5 - 12i| \)
\( |5 - 12i| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} \)
Result: \( |5 - 12i| = 13 \)
7. Powers of i
The Cycle:
Powers of i repeat in a cycle of 4:
| Power | Value | Calculation |
|---|---|---|
| \( i^1 \) | \( i \) | - |
| \( i^2 \) | \( -1 \) | \( i \cdot i = -1 \) |
| \( i^3 \) | \( -i \) | \( i^2 \cdot i = -1 \cdot i = -i \) |
| \( i^4 \) | \( 1 \) | \( i^2 \cdot i^2 = (-1)(-1) = 1 \) |
| \( i^5 \) | \( i \) | \( i^4 \cdot i = 1 \cdot i = i \) |
Pattern Summary:
\[ i^1 = i \quad i^2 = -1 \quad i^3 = -i \quad i^4 = 1 \]
Then the pattern repeats: \( i^5 = i, i^6 = -1, i^7 = -i, i^8 = 1, \ldots \)
Method to Find \( i^n \):
1. Divide the exponent by 4
2. Find the remainder (0, 1, 2, or 3)
3. Use the remainder to determine the value:
• Remainder 0: \( i^n = 1 \)
• Remainder 1: \( i^n = i \)
• Remainder 2: \( i^n = -1 \)
• Remainder 3: \( i^n = -i \)
Examples:
Find \( i^{17} \)
Divide: 17 ÷ 4 = 4 remainder 1
Result: \( i^{17} = i \)
Find \( i^{50} \)
Divide: 50 ÷ 4 = 12 remainder 2
Result: \( i^{50} = -1 \)
Find \( i^{99} \)
Divide: 99 ÷ 4 = 24 remainder 3
Result: \( i^{99} = -i \)
8. Quick Reference Summary
Essential Formulas:
Standard Form: \( z = a + bi \)
Addition: \( (a+bi) + (c+di) = (a+c) + (b+d)i \)
Multiplication: \( (a+bi)(c+di) = (ac-bd) + (ad+bc)i \)
Conjugate: \( \overline{a+bi} = a-bi \)
Absolute Value: \( |a+bi| = \sqrt{a^2+b^2} \)
Division: Multiply by conjugate of denominator
Powers of i: Divide exponent by 4, use remainder
📚 Study Tips
✓ Remember: i² = -1 is the foundation of all complex number operations
✓ To divide: multiply by conjugate to make denominator real
✓ Powers of i cycle every 4: i, -1, -i, 1, then repeat
✓ Conjugate changes sign of imaginary part only
✓ Absolute value is always non-negative (like distance)