Complex Numbers
Maharashtra Board · Class 11 · Mathematics & Statistics
Quick revision notes for Complex Numbers — Maharashtra Board Class 11 Mathematics & Statistics. Key concepts, formulas, and definitions for last-minute revision.
Key Topics to Revise
Introduction to Complex Numbers and Imaginary Unit
- The imaginary unit i is defined as i = √(-1), so i² = -1
- Powers of i follow a cyclical pattern: i¹ = i, i² = -1, i³ = -i, i⁴ = 1
- For any positive integer n: i⁴ⁿ = 1, i⁴ⁿ⁺¹ = i, i⁴ⁿ⁺² = -1, i⁴ⁿ⁺³ = -i
Operations on Complex Numbers
- Addition: (a + ib) + (c + id) = (a + c) + i(b + d)
- Subtraction: (a + ib) - (c + id) = (a - c) + i(b - d)
- Multiplication: (a + ib)(c + id) = (ac - bd) + i(ad + bc)
Square Root of Complex Numbers
- If √(x + iy) = a + ib, then squaring both sides gives x + iy = (a + ib)²
- This leads to: x = a² - b² and y = 2ab
- Solve simultaneously to find a and b
Quadratic Equations with Complex Numbers
- For ax² + bx + c = 0, roots are x = (-b ± √(b² - 4ac))/(2a)
- If discriminant D = b² - 4ac < 0, roots are complex conjugates
- Complex roots always occur in conjugate pairs when coefficients are real
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