Mathematics · Class 11
Complex Numbers
Learn Complex Numbers with notes, examples, and practice questions.
3 sections15–25 min read6 MCQs · 2 examples
Chapter Overview
Complex numbers extend reals by introducing i where i² = −1. They enable solutions to x²+1=0 and simplify trigonometry and polynomials.
Forms
Standard
z = a + ib
Modulus
|z| = √(a²+b²)
Argument
arg(z) = θ, tan θ = b/a
Polar
z = r(cos θ + i sin θ)
Euler & Operations
Euler
e^(iθ) = cos θ + i sin θ
De Moivre
(cos θ + i sin θ)ⁿ = cos nθ + i sin nθ
Conjugate
z̄ = a − ib; z·z̄ = |z|²
Solved Examples
Step-by-step solutions — read each step before checking the final answer.
Modulus
|3+4i|
- 1√(9+16)=5
Answer
5
Multiply
(1+i)²
- 11+2i+i²=2i
Answer
2i
Key Points to Remember
- ✓ i⁴ = 1; powers of i cycle every 4
- ✓ Quadratic with D<0 has complex roots
- ✓ Argand plane plots a + ib
Exam Tips
- • Rationalise using conjugate
- • Use polar form for multiplication/division
Practice MCQs — Complex Numbers
Test your understanding with topic-wise multiple choice questions. Explanations appear after each answer.
Question 1 of 6Score: 0/0
i² equals: