Arithmetic with complex numbers: Class 11 practice
Treat i like a variable but replace i² by −1. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Algebraic expansion
The idea behind arithmetic with complex numbers
Treat i like a variable but replace i² by −1. Add real and imaginary parts separately; to divide, multiply top and bottom by the conjugate of the denominator.
Step-by-step method
- Expand brackets.
- Replace i² by −1.
- Collect real and imaginary parts.
- For division multiply by the conjugate.
Worked example 1 (Basic)
Find (-2 + 2i) × (-5 + 2i).
- = 10 + -4i + -10i + 4i²
- = 6 − 14i
- Answer: 6 − 14i
Worked example 2 (Standard)
Find the modulus of 0 + 4i and its conjugate.
- |z| = √(0 + 16) = √16
- z̄ = 0 − 4i
- Answer: |z| = √16, z̄ = 0 − 4i
Worked example 3 (Challenge)
Write (4 + 1i) ÷ (5 + 1i) in the form x + iy.
- Multiply by 5 − 1i
- Denominator = 26
- = 21/26 + 1/26i
- Answer: 21/26 + 1/26i
Common mistakes and misconceptions
- Writing i² = 1
- Multiplying only the denominator by the conjugate
Practice questions
- (Basic) Find (1 − 6i) × (6 + 4i).
- (Basic) Find (-5 − 5i) × (-3 − 1i).
- (Standard) Find the modulus of -1 − 6i and its conjugate.
- (Standard) Find the modulus of 5 − 1i and its conjugate.
- (Challenge) Write (-5 + 4i) ÷ (-1 + 3i) in the form x + iy.
- (Challenge) Write (3 − 1i) ÷ (1 − 4i) in the form x + iy.
Answer key with full working
- Q1: = 6 + 4i + -36i + -24i² → = 30 − 32i → Answer: 30 − 32i
- Q2: = 15 + 5i + 15i + 5i² → = 10 + 20i → Answer: 10 + 20i
- Q3: |z| = √(1 + 36) = √37 → z̄ = -1 + 6i → Answer: |z| = √37, z̄ = -1 + 6i
- Q4: |z| = √(25 + 1) = √26 → z̄ = 5 + 1i → Answer: |z| = √26, z̄ = 5 + 1i
- Q5: Multiply by -1 − 3i → Denominator = 10 → = 17/10 + 11/10i → Answer: 17/10 + 11/10i
- Q6: Multiply by 1 + 4i → Denominator = 17 → = 7/17 + 11/17i → Answer: 7/17 + 11/17i
Syllabus note
This skill is listed under "Complex Numbers" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is arithmetic with complex numbers in?
"Complex Numbers", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.