Class 11 Maths Notes

~10 min read

Complex Numbers Class 11 Notes

Complete, exam-ready notes on complex numbers: the imaginary unit i, the form z = a + ib, addition, subtraction, multiplication and division, conjugates and moduli, the Argand plane and polar form, quadratic equations with real coefficients, and cube roots of unity — written for CBSE, JEE and NEET revision.

Class:11Subject:MathematicsCovers:CBSE · JEE

7 Key Formulas4 Practice MCQs

DWritten byDeep Narayan

UpdatedSeptember 2026

Key Concept Summary

What is this chapter about in one line?

Complex numbers extend the real number line to a plane using the imaginary unit i, giving solutions to every quadratic and rich tools like modulus and polar form.

The Imaginary Unit and Standard Form

Definition

Imaginary unit i and complex number z = a + ib

The imaginary unit i satisfies i² = −1. A complex number is written z = a + ib, where a is the real part Re(z) and b is the imaginary part Im(z). Real numbers are the case b = 0; pure imaginary numbers have a = 0.

Imaginary unit and standard form

i^{2} = - 1, z = a + ib, a = Re (z), b = Im (z)

Powers of i cycle in fours: i¹ = i, i² = −1, i³ = −i, i⁴ = 1.
Two complex numbers are equal iff their real and imaginary parts are equal.
Zero in complex form is 0 + 0i.

Addition, Subtraction, Multiplication and Division

Definition

Operations on complex numbers

Add or subtract like terms: add real to real and imaginary to imaginary. Multiply by expanding (foil), using i² = −1. Divide by multiplying numerator and denominator by the conjugate of the denominator to make it real.

Addition: (a + ib) + (c + id) = (a + c) + i(b + d).
Multiplication: (a + ib)(c + id) = (ac − bd) + i(ad + bc).
Division: (a + ib)/(c + id) = (a + ib)(c − id)/(c² + d²).

Conjugate and Modulus

Definition

Conjugate z̄ and modulus |z|

The conjugate of z = a + ib is z̄ = a − ib; geometrically it reflects z across the real axis. The modulus |z| is the distance of z from the origin in the Argand plane. Multiplying a number by its conjugate gives the square of its modulus.

Conjugate and modulus

\overset{z}{ˉ} = a - ib, ∣ z ∣ = a^{2} + b^{2}

Conjugate times the number

z \overset{z}{ˉ} = ∣ z ∣^{2}

Argand Plane and Polar Form

Definition

Argand plane and polar form

In the Argand plane, the real part is the x-axis and the imaginary part the y-axis. Polar form writes z = r(cosθ + i sinθ) where r = |z| is the modulus and θ the argument, the angle the vector makes with the positive real axis.

Polar form of a complex number

z = r (cos θ + i sin θ), r = ∣ z ∣ = a^{2} + b^{2}

Product of moduli

Moduli multiply in product and division: |z₁z₂| = |z₁||z₂| and |z₁/z₂| = |z₁|/|z₂|. Arguments add for a product.

Quadratic Equations with Complex Roots

Definition

Quadratic with real coefficients

A quadratic ax² + bx + c = 0 with real coefficients has complex (non-real) roots exactly when its discriminant is negative. Such roots always appear as a conjugate pair a ± ib.

Discriminant D = b² − 4ac; if D < 0 the roots are complex conjugates.
Roots: x = (−b ± √(b² − 4ac))/(2a), using i for the square root of a negative number.
Sum of roots = −b/a and product = c/a for any quadratic.

Cube Roots of Unity

Definition

Cube roots of unity

The three cube roots of 1 are 1, ω and ω². They always satisfy ω³ = 1 and 1 + ω + ω² = 0, and they lie at the vertices of an equilateral triangle in the Argand plane.

Cube roots of unity relations

1 + ω + ω^{2} = 0, ω^{3} = 1

Simplify high powers

Because ω³ = 1, any power of ω reduces by its remainder when divided by 3. Together with 1 + ω + ω² = 0 this collapses many JEE-style sums fast.

Solved Examples

Example: Find the conjugate and modulus of z = 3 + 4i, and compute z·z̄.

Solution: z̄ = 3 − 4i; |z| = √(9 + 16) = 5; z·z̄ = 9 + 16 = 25 = |z|².

Example: Write z = 1 + i in polar form.

Solution: r = √(1 + 1) = √2 and θ = tan⁻¹(1/1) = π/4, so z = √2(cos(π/4) + i sin(π/4)).

NCERT Exercise Questions

From the textbook

All NCERT questions for this chapter — NCERT Maths Part I, Chapter 4 (Complex Numbers and Quadratic Equations). Questions follow the textbook order so you can match them to the book.

Exercises (NCERT Q1–Q14)

1) Express the complex number (5i)(−3/5 i) in the form a + ib.
2) Express the complex number i⁹ + i¹⁹ in the form a + ib.
3) Express the complex number i⁻³⁹ in the form a + ib.
4) Express the complex number 3(7 + 7i) + i(7 + 7i) in the form a + ib.
5) Express the complex number (1 − i) − (−1 + 6i) in the form a + ib.
6) Express the complex number (1/5 + 2/5 i) − (4 + 5/2 i) in the form a + ib.
7) Express the complex number [(1/3 + 7/3 i) + (4 + 1/3 i)] − (−4/3 + i) in the form a + ib.
8) Express the complex number (1 − i)⁴ in the form a + ib.
9) Express the complex number (1/3 + 3i)³ in the form a + ib.
10) Express the complex number (−2 − 1/3 i)³ in the form a + ib.
11) Find the multiplicative inverse of the complex number 4 − 3i.
12) Find the multiplicative inverse of the complex number √5 + 3i.
13) Find the multiplicative inverse of the complex number −i.
14) Express the expression (3 + i√5)(3 − i√5) / [(√3 + √2 i) − (√3 − √2 i)] in the form a + ib.

Answers at a glance

1
Q1 → 3 + 0i.
2
Q2 → 0 + 0i.
3
Q3 → 0 + i.
4
Q4 → 14 + 28i.
5
Q5 → 2 − 7i.
6
Q6 → −19/5 − 21/10 i.
7
Q7 → 17/3 + 5/3 i.
8
Q8 → −4 + 0i.
9
Q9 → −242/27 − 26i.
10
Q10 → −22/3 − 107/27 i.
11
Q11 → 4/25 + 3/25 i.
12
Q12 → √5/14 − 3/14 i.
13
Q13 → i.
14
Q14 → 0 − (7/√2)i.

CBSE Pattern Questions

How CBSE asks this chapter

CBSE Class 11 Mathematics board paper: Section A (20 MCQs × 1M), Section B (5 × 2M), Section C (6 × 3M) and Section D (4 × 4M). Complex Numbers and Quadratic Equations tests 1-mark MCQs on imaginary unit, modulus and conjugates, 2/3-mark operations reducing to a + ib form, and 4-mark questions on polar form, argument and solving quadratic equations over C.

MCQ Questions

The value of i² is

The modulus of 3 + 4i is

The conjugate of 3 − 2i is

The argument of 1 + i is

1 Mark Questions

1
Find the modulus of −2i. (Ans: 2)
2
Write i⁴. (Ans: 1)
3
State the condition for two complex numbers to be equal.
4
Find the conjugate of 5 + 2i. (Ans: 5 − 2i)

2 Mark Questions

1
Express (2 + 3i)(1 − i) in the form a + ib. (Ans: 5 + i)
2
Find the real and imaginary parts of (1 + 2i)/(1 − i). (Ans: multiply by conjugate: (−1 + 3i)/2, real −1/2, imaginary 3/2)
3
Find (1 + i)² and write its modulus. (Ans: 2i, modulus 2)

3 Mark Questions

1
Find the multiplicative inverse of 1 − i. (Ans: (1 + i)/2 = 1/2 + (1/2)i)
2
Write 1 + √3 i in polar form. (Ans: r = 2, θ = π/3, so 2(cos π/3 + i sin π/3))
3
Solve x² + 2 = 0 over C. (Ans: x = ±√2 i)

4 Mark Questions

1
Derive the formula for the square root of a complex number a + ib and use it to find √(7 + 24i). (Ans: ±(4 + 3i))
2
Find the modulus and argument of (1 + i)/(1 − i), and write the result in polar form. (Ans: (−1)²+... result is i; modulus 1, argument π/2)

PDF: Assignments & Tests

Printable study material

Free downloadable PDFs for Complex Numbers and Quadratic Equations — two topic-wise assignments covering every concept of the chapter, and two full CBSE-pattern tests (each 90 minutes, 40 marks) mirroring the board paper. No signup required.

Free PDF Downloads & Assignments

No Signup Required

Practice path: complete Assignment 1 first (topic-by-topic), then Assignment 2 (mixed difficulty with full solutions), then attempt the timed tests.

Complex Numbers Assignment 1 — Topic-wise Questions

Covers all six topics: the imaginary unit and standard form; addition, subtraction, multiplication and division; conjugate and modulus; Argand plane and polar form; quadratic equations with complex roots; and cube roots of unity.

complex-numbers-class-11-notes-assignment-1.pdf

Download Free PDF(No signup)

Complex Numbers Assignment 2 — Challenge + Integrated Problems

Higher-difficulty set with multi-step problems (powers of i, division in a + ib form, modulus/argument of quotients, polar conversion, complex quadratic roots, ω identities) and full worked answers. Includes an answer key.

complex-numbers-class-11-notes-assignment-2.pdf

Download Free PDF(No signup)

Complex Numbers Test 1 — CBSE Pattern (90 min, 40 marks)

Full board-pattern test: Section A (MCQ/VSA 1M × 10), Section B (2M × 4), Section C (3M × 4), Section D (5M × 2) = 40 marks. Time: 90 minutes. Includes answer key.

complex-numbers-class-11-notes-test-1-cbse.pdf

Download Free PDF(No signup)

Complex Numbers Test 2 — CBSE Pattern (90 min, 40 marks)

Second full paper with different questions, same 40-mark / 90-minute blueprint. Includes answer key.

complex-numbers-class-11-notes-test-2-cbse.pdf

Download Free PDF(No signup)

Quick Revision

Key formulas at a glance

Memorise these equations — direct application numericals and derivations in CBSE & JEE frequently hinge on these.

Imaginary unit

i^{2} = - 1

Conjugate squared

z \overset{z}{ˉ} = ∣ z ∣^{2}

Product of moduli

∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣ ∣ z_{2} ∣

Conjugate

\overset{z}{ˉ} = a - ib

Modulus

∣ z ∣ = a^{2} + b^{2}

Polar form

z = r (cos θ + i sin θ)

Cube roots of unity

1 + ω + ω^{2} = 0, ω^{3} = 1

Exam Strategy

How this chapter is asked

High-yield question patterns observed across CBSE boards, JEE Main & Advanced, and NEET.

i² = −1; powers of i cycle through i, −1, −i, 1.
Two complex numbers are equal iff real parts and imaginary parts match.
Divide by multiplying the top and bottom by the conjugate.
The modulus |z| = √(a² + b²) is the distance from the origin.
z·z̄ = |z|² is handy for dividing and simplifying.
Polar form z = r(cosθ + i sinθ) simplifies products and powers.
Cube roots of unity: 1 + ω + ω² = 0 and ω³ = 1.

FAQ

Frequently asked questions

What is the imaginary unit i?

i is defined by i² = −1. Any complex number is written z = a + ib, with a the real part and b the imaginary part.

How do you divide two complex numbers?

Multiply both numerator and denominator by the conjugate of the denominator. For (a + ib)/(c + id), this gives (a + ib)(c − id)/(c² + d²), making the denominator a real number.

What are the conjugate and modulus of a complex number?

The conjugate of a + ib is a − ib. The modulus is |z| = √(a² + b²), the distance from the origin. Their product z·z̄ = |z|² is always a non-negative real number.

What are the cube roots of unity?

The three cube roots of 1 are 1, ω and ω². They satisfy ω³ = 1 and 1 + ω + ω² = 0, and they are placed at the vertices of an equilateral triangle in the Argand plane.

Next Chapters

Continue learning

Class 11 Maths Notes Sets Notes All Class 11 & 12 Notes Class 12 Maths Notes Relations and Functions Notes Maths Tutor in Mohali Class 11 Maths NCERT Solutions Class 11 Maths NCERT Solutions — Chapter 4

Same chapter

Read alongside

Complex Numbers and Quadratic Equations Class 11 Maths NCERT Solutions

Interactive Quiz

Chapter MCQ practice test

Instant scoring with complete solutions — test your mastery in under 15 minutes.

Active Recall Practice

Chapter MCQ Mock Test

Evaluate how well you have retained the concepts, formulas, and reaction mechanisms from this chapter. Questions adhere strictly to latest CBSE, JEE & NEET trends.

4 questions~6 minutesInstant Score & Solutions

Master this chapter with expert live guidance

Self-study notes lay the ground, but conceptual doubts clear fastest in an interactive classroom. Narayan Gurukul Academy (ClassApna) conducts small-batch CBSE, JEE & NEET coaching with daily doubt solving and rigorous mock tests.

Book a Free Demo Class Explore Plans & Batches

Small batches · 1-on-1 personal mentorship · Live online & offline centre