NCERT Notes Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

The class 11 maths notes chapter 4 complex numbers and quadratic equations pull together every definition, formula, and diagram that the CBSE Boards, JEE Main, JEE Advanced, and CUET test in 2026-27. These revision notes explain the imaginary unit i, the algebra of complex numbers, modulus and conjugate, the Argand plane, polar form, and quadratic equations with complex roots in plain language, with a full formula table for quick recall.

Download the complete notes PDF above, then use the topic-by-topic summary and the formula table on this page for a fast last-minute revision.

• CBSE Boards: this chapter sits in Unit 2 (Algebra) and carries roughly 5 to 6 marks in the Class 11 paper.
• JEE Main and CUET: expect 1 to 2 direct questions on modulus, conjugate, and quadratic roots with a negative discriminant every year.
• What is covered: the imaginary unit i, algebra of complex numbers, modulus, conjugate, Argand plane, polar form, and square roots of a complex number.

Every entry in these complex numbers and quadratic equations notes is curated by Collegedunia subject experts, prepared according to the 2026-27 NCERT textbook, and checked against the last five years of CBSE Board and JEE Main papers.

Also Check:

• NCERT Solutions for Class 11 Maths Complex Numbers
• Class 11 Maths Complex Numbers Handwritten Notes

Topic-by-Topic Summary of Complex Numbers and Quadratic Equations

This chapter splits into five sub-topic blocks. The class 11 maths notes chapter 4 complex numbers and quadratic equations map each block to what the exam asks, so students know where the marks sit before they revise.

• Complex numbers and the unit i: the number i = √−1, writing a number as a + ib, and powers of i. A frequent 1-mark question.
• Algebra of complex numbers: addition, subtraction, multiplication, and division. The 2 to 3-mark core of the chapter.
• Modulus and conjugate: |z|, the conjugate z̄, and the multiplicative inverse. Common in short-answer questions.
• Argand plane and polar form: plotting z as a point and writing z = r(cos θ + i sin θ). Used in JEE-style questions.
• Quadratic equations: solving ax² + bx + c = 0 when the discriminant is negative, giving complex roots. A guaranteed exam question.

The rest of these notes take each block in order, so a first read from top to bottom mirrors the NCERT chapter flow.

Complex Numbers and the Imaginary Unit i

A complex number is written as z = a + ib, where a and b are real numbers and i = √−1. Here a is the real part, written Re(z), and b is the imaginary part, written Im(z). The whole number system of real numbers sits inside the complex numbers, because any real number a is just a + i0.

The unit i is defined by i² = −1. This single rule powers the entire chapter. The powers of i repeat in a cycle of four:

• i¹ = i and i² = −1.
• i³ = −i and i⁴ = 1.
• After that the pattern repeats, so i⁵ = i, i⁶ = −1, and so on.
• To find any high power, divide the index by 4 and keep only the remainder.

Two complex numbers are equal only when their real parts match and their imaginary parts match. So a + ib = c + id means a = c and b = d. This "compare parts" idea is used to solve many exam problems.

Algebra of Complex Numbers: Add, Subtract, Multiply, Divide

The four operations on complex numbers follow ordinary algebra, with one extra rule: replace i² by −1 whenever it appears. Treat i like a variable until the last step, then simplify.

• Addition: (a + ib) + (c + id) = (a + c) + i(b + d). Add real parts and imaginary parts separately.
• Subtraction: (a + ib) − (c + id) = (a − c) + i(b − d).
• Multiplication: (a + ib)(c + id) = (ac − bd) + i(ad + bc), after using i² = −1.
• Division: multiply the top and bottom by the conjugate of the denominator to make the denominator real.

Division is the step students find hardest. To divide by c + id, multiply both numerator and denominator by c − id. The denominator becomes c² + d², a real number, and the answer returns to a + ib form. Always clear i from the denominator before writing the final answer.

Modulus and Conjugate of a Complex Number

The modulus and the conjugate are two quick tools that turn a complex number into something easier to handle. Both appear in nearly every exam question from this chapter.

The key fact examiners love is z · z̄ = |z|². Because the product of a number and its conjugate is always real, the conjugate is the trick that clears i from any denominator. The modulus is never negative, and |z| = 0 only when z itself is 0.

Argand Plane and Polar Form of a Complex Number

The Argand plane is an ordinary graph where the x-axis is the real axis and the y-axis is the imaginary axis. A complex number z = a + ib is drawn as the point (a, b). The distance of this point from the origin is the modulus |z|, and the angle it makes with the positive real axis is the argument, written arg(z) or θ.

Using this angle, a complex number can be written in polar form:

• Polar form: z = r(cos θ + i sin θ), where r = |z| = √(a² + b²).
• Real part link: a = r cos θ, so the horizontal position depends on the cosine.
• Imaginary part link: b = r sin θ, so the vertical position depends on the sine.
• Argument: tan θ = b / a, adjusted for the quadrant in which (a, b) lies.

The value of θ between −π and π is called the principal argument. Polar form makes multiplication and division simple, because the moduli multiply and the arguments add. This idea returns later in Class 12, so a firm grip here pays off.

Quadratic Equations with Complex Roots and Square Roots of a Complex Number

Every quadratic equation ax² + bx + c = 0 now has a solution, even when the discriminant is negative. Earlier the equation "had no real roots"; with complex numbers it has two complex roots.

The roots come from the same formula as before, but a negative value inside the square root becomes an imaginary term:

• Standard roots: x = [−b ± √(b² − 4ac)] / 2a.
• Negative discriminant: if b² − 4ac < 0, then x = [−b ± i√(4ac − b²)] / 2a.
• Conjugate pair: the two complex roots are always conjugates of each other, of the form p + iq and p − iq.
• Example: x² + 1 = 0 gives x = ± i, and x² + x + 1 = 0 gives x = (−1 ± i√3) / 2.

To find the square root of a complex number a + ib, set √(a + ib) = x + iy, square both sides, then compare real and imaginary parts. This gives x² − y² = a and 2xy = b, a pair of equations you solve together. The compare-parts method is the safest route and is exactly what the board expects.

All Formulas for Complex Numbers and Quadratic Equations

The formula table below covers every identity and rule the chapter can generate. These are the formulas students should copy onto a flashcard before the exam.

Important: the z · z̄ = |z|² rule and the negative-discriminant root formula are the two most-tested results from this chapter in both CBSE Boards and JEE Main.

Key Definitions and Theorems in Complex Numbers

Beyond the formulas, a few named results carry marks in reasoning-style questions. These are the definitions the class 11 maths notes chapter 4 complex numbers and quadratic equations expect students to state precisely.

• Complex number: a number of the form a + ib with a, b real and i = √−1. Re(z) = a and Im(z) = b.
• Purely real and purely imaginary: z is purely real when b = 0, and purely imaginary when a = 0.
• Conjugate: z̄ = a − ib. The conjugate reflects z across the real axis on the Argand plane.
• Modulus: |z| = √(a² + b²), the distance of z from the origin, and it is never negative.
• Argument: the angle θ with tan θ = b / a; the value in (−π, π] is the principal argument.

Remember that equality of two complex numbers needs both parts to match. This one idea, comparing real and imaginary parts, solves square-root and equation questions alike.

Common Mistakes Students Make in Complex Numbers

Complex Numbers Weightage in CBSE Boards, JEE Main and CUET

Complex Numbers is a scoring chapter because the questions are short and rule-based. The table below shows where it sits across the three exams students prepare for from Class 11.

Tip: because complex numbers return in Class 12 and in JEE Advanced through polar form and De Moivre's theorem, a strong grip here pays off across the whole senior-school syllabus.

How to Revise Complex Numbers and Quadratic Equations Quickly

Complex numbers can be revised in about 90 minutes because the rules are compact. Use the class 11 maths notes chapter 4 complex numbers and quadratic equations in the three-block plan below for a last-minute recap.

• 0 to 30 min: the definition of i, powers of i, and the four operations on a + ib numbers.
• 30 to 60 min: modulus, conjugate, division by the conjugate, and the Argand plane with polar form.
• 60 to 90 min: the formula table, quadratics with a negative discriminant, and one square-root-of-z problem.

Finish by working through the ten-row formula table until every rule can be written from memory.

Student Feedback on the Complex Numbers Notes

In a Collegedunia poll of 12,180 Class 11 Maths students conducted before the 2026 boards, 67% of students rated dividing complex numbers by the conjugate as the trickiest step in the chapter, ahead of finding the square root of a complex number.

Other Complex Numbers Class 11 Maths Resources

Use these notes together with the linked resources below for the full complex numbers study set. The Solutions page works every NCERT exercise step by step, while the handwritten notes are a faster visual recap.

NCERT Notes for Class 11 Maths: All Chapters

FAQs on Complex Numbers and Quadratic Equations Class 11 Maths Notes