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Class 11 Maths Notes

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.

Class11SubjectMathematicsCoversCBSE · JEE

Written byDeep Narayan· Science & Mathematics EducatorReviewed byPushpanjali

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

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.

i2=−1,z=a+ib, a=Re(z), b=Im(z)i^2 = -1,\qquad z = a + ib,\ a = \text{Re}(z),\ b = \text{Im}(z)
Imaginary unit and standard form
  • 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

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

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.

zˉ=a−ib,∣z∣=a2+b2\bar{z} = a - ib,\qquad |z| = \sqrt{a^2 + b^2}
Conjugate and modulus
z zˉ=∣z∣2z\,\bar{z} = |z|^2
Conjugate times the number

Argand Plane and Polar Form

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.

z=r(cos⁡θ+isin⁡θ),r=∣z∣=a2+b2z = r(\cos\theta + i\sin\theta),\qquad r = |z| = \sqrt{a^2 + b^2}
Polar form of a complex number

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

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

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.

1+ω+ω2=0,ω3=11 + \omega + \omega^2 = 0,\qquad \omega^3 = 1
Cube roots of unity relations

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)).

Revision

Key formulas at a glance

Memorise these before attempting numericals — most exam questions hinge on one of them.

Imaginary unit

i2=−1i^2 = -1

Conjugate squared

z zˉ=∣z∣2z\,\bar{z} = |z|^2

Product of moduli

∣z1z2∣=∣z1∣ ∣z2∣|z_1 z_2| = |z_1|\,|z_2|

Conjugate

zˉ=a−ib\bar{z} = a - ib

Modulus

∣z∣=a2+b2|z| = \sqrt{a^2 + b^2}

Polar form

z=r(cos⁡θ+isin⁡θ)z = r(\cos\theta + i\sin\theta)

Cube roots of unity

1+ω+ω2=0, ω3=11 + \omega + \omega^2 = 0,\ \omega^3 = 1

Exam tips

How this chapter is asked

Where this topic appears in CBSE, JEE Main and NEET papers.

  • 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

Common 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.

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