Complex Numbers and Quadratic Equations: CBSE Class 11 Maths Chapter 5 Notes

Key Terms and Definitions

Imaginary Number (iota)

A number whose square is negative. The fundamental imaginary unit is iota (i), defined as i = √(-1).

Complex Number (z)

A number of the form a + ib, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. 'a' is the real part and 'b' is the imaginary part.

Real Part of z (Re(z))

For a complex number z = a + ib, the real part is 'a'.

Imaginary Part of z (Im(z))

For a complex number z = a + ib, the imaginary part is 'b'.

Modulus of z (|z|)

The non-negative real number √(a² + b²) for a complex number z = a + ib. It represents the distance of the point (a,b) from the origin on the Argand plane.

Conjugate of z (z̄)

For a complex number z = a + ib, its conjugate is z̄ = a - ib. It is obtained by changing the sign of the imaginary part.

Argument of z (arg(z))

The angle θ made by the line segment connecting the origin to the point representing z on the Argand plane with the positive real axis. The value of θ such that -π < θ ≤ π is called the Principal Argument.

Polar Form

A representation of a complex number z using its modulus (r) and argument (θ) as z = r(cosθ + isinθ).

Must-Remember Formulas and Properties

• Powers of Iota (i): i¹ = i, i² = -1, i³ = -i, i⁴ = 1. The cycle repeats every 4 powers. For any integer k, i⁴ᵏ = 1, i⁴ᵏ⁺¹ = i, i⁴ᵏ⁺² = -1, i⁴ᵏ⁺³ = -i.
• Equality: Two complex numbers z₁ = a + ib and z₂ = c + id are equal if and only if a = c and b = d.
• Modulus and Conjugate Relation: For any complex number z, z * z̄ = |z|².
• Properties of Modulus: |z₁z₂| = |z₁||z₂| and |z₁/z₂| = |z₁|/|z₂| (where z₂ ≠ 0).
• Properties of Conjugate: (z₁ ± z₂)̄ = z₁̄ ± z₂̄, (z₁z₂)̄ = z₁̄z₂̄, and (z₁/z₂)̄ = z₁̄/z₂̄ (where z₂ ≠ 0).
• Multiplicative Inverse: The multiplicative inverse of a non-zero complex number z = a + ib is z⁻¹ = z̄ / |z|² = (a - ib) / (a² + b²).
• Quadratic Formula (for D < 0): For ax² + bx + c = 0, where a, b, c ∈ R and b² - 4ac < 0, the roots are x = [-b ± i√(4ac - b²)] / 2a.
• Polar Form Multiplication: If z₁ = r₁(cosθ₁ + isinθ₁) and z₂ = r₂(cosθ₂ + isinθ₂), then z₁z₂ = r₁r₂(cos(θ₁ + θ₂) + isin(θ₁ + θ₂)).

The Argand Plane and Polar Representation

Until now, we've visualized numbers on a single line—the number line. However, a complex number z = x + iy has two components: a real part (x) and an imaginary part (y). This structure naturally lends itself to a two-dimensional representation. The plane used to plot complex numbers is called the Argand plane or the complex plane. In this plane, the horizontal axis is the Real Axis (representing the real part x) and the vertical axis is the Imaginary Axis (representing the imaginary part y). So, the complex number z = x + iy corresponds to the unique point P(x, y) in this Cartesian coordinate system.

This geometric view leads to a powerful alternative way to express complex numbers: the Polar Form. Instead of using Cartesian coordinates (x, y), we can describe the point P using its distance from the origin (O) and the angle it makes with the positive real axis. The distance OP is called the modulus of z, denoted by r = |z| = √(x² + y²). The angle, measured counter-clockwise from the positive real axis, is the argument or amplitude of z, denoted by θ. Using basic trigonometry, we can see that x = r cos(θ) and y = r sin(θ). Substituting these back into z = x + iy gives the polar representation: z = r(cosθ + isinθ). This form is extremely useful for multiplication and division of complex numbers, as well as for understanding roots and powers via De Moivre's Theorem.

How to Find the Square Root of a Complex Number

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Worked Examples

• Find the modulus and argument of the complex number z = -1 - i√3. Here, a = -1 and b = -√3. Modulus: |z| = r = √((-1)² + (-√3)²) = √(1 + 3) = √4 = 2. Argument: Since both x and y are negative, the point lies in the 3rd quadrant. tanα = |b/a| = |-√3/-1| = √3. So, α = π/3. For the 3rd quadrant, the principal argument θ = -(π - α) = -(π - π/3) = -2π/3. Thus, |z| = 2 and arg(z) = -2π/3.
• Solve the quadratic equation: x² + x + 1 = 0. Here, a=1, b=1, c=1. Discriminant (D): D = b² - 4ac = 1² - 4(1)(1) = 1 - 4 = -3. Since D < 0, the roots are complex. Using the formula x = [-b ± i√(4ac - b²)] / 2a: x = [-1 ± i√(4(1)(1) - 1²)] / 2(1) x = [-1 ± i√(3)] / 2 The roots are x = -1/2 + i√3/2 and x = -1/2 - i√3/2.

Exam Traps and Scoring Tips

Principal Argument: A very common error is calculating the argument θ without considering the quadrant. Always check the signs of the real part (x) and imaginary part (y) to determine the correct quadrant for your point. Remember the principal argument θ must be in the range -π < θ ≤ π.

• Quadrant 1: θ = α
• Quadrant 2: θ = π - α
• Quadrant 3: θ = -(π - α)
• Quadrant 4: θ = -α

(where α is the acute angle tan⁻¹|y/x|)

Final Form: When asked to express a complex number in a + ib form, ensure your final answer is simplified to that format. For example, don't leave your answer as (2+3i)/(1-i). Rationalize the denominator to get the final a + ib form. This shows completeness and secures full marks.

Quick Revision Check

• What is the value of i⁹⁹? i⁹⁹ = i^(4*24 + 3) = i³ = -i.
• Find the multiplicative inverse of z = 4 - 3i. z⁻¹ = z̄ / |z|². Here z̄ = 4 + 3i and |z|² = 4² + (-3)² = 16 + 9 = 25. So, z⁻¹ = (4 + 3i) / 25 = 4/25 + (3/25)i.
• What are the roots of the equation x² + 16 = 0? x² = -16 => x = √(-16) => x = √(16 * -1) => x = ±4i.
• What is the conjugate of the complex number z = (2+i)²? First, expand z = (2+i)² = 4 + i² + 4i = 4 - 1 + 4i = 3 + 4i. The conjugate is z̄ = 3 - 4i.

Frequently Asked Questions