Class 11 — Mathematics (NCERT)

Complex Numbers and Quadratic Equations: Argand Plane, Modulus and Argument

Class 11

✓By the end of this lesson students will be able to represent a complex number geometrically on the Argand plane.
✓By the end of this lesson students will be able to calculate the modulus of a given complex number.
✓By the end of this lesson students will be able to determine the principal argument of a complex number.
✓By the end of this lesson students will be able to understand the geometric interpretation of modulus and argument.

Key concepts

Argand Plane

A complex number z = x + iy can be uniquely represented by a point P(x, y) in the Cartesian plane. The plane representing complex numbers is called the Argand plane or the complex plane. The x-axis is called the real axis and the y-axis is called the imaginary axis. For example, the complex number 3 + 4i is represented by the point (3, 4) on the Argand plane.

Modulus of a Complex Number

The modulus of a complex number z = x + iy, denoted by |z|, is the distance of the point P(x, y) from the origin O(0, 0) in the Argand plane. It is always a non-negative real number. Geometrically, it represents the length of the vector from the origin to the point representing the complex number.

If z = x + iy, then |z| = √(x² + y²)

Argument of a Complex Number

The argument (or amplitude) of a non-zero complex number z = x + iy, denoted by arg(z) or amp(z), is the angle θ that the line segment OP (where O is the origin and P is the point (x, y)) makes with the positive direction of the real axis. The value of θ such that -π < θ ≤ π is called the principal argument of z. The argument of 0 is undefined. To find the principal argument, we first find the acute angle α such that tan α = |y/x|. Then, based on the quadrant in which the point (x, y) lies, the principal argument θ is determined as follows:

Let z = x + iy. First, find α such that tan α = |y/x| (where α is an acute angle).\n1. If x > 0, y > 0 (1st Quadrant): θ = α\n2. If x < 0, y > 0 (2nd Quadrant): θ = π - α\n3. If x < 0, y < 0 (3rd Quadrant): θ = -(π - α) or α - π\n4. If x > 0, y < 0 (4th Quadrant): θ = -α\nSpecial Cases:\n- If z is purely real (y=0): θ = 0 if x > 0, θ = π if x < 0.\n- If z is purely imaginary (x=0): θ = π/2 if y > 0, θ = -π/2 if y < 0.

Key facts to remember

1A complex number z = x + iy is represented by the point (x, y) on the Argand plane, where the x-axis is the real axis and the y-axis is the imaginary axis.
2The modulus of z = x + iy is |z| = √(x² + y²), which represents the distance of the point (x, y) from the origin.
3The argument of z, arg(z), is the angle made by the line segment from the origin to (x, y) with the positive real axis.
4The principal argument θ of a complex number z satisfies the condition -π < θ ≤ π.
5To find the principal argument, first calculate the acute angle α = tan⁻¹(|y/x|). Then adjust α based on the quadrant of (x, y).
6The argument of the complex number 0 + 0i is undefined.

Worked examples

Example 1

Represent the complex number z = 3 + 4i on the Argand plane and find its modulus.

IStep 1: Identify the real and imaginary parts. For z = 3 + 4i, the real part is x = 3 and the imaginary part is y = 4.

IIStep 2: Plot the point (x, y) = (3, 4) on the Argand plane. Mark the x-axis as the real axis and the y-axis as the imaginary axis.

IIIStep 3: Calculate the modulus using the formula |z| = √(x² + y²).

IV|z| = √(3² + 4²)

V|z| = √(9 + 16)

VI|z| = √25

VII|z| = 5

Answer

The complex number z = 3 + 4i is represented by the point (3, 4) on the Argand plane. Its modulus is |z| = 5.

The modulus represents the distance of the point (3, 4) from the origin (0, 0).

Example 2

Find the modulus and principal argument of the complex number z = 1 - i.

IStep 1: Identify the real and imaginary parts. For z = 1 - i, x = 1 and y = -1.

IIStep 2: Calculate the modulus.

III|z| = √(x² + y²) = √(1² + (-1)²)

IV|z| = √(1 + 1) = √2

VStep 3: Determine the acute angle α such that tan α = |y/x|.

VItan α = |-1/1| = 1

VIISince tan(π/4) = 1, we have α = π/4.

VIIIStep 4: Determine the quadrant and apply the argument formula. The point (1, -1) lies in the 4th Quadrant (x > 0, y < 0).

9Therefore, the principal argument θ = -α = -π/4.

Answer

The modulus of z = 1 - i is |z| = √2 and its principal argument is arg(z) = -π/4.

Always ensure the principal argument lies in the range (-π, π].

Example 3

Find the modulus and principal argument of the complex number z = -√3 + i.

IStep 1: Identify the real and imaginary parts. For z = -√3 + i, x = -√3 and y = 1.

IIStep 2: Calculate the modulus.

III|z| = √(x² + y²) = √((-√3)² + 1²)

IV|z| = √(3 + 1) = √4 = 2

VStep 3: Determine the acute angle α such that tan α = |y/x|.

VItan α = |1/-√3| = 1/√3

VIISince tan(π/6) = 1/√3, we have α = π/6.

VIIIStep 4: Determine the quadrant and apply the argument formula. The point (-√3, 1) lies in the 2nd Quadrant (x < 0, y > 0).

9Therefore, the principal argument θ = π - α = π - π/6 = (6π - π)/6 = 5π/6.

Answer

The modulus of z = -√3 + i is |z| = 2 and its principal argument is arg(z) = 5π/6.

A rough sketch on the Argand plane helps in visualising the quadrant correctly.

Common mistakes

✗Incorrectly identifying the quadrant of the complex number, leading to an incorrect sign or formula for the argument.
✗Confusing the general argument (2nπ + θ) with the principal argument, which has a specific range (-π, π].
✗Calculating tan α using y/x directly instead of |y/x|, which should always yield an acute angle.
✗Errors in basic trigonometric values for standard angles (e.g., π/6, π/4, π/3).
✗Forgetting to square the real and imaginary parts when calculating the modulus, or making sign errors with negative numbers.

Exam tips

★Always draw a rough sketch of the complex number on the Argand plane to correctly identify its quadrant before calculating the argument. This helps prevent sign errors.
★Memorise the quadrant-specific formulas for the principal argument and the range (-π, π].
★Be thorough with calculations, especially when dealing with square roots and negative numbers for the modulus.
★Practise finding the argument for complex numbers in all four quadrants and on the axes (purely real or purely imaginary) to ensure full understanding.

Ready to practise?

Try a problem on this topic

Snap a photo or type a question — get step-by-step working instantly.

Solve a problem →← Maths curriculum