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MathematicsClass 11Mathematics

Mathematics | Class 11 Mathematics Notes

By ConceptScroll Team · Published on 17 July 2026 · 3 min read

Mathematics – this guide gives you a concise, exam-ready overview of Mathematics from Class 11 Mathematics, written by ConceptScroll editors and reviewed against the latest NCERT textbook.

4.4 The Modulus and the Conjugate of a Complex Number

This section introduces two important concepts related to complex numbers: modulus and conjugate.

Modulus: For a complex number z = a + ib, the modulus |z| is defined as the non-negative real number √(a² + b²). Geometrically, it represents the distance of the point (a, b) in the complex plane from the origin (0, 0).

Conjugate: The conjugate of z = a + ib, denoted by z̅, is defined as a - ib. Geometrically, the conjugate corresponds to the reflection of the point (a, b) about the real axis in the Argand plane.

The multiplicative inverse of a non-zero complex number z can be expressed using its conjugate and modulus as z⁻¹ = z̅ / |z|². This formula simplifies division and inversion of complex numbers.

Several properties involving modulus and conjugates are presented: (i) |z₁ z₂| = |z₁| × |z₂| (ii) |z₁ / z₂| = |z₁| / |z₂|, provided z₂ ≠ 0 (iii) Conjugate of product: (z₁ z₂)̅ = z̅₁ z̅₂ (iv) Conjugate of sum/difference: (z₁ ± z₂)̅ = z̅₁ ± z̅₂

Examples illustrate finding multiplicative inverses and expressing complex fractions in a + ib form using conjugates and moduli.

📊 Diagram: No new diagrams introduced here; references to Argand plane geometry are made in section 4.5.

🧪 Activity: No activity in this section.

🔗 Connection: Prepares for geometric representation of complex numbers in the Argand plane in section 4.5.

Frequently asked questions

1. $(5i)\left(-\frac{3}{5}i\right)$

Given: (5i)\left(-\frac{3}{5}i\right)

Step 1: Multiply the constants: 5 (-3/5) = -3 Step 2: Multiply the imaginary units: i i = i^2 = -1 Step 3: So, (5i)(-3/5 i) = -3 * (-1) = 3

Answer: 3

2. $i^9 + i^{19}$

Recall powers of i cycle every 4: i^1 = i i^2 = -1 i^3 = -i i^4 = 1

Calculate i^9: 9 mod 4 = 1, so i^9 = i

Calculate i^{19}: 19 mod 4 = 3, so i^{19} = -i

Sum: i + (-i) = 0

Answer: 0

3. $i^{-39}$

Recall powers of i cycle every 4. First, convert negative power: i^{-39} = 1 / i^{39}

Calculate 39 mod 4: 39 mod 4 = 3 So, i^{39} = i^3 = -i

Therefore, i^{-39} = 1 / (-i) = -1 / i

Multiply numerator and denominator by i: = (-1 / i) * (i / i) = (-i) / (i^2) = (-i) / (-1) = i

Answer: i

4. $3(7 + i7) + i(7 + i7)$

Given expression: 3(7 + 7i) + i(7 + 7i)

Step 1: Expand each term: 3(7 + 7i) = 21 + 21i

Step 2: i(7 + 7i) = 7i + 7i^2 = 7i + 7(-1) = 7i - 7

Step 3: Add the two results: (21 + 21i) + (7i - 7) = (21 - 7) + (21i + 7i) = 14 + 28i

Answer: 14 + 28i

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