Mathematics | Class 11 Mathematics Notes

4.3 Algebra of Complex Numbers

This section develops the algebraic operations on complex numbers, including addition, subtraction, multiplication, and division, along with their properties.

Addition: For two complex numbers z₁ = a + ib and z₂ = c + id, their sum is defined as z₁ + z₂ = (a + c) + i(b + d), which is also a complex number. Addition satisfies closure, commutativity, associativity, existence of additive identity (0 + i0), and additive inverses (-a - ib).

Subtraction: Defined as z₁ - z₂ = z₁ + (-z₂), where -z₂ is the additive inverse of z₂.

Multiplication: For z₁ = a + ib and z₂ = c + id, the product is z₁z₂ = (ac - bd) + i(ad + bc). Multiplication also satisfies closure, commutativity, associativity, existence of multiplicative identity (1 + i0), multiplicative inverses for non-zero complex numbers, and distributive laws.

Division: For z₂ ≠ 0, division is defined as z₁ / z₂ = z₁ × (1 / z₂), where the multiplicative inverse of z₂ is (a / (a² + b²)) + i(-b / (a² + b²)) for z₂ = a + ib.

Power of i: Powers of i cycle every four steps: i⁰ = 1, i¹ = i, i² = -1, i³ = -i, i⁴ = 1, and so on. Negative powers are defined similarly.

Square roots of negative real numbers: The square roots of -a (a > 0) are ±√a i. The symbol √(-a) is taken as √a i.

Identities: Algebraic identities such as (z₁ + z₂)² = z₁² + 2z₁z₂ + z₂² hold for complex numbers, and similar expansions for cubes and differences are valid.

Several examples illustrate these operations and identities, including expressing complex expressions in a + ib form and computing powers.

📊 Diagram: No specific diagrams in this section.

🧪 Activity: No activity in this section.

🔗 Connection: Leads to the geometric interpretation of complex numbers in the Argand plane in section 4.5.