Operations with Complex Numbers in AC/Signal Analysis Assess the statement: “Complex numbers can be added and subtracted but not multiplied or divided.”

Introduction / Context:
Complex arithmetic underpins phasor analysis, impedance calculations, and control systems. Engineers routinely add, subtract, multiply, and divide complex quantities representing amplitudes and phase angles. The statement challenges whether all four operations are valid.

Given Data / Assumptions:

• Complex numbers expressed in rectangular (a + j b) or polar (r ∠ θ) form.
• Nonzero denominator when performing division.
• Standard algebraic rules for complex arithmetic apply.

Concept / Approach:

The field of complex numbers supports addition, subtraction, multiplication, and division (except division by zero). In rectangular form, multiplication and division use distributive/associative properties and complex conjugates. In polar form, multiplication multiplies magnitudes and adds angles; division divides magnitudes and subtracts angles.

Step-by-Step Solution:

Verification / Alternative check:

Phasor operations in AC circuits rely on these rules: multiplying by j rotates a vector by +90 degrees; impedance division calculates current from voltage and impedance; transfer functions use complex rational expressions routinely.

Why Other Options Are Wrong:

Claims restricting multiplication or division contradict the algebraic structure of complex numbers. Both rectangular and polar forms fully support all operations (with the usual caveat of nonzero divisors).

Common Pitfalls:

Confusing angle units (degrees vs. radians) in polar form; forgetting to rationalize the denominator in rectangular form; or mishandling signs when using the conjugate.

Final Answer:

False