Definition

  • Complex numbers are numbers that have both a real part and an imaginary part.
  • A complex number is written in the form a + bi, where:
    • a is the real part
    • b is the coefficient of the imaginary part
    • i is the imaginary unit and i² = −1
  • The set of complex numbers is denoted by C.

Components

  • Real part: The number a in a + bi
  • Imaginary part: The number b in a + bi
  • If b = 0, the number is purely real.
  • If a = 0 and b ≠ 0, the number is purely imaginary.

Examples

  • 2 + 3i is a complex number (real part = 2, imaginary part = 3i)
  • −5i is a purely imaginary number
  • 7 is a purely real number (equivalent to 7 + 0i)
  • 0 + 4i is a purely imaginary number

Properties

  • Complex numbers are closed under addition, subtraction, multiplication, and division.
  • They follow commutative, associative, and distributive laws of algebra.
  • Every complex number has a unique conjugate.
  • The conjugate of a + bi is a − bi.
  • Modulus of a complex number |a + bi| = √(a² + b²)

Graphical Representation

  • Complex numbers can be represented on the Argand plane.
  • X-axis represents the real part, and Y-axis represents the imaginary part.
  • Each complex number corresponds to a unique point on this plane.

Important Identities

  • i² = −1
  • i³ = −i
  • i⁴ = 1
  • i⁵ = i (and so on, cyclic every 4 powers)

Classification

  • Purely real numbers: b = 0 (e.g., 6)
  • Purely imaginary numbers: a = 0 (e.g., 4i)
  • Non-real complex numbers: both a and b ≠ 0 (e.g., 3 + 2i)

Applications

  • Used in electrical engineering (AC circuits)
  • Used in quantum mechanics and control systems
  • Applied in signal processing, fluid dynamics, and complex analysis
  • Essential in solving quadratic equations with negative discriminants

Set Relationships

  • Real numbers ⊂ Complex numbers
  • All real and imaginary numbers are part of the complex number system.
  • Complex numbers are the most generalized form of numbers in elementary mathematics.

Miscellaneous Facts

  • The equation x² + 1 = 0 has no real solution, but it has complex solutions ±i.
  • Every polynomial equation has at least one complex root (Fundamental Theorem of Algebra).
  • Multiplying a complex number by its conjugate gives a real number.
  • Complex numbers help solve real-world problems that involve two dimensions (real and imaginary parts).

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Questions