Some Basic Properties

• Commutative law for Addition :
  • z₁ + z₂ = z₂ + z₁


• Commutative law for Multiplication :
  • z₁.z₂ = z₂.z₁


• Associative law for addition :
  • (z₁ + z₂) + z₃ = z₁ + (z₂ + z₃)


• Associative law for Multiplication :
  • (z₁.z₂).z₃ = z₁.(z₂.z₃)


• Distributive Law :
  • z₁.(z₁ + z₃) = z₁.z₂ + z₁.z₃

Some Basic Proofs

• The sum of two conjugate complex numbers is real :
  • Let, z = a + ib (a, b are real numbers) be a complex number. Then, conjugate of z is z' = a - ib.
    Now, z + z' = a + ib + a - ib = 2a, which is real.


• If z₁ and z₂ are two complex numbers, then always :
  • |z₁ + z₂| ≤ |z₁| + |z₂|


• The product of two conjugate complex numbers is real :
  • Let, z = a + ib (a, b are real number) be a complex number. Then, conjugate of z is z' = a - ib.
    z ∙ z' = (a + ib)(a - ib) = a² - i²b² = a² + b²,
    (Since i² = -1), which is real.

De Moivre's Theorem

• The formula for this theorem is as :
  • (cosx + isinx)ⁿ = cos(nx) + isin(nx)