Complex Conjugation and Arithmetic Properties

Introduction

• Complex conjugation and its arithmetic properties: multiplication, addition, subtraction, division.
• Complex number z = x + yi and its conjugate z̅ = x - yi.

Real and Imaginary Parts Using Complex Conjugation

• Real Part:
  • Sum of a complex number and its conjugate:
    • z + z̅ = 2x (twice the real part).
    • Real part: x = 1/2(z + z̅).
• Imaginary Part:
  • Difference between a complex number and its conjugate:
    • z - z̅ = 2iy.
    • Imaginary part: y = 1/(2i)(z - z̅).

Properties of Complex Conjugation

Addition

• Complex Conjugate of Sum:
  • Given two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2, the conjugate of their sum:
  • (z1 + z2)̅ = z̅1 + z̅2.

Multiplication

• Complex Conjugate of Product:
  • (z1 * z2)̅ = z̅1 * z̅2.
  • Multiplying the conjugates directly results in the conjugate of the product.

Further Arithmetic Properties

• Subtraction:
  • Conjugate of a difference: (z1 - z2)̅ = z̅1 - z̅2.
• Division:
  • Conjugate of a quotient: (z1 / z2)̅ = z̅1 / z̅2.
  • Calculate by rationalizing the denominator first.

Conclusion

• Explored basic arithmetic operations with complex conjugates.
• Exercises to verify subtraction and division properties.