Lecture on Complex Numbers

Introduction to Complex Numbers

• Real numbers squared are always ≥ 0.
• Equation ( z^2 = -1 ) has no real number solution.
• Complex numbers are introduced to solve such equations.
• Applications: Electrical circuits, fluid mechanics.

Imaginary Unit

• Imaginary unit ( i ) is defined as a solution to ( i^2 = -1 ).
• Engineers sometimes use ( j ) instead of ( i ).
• ( i ) is the square root of (-1), hence called the imaginary unit.

Complex Numbers

• Complex number form: ( x + iy ) or ( x + yi ), where ( x, y ) are real numbers.
• ( x ) is the real part, ( y ) is the imaginary part.
  • Real part of ( z ): ( x )
  • Imaginary part of ( z ): ( y )
• Example: For ( z = 3 - 4i )
  • Real part: 3
  • Imaginary part: -4 (not (-4i), just (-4))

Arithmetic with Complex Numbers

• Equality of Complex Numbers:

  • Two complex numbers ( z_1 = x_1 + iy_1 ) and ( z_2 = x_2 + iy_2 ) are equal if:
    • ( x_1 = x_2 )
    • ( y_1 = y_2 )

• Addition & Subtraction:

  • Addition: ( z_1 + z_2 = (x_1 + x_2) + i(y_1 + y_2) )
  • Subtraction: ( z_1 - z_2 = (x_1 - x_2) + i(y_1 - y_2) )

• Multiplication:

  • Formula: ( (x_1 + iy_1)(x_2 + iy_2) )
  • Expand the expression:
    • ( x_1x_2 + x_1iy_2 + iy_1x_2 + i^2y_1y_2 )
  • Simplification:
    • ( i^2 = -1 ) ( \rightarrow - y_1y_2 )
  • Result: ( (x_1x_2 - y_1y_2) + i(x_1y_2 + x_2y_1) )

Conclusion

• Complex numbers extend real numbers by introducing an imaginary component.
• Fundamental operations (addition, subtraction, multiplication) are well-defined.
• Imaginary and real components play distinct roles in calculations.

Note:

• Imaginary part of complex numbers is a real number, not multiplied by ( i ).