Lecture on Imaginary and Complex Numbers

Understanding Imaginary Numbers

• Imaginary numbers are complex numbers with the imaginary unit 'i'.
• Definition of 'i':
  • (i = \sqrt{-1})
  • (i^2 = -1)
  • (i^3 = -i)
  • (i^4 = 1)

Simplifying Powers of 'i'

• Express higher powers of 'i' as multiples of four plus a remainder:
  • (i^7 = i^4 \times i^3 = 1 \times (-i) = -i)
  • (i^{26} = i^{24} \times i^2 = 1 \times (-1) = -1)
  • (i^{33} = i^{32} \times i = 1 \times i = i)
  • (i^{43} = i^{40} \times i^3 = 1 \times (-i) = -i)

Adding and Subtracting Imaginary Numbers

• Example: Simplify (5(2 + 3i) - 4(7 - 2i))
  • Distribute:
    • (5 \times 2 = 10)
    • (5 \times 3i = 15i)
    • (-4 \times 7 = -28)
    • (-4 \times -2i = 8i)
  • Combine like terms:
    • Real: (10 - 28 = -18)
    • Imaginary: (15i + 8i = 23i)
  • Result: (-18 + 23i)

Multiplying Complex Numbers

• Use FOIL method:
  • Example: ((5 - 2i)(8 + 3i))
    • (5 \times 8 = 40)
    • (5 \times 3i = 15i)
    • (-2i \times 8 = -16i)
    • (-2i \times 3i = -6i^2 = 6) (since (i^2 = -1))
  • Combine terms:
    • (40 + 6 = 46)
    • (15i - 16i = -i)
  • Result: (46 - i)

Dividing Complex Numbers

• Multiply numerator and denominator by the conjugate of the denominator:
  • Example: (\frac{3 + 2i}{4 - 3i})
    • Conjugate: (4 + 3i)
    • Multiply top and bottom by conjugate.
    • Simplify using FOIL and combine:
      • Numerator: (12 + 17i - 6)
      • Denominator: (16 + 9 = 25)
    • Result: (\frac{6}{25} + \frac{17}{25}i)

Solving Equations with Complex Numbers

• Example: Solve (4x + 3i = 12 - 15yi)
  • Real parts: (4x = 12 \Rightarrow x = 3)
  • Imaginary parts: (3i = -15yi \Rightarrow y = -\frac{1}{5})

Solving Algebraic Equations

• Example: (x^2 + 36 = 0)
  • Solution involves imaginary numbers:
    • (x^2 = -36)
    • (x = \pm 6i) (using (\sqrt{-1} = i))

Plotting Complex Numbers

• Plotting (4 + 3i):
  • Real part (4) on x-axis, Imaginary part (3) on y-axis.
  • Absolute value: (\sqrt{4^2 + 3^2} = 5)

Key Takeaways

• Understand the arithmetic operations: addition, subtraction, multiplication, division of complex numbers.
• Simplify powers of 'i' using cycle of four.
• Solve equations involving real and imaginary parts separately.
• Plot and find the absolute value of complex numbers.