Special Cases Involving Complex Numbers

Key Concepts

• Imaginary Numbers
  • Imaginary unit: ( i = \sqrt{-1} )
  • Operations involving imaginary numbers require manipulation to isolate ( i ).

Example Problems

Example 1: ( \sqrt{-7} \times \sqrt{-2} )

• Do not combine directly into ( \sqrt{14} ).
• Breakdown:
  • ( \sqrt{-7} = i \sqrt{7} )
  • ( \sqrt{-2} = i \sqrt{2} )
• Multiply:
  • ( i \times i = i^2 )
  • ( \sqrt{7} \times \sqrt{2} = \sqrt{14} )
• Result:
  • ( i^2 = -1 ) hence, (-\sqrt{14}).

Example 2: ( \sqrt{-49} \times \sqrt{-16} )

• Breakdown:
  • ( \sqrt{-49} = 7i )
  • ( \sqrt{-16} = 4i )
• Multiply:
  • ( 7 \times 4 = 28 )
  • ( i \times i = i^2 )
• Result:
  • ( i^2 = -1 ) hence, (-28).

Powers of ( i ) Pattern

• Recognizing powers of ( i ):
  • ( i^1 = i )
  • ( i^2 = -1 )
  • ( i^3 = -i )
  • ( i^4 = 1 )
  • Pattern repeats every four terms.

Examples of Powers:

• ( i^5 ) to ( i^{12} )
  • ( i^5 = i )
  • ( i^6 = -1 )
  • ( i^7 = -i )
  • ( i^8 = 1 )
  • ( i^9 = i )
  • ( i^{10} = -1 )
  • ( i^{11} = -i )
  • ( i^{12} = 1 )

Solving Powers of ( i )

• Method: Use division and remainder.
• Formula: Divide the power by 4 and determine the remainder.

Example Calculation

• ( i^{73} )
  • Divide 73 by 4.
  • Remainder is 1, thus ( i^{73} = i^1 = i ).

Operations Involving Powers

• Subtraction

  • ( i^{15} - i^5 )
  • Calculate remainders:
    • ( i^{15} \equiv i^3 = -i )
    • ( i^5 = i )
  • Result: (-i - i = -2i)

• Division

  • ( i^{18} / i^{10} )
  • Subtract exponents:
    • ( i^8 \equiv i^4 = 1 )

Conclusion

• Understanding the manipulation and pattern of ( i ) powers is essential for dealing with complex numbers effectively.