Operations with Complex Numbers

Key Concepts

• Complex Numbers: Numbers that include a real part and an imaginary part, typically written as ( a + bi ) where ( a ) is the real part and ( bi ) is the imaginary part.
• Imaginary Unit ( i ): Defined as ( i = \sqrt{-1} ).
• ( i^2 ): Important property to remember is ( i^2 = -1 ).

Operations

Addition and Subtraction

• Treat imaginary units like variables when adding or subtracting.
• Example Addition:
  • ( (3 + 4i) + (-2i + 5i) = 7 + 3i )
  • Combine real terms (3 + 4) and imaginary terms ((-2i + 5i)).
• Example Subtraction:
  • Distribute the negative across the parentheses.
  • ( (1 - 7i) - (1 - 5i) = 0 - 2i )
• When adding or subtracting, ( i^2 ) typically doesn't appear.

Multiplication

• Use distribution (or FOIL for binomials) to multiply.
• Example Single Distribution:
  • ( 5i(4 + 3i) )
  • Distribute ( 5i ):
    • ( 20i + 15i^2 )
  • Remember ( i^2 = -1 ):
    • Simplify to ( 20i - 15 )
    • Write in standard form: ( -15 + 20i )
• Example FOIL Method:
  • ( (3 + i)(2 + 2i) )
  • First: ( 3 \times 2 = 6 )
  • Outer and Inner: ( 3 \times 2i + i \times 2 = 6i + 2i = 8i )
  • Last: ( i \times 2i = 2i^2 = 2(-1) = -2 )
  • Combine: ( 6 - 2 = 4 )
  • Final form: ( 4 + 8i )

Important Tips

• Always Simplify( i^2 ) to (-1): Essential for correct simplification.
• Standard Form: Always express results as ( a + bi ).
• Common Mistakes: Forgetting to replace ( i^2 ) with (-1), improper ordering of terms.

Remember to keep practicing these operations and watch for opportunities to simplify. You can master working with complex numbers by applying these consistent methods and principles.