When you're working with complex numbers, especially when adding, subtracting, or multiplying them, there are some important rules that can make everything easier to understand:

1. Addition and Subtraction:

   • Think of complex numbers like regular polynomials! Just separate the real parts and the imaginary parts.
     For example, if you have two complex numbers, ( z_1 = a + bi ) and ( z_2 = c + di ), here’s what to do:
     • For addition:
       ( z_1 + z_2 = (a + c) + (b + d)i )
     • For subtraction:
       ( z_1 - z_2 = (a - c) + (b - d)i )

2. Multiplication:

   • Use the distributive property, which is sometimes called the FOIL method.
     It's like mixing two groups together. Multiply every part of the first complex number by every part of the second.
     For our example ( z_1 = a + bi ) and ( z_2 = c + di ), you get:
     • For multiplication:
       ( z_1 \cdot z_2 = (ac - bd) + (ad + bc)i )

3. Commutative and Associative Properties:

   • When adding or multiplying, remember you can change the order of the numbers (commutative), and you can group them differently (associative) without changing the answer.

Keeping these rules in mind can make working with complex numbers much easier!