When we multiply complex numbers, we use a couple of helpful rules. One important rule is that (i² = -1). Complex numbers look like this: (a + bi), where (a) and (b) are regular numbers, and (i) is something called the imaginary unit.

Steps for Multiplication

1. Identify the Complex Numbers: Let's say we have two complex numbers: (z_1 = a + bi) and (z_2 = c + di).

2. Apply the Distributive Property: We need to multiply each part of the first complex number with each part of the second one: [ z_1 \cdot z_2 = (a + bi)(c + di) = ac + adi + bci + bidi ]

3. Combine Like Terms: Now we need to organize what we have:

   • The real part is (ac).
   • The imaginary parts combine to make: (adi + bci = (ad + bc)i).
   • For the term (bidi), we use the rule that (i² = -1): (bdi² = bd(-1) = -bd).

4. Final Expression: Now we put everything together: [ z_1 \cdot z_2 = (ac - bd) + (ad + bc)i ]

Conclusion

So, when we multiply the two complex numbers, we get: [ z_1 \cdot z_2 = (ac - bd) + (ad + bc)i ]

This method works every time! It shows how the rule (i² = -1) helps us with our calculations when we multiply complex numbers.