The Distributive Property is an important math rule that helps us do multiplication in a simple way. It states:

$$a(b + c) = ab + ac$$

This means we can multiply a number by a group of numbers added together. This rule is super helpful when we multiply expressions that involve complex numbers.

What Are Complex Numbers?

Complex numbers look like this:

$$a + bi$$

Here, $a$ is the real part and $b$ is the imaginary part. The letter $i$ stands for the imaginary unit, which is defined as $i = \sqrt{-1}$. When we multiply complex numbers, we use the Distributive Property and remember that $i^2 = -1$.

Example: Multiplying Complex Numbers

Let's look at how to multiply two complex numbers:

$$(2 + 3i)(4 + 5i)$$

We can use the Distributive Property in these steps:

1. Distribute each part of the first complex number:

   • First, multiply $2$ by both parts of the second complex number:
     • $2 \cdot 4 = 8$
     • $2 \cdot 5i = 10i$
   • Next, multiply $3i$ by both parts of the second complex number:
     • $3i \cdot 4 = 12i$
     • $3i \cdot 5i = 15i^2$

2. Add everything together:

   • We get: $$8 + 10i + 12i + 15i^2$$

3. Change $i^2$ to $-1$:

   • So, $15i^2$ becomes $15(-1) = -15$.

4. Combine everything:

   • Now, let's put together the real and imaginary parts: $$8 - 15 + (10i + 12i) = -7 + 22i$$

So, we find that multiplying $(2 + 3i)(4 + 5i)$ gives us $-7 + 22i$.

Conclusion

Knowing how to use the Distributive Property is really important for multiplying complex numbers. This skill is essential in Year 9 Mathematics.