When we multiply complex numbers, we can use a helpful tool called the distributive property.

What Are Complex Numbers?

Complex numbers look like this: $a + bi$. Here, $a$ and $b$ are regular numbers. The "i" stands for an imaginary number. It’s special because it follows the rule that $i^2 = -1$.

The Distributive Property

The distributive property says that if you have a number or expression multiplied by a sum, you can break it apart.

For example, if you have $a(b + c)$, you can write it as $ab + ac$.

We will use this property to multiply complex numbers.

Example of Multiplying Complex Numbers

Let's multiply two complex numbers: $(3 + 2i)$ and $(1 + 4i)$.

We can separate this using the distributive property:

$(3 + 2i)(1 + 4i) = 3(1 + 4i) + 2i(1 + 4i)$

Now, let’s do the math step by step:

1. For $3(1 + 4i)$:

   • First, $3 \cdot 1 = 3$
   • Then, $3 \cdot 4i = 12i$
   • So, $3(1 + 4i) = 3 + 12i$.

2. For $2i(1 + 4i)$:

   • First, $2i \cdot 1 = 2i$
   • Then, $2i \cdot 4i = 8i^2$.

Now, we remember that $i^2 = -1$. So, $8i^2$ becomes $8(-1) = -8$. Therefore,

$2i(1 + 4i) = 2i - 8.$

Combine the Results

Now, we put both results together:

$3 + 12i + 2i - 8$

Let’s simplify this:

$(3 - 8) + (12i + 2i) = -5 + 14i.$

So, when we multiply $(3 + 2i)(1 + 4i)$, we get $-5 + 14i$.

Summary

In simple terms, here is how we can multiply complex numbers:

1. Break It Down: Use the distributive property to separate the complex numbers.
2. Do The Math: Multiply each part.
3. Remember $i^2$: Substitute $i^2$ with $-1$ when you see it.
4. Combine: Group similar parts together.

By following these steps, multiplying complex numbers becomes easy. The next time you face this, just remember to distribute and combine—it really makes things clearer!