To multiply complex numbers, let’s go through it step by step.

Step 1: What Are Complex Numbers?

A complex number looks like this: $a + bi$. Here, $a$ and $b$ are regular numbers, and $i$ stands for an imaginary number. We define $i$ by the rule that $i^2 = -1$.

Step 2: Setting Up the Problem

Let’s try to multiply two complex numbers: $(2 + 3i)$ and $(4 + 5i)$.

Step 3: Use the Distributive Property

Now, we will use the distributive property. This means we multiply each part of the first number by each part of the second number:

$(2 + 3i)(4 + 5i) = 2 \cdot 4 + 2 \cdot 5i + 3i \cdot 4 + 3i \cdot 5i$

Step 4: Calculate Each Part

Let’s do the calculations:

• $2 \cdot 4 = 8$
• $2 \cdot 5i = 10i$
• $3i \cdot 4 = 12i$
• $3i \cdot 5i = 15i^2$

Step 5: Replace $i^2$

Next, we need to use the rule about $i$:

Replace $15i^2$ with $-15$:

$8 + 10i + 12i - 15$

Step 6: Combine Like Terms

Now we combine the regular numbers and the imaginary numbers:

$(8 - 15) + (10i + 12i) = -7 + 22i$

Conclusion

So when we multiply $(2 + 3i)$ and $(4 + 5i)$, we get $-7 + 22i$. That's how to multiply complex numbers using the distributive property!