When it comes to multiplying complex numbers, don’t worry! It’s not too hard if you take it one step at a time. In this article, we’ll break down how to multiply complex numbers in a clear and simple way.

What Are Complex Numbers?

Complex numbers are numbers that have two parts: a real part and an imaginary part. They are usually written in this form: $a + bi$, where:

• $a$ is the real part,
• $b$ is the imaginary part, and
• $i$ stands for the imaginary unit, which means $i = \sqrt{-1}$.

How to Multiply Complex Numbers

To multiply two complex numbers, let’s say $z_1 = a + bi$ and $z_2 = c + di$, we can use a method called the distributive property (sometimes called FOIL for binomials). Here’s how it goes:

1. Distribute the terms:

   $z_1 \cdot z_2 = (a + bi)(c + di)$

   When you distribute, you get:

   $z_1 \cdot z_2 = ac + adi + bci + bdi^2$

2. Combine the like terms: Remember, $i^2 = -1$. So, you can switch $i^2$ for $-1$ in your math:

   $z_1 \cdot z_2 = ac + (ad + bc)i + bd(-1)$

   This simplifies to:

   $z_1 \cdot z_2 = (ac - bd) + (ad + bc)i$

Example of Multiplying Complex Numbers

Let’s see an example. Suppose we want to multiply $z_1 = 3 + 2i$ and $z_2 = 4 + 5i$.

1. Set up the expression:

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

2. Use the distributive property:

   $= 3 \cdot 4 + 3 \cdot 5i + 2i \cdot 4 + 2i \cdot 5i$ $= 12 + 15i + 8i + 10(-1)$

3. Combine the terms:

   $= 12 + 23i - 10$ $= (12 - 10) + 23i = 2 + 23i$

So, $z_1 \cdot z_2 = 2 + 23i$.

Special Cases

1. Multiplying by $i$: If you multiply a complex number by $i$, it spins the number 90 degrees to the left on the complex plane. For instance, if you multiply $z = 1 + 2i$ by $i$, it turns into:

   $z \cdot i = (1 + 2i)i = 1i + 2i^2 = i - 2 = -2 + i$

2. Conjugates: If you multiply a complex number by its conjugate, you’ll get a real number. The conjugate of $z = a + bi$ is written as $\overline{z} = a - bi$.

   $z \cdot \overline{z} = (a + bi)(a - bi) = a^2 - b^2i^2 = a^2 + b^2$

Conclusion

Multiplying complex numbers doesn’t have to be scary. If you follow the steps of the distributive property, remember that $i^2 = -1$, and practice with different examples, you’ll get the hang of it in no time. Keep working on it, and soon you’ll be multiplying complex numbers like a pro!