When I first learned about adding complex numbers, I found it a bit confusing. But once you understand it, it’s actually quite simple!

What Are Complex Numbers?

Complex numbers look like this: $a + bi$, where:

• $a$ is the real part
• $b$ is the imaginary part
• $i$ is a special number that stands for $\sqrt{-1}$

When we need to add or subtract these numbers, we deal with the real and imaginary parts separately.

How to Add Complex Numbers

Let’s say you have two complex numbers:

$$z_1 = a + bi$$ $$z_2 = c + di$$

To add these together, just add the real parts and the imaginary parts like this:

$$z_1 + z_2 = (a + c) + (b + d)i$$

Here’s how it works:

• Add the Real Parts: This means you add the $a$ and $c$ numbers.
• Add the Imaginary Parts: You do the same with the $b$ and $d$ numbers.

Example

Let’s use some actual numbers:

$$z_1 = 3 + 4i$$ $$z_2 = 1 + 2i$$

To add these together:

1. For the real parts: $3 + 1 = 4$
2. For the imaginary parts: $4 + 2 = 6$

So the answer is:

$$z_1 + z_2 = 4 + 6i$$

How to Subtract Complex Numbers

Subtracting complex numbers is similar. You still keep the real and imaginary parts separate.

For example, if we use the same complex numbers but want to subtract:

$$z_1 - z_2 = (3 + 4i) - (1 + 2i)$$

You would do:

1. For the real parts: $3 - 1 = 2$
2. For the imaginary parts: $4 - 2 = 2$

The answer would be:

$$z_1 - z_2 = 2 + 2i$$

Conclusion: The Easy Way to Handle Complex Numbers

In the end, it’s pretty neat how just separating the real and imaginary parts makes adding and subtracting complex numbers easy.

Think of it like doing two simple math problems at the same time. Once you get used to this way of thinking, using complex numbers feels simple, almost like mixing colors in art. Just blend the parts to create something new!