Visualizing complex numbers can really help us understand how to add and subtract them. Let’s make it simpler!

Complex Numbers on the Plane

Think of a flat surface called the complex plane.

On this plane:

• The x-axis shows the real part.
• The y-axis shows the imaginary part.

For example, a complex number like $z = 3 + 4i$ can be shown as the point (3, 4) on this plane.

Adding Complex Numbers

When you add two complex numbers, like $z_1 = 2 + 3i$ and $z_2 = 1 + 2i$, it’s easy! Just add the parts that match:

• Real parts: $2 + 1 = 3$
• Imaginary parts: $3 + 2 = 5$

So, $z_1 + z_2 = 3 + 5i$.

On the complex plane, you can see this as moving from point (2, 3) to (3, 5).

Subtracting Complex Numbers

Subtraction is pretty similar. Let’s take $z_1 = 4 + 5i$ and $z_2 = 2 + 3i$:

• Real parts: $4 - 2 = 2$
• Imaginary parts: $5 - 3 = 2$

So, $z_1 - z_2 = 2 + 2i$.

On the complex plane, you can picture this as moving left and down.

Conclusion

Seeing complex numbers on the plane makes it easier to do calculations. It also helps us understand them better.

When you add or subtract these points, it gives you a clear picture of what is happening. This makes working with complex numbers much less scary!