To understand how complex numbers work on a special graph called the complex plane, we first need to know how to write these numbers.

Complex numbers look like this: $a + bi$. Here, $a$ is the real part, and $b$ is the part that we call imaginary. On the complex plane, we plot them like this: $a$ goes on the x-axis (which is the real axis), and $b$ goes on the y-axis (which is the imaginary axis).

Adding Complex Numbers

When you want to add two complex numbers, let’s say $z_1 = a + bi$ and $z_2 = c + di$, you just add the real parts together and the imaginary parts together:

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

Think of it like this: In the complex plane, addition looks like moving from one point to another. For example:

• If $z_1 = 1 + 2i$ (which is the point $(1, 2)$) and $z_2 = 2 + 3i$ (the point $(2, 3)$), then when you add them together, you get $z_1 + z_2 = 3 + 5i$ (the point $(3, 5)$). This means you are moving from point $z_1$ by a certain distance towards point $z_2$.

Subtracting Complex Numbers

Subtracting complex numbers works in a similar way. For our numbers $z_1$ and $z_2$, it looks like this:

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

Here’s what happens: This movement is like going backwards by the coordinates of $z_2$. So if you take $z_2$ away from $z_1$, you end up with a new point showing where you land.

Multiplying Complex Numbers

Multiplying complex numbers is a bit more exciting. For $z_1$ and $z_2$, multiplication is shown by:

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

What to picture here: When you multiply, it’s about changing size and twisting around in the complex plane. For example, if you multiply any complex number by $i$ (which is $0 + 1i$), it rotates the point by 90 degrees in a counter-clockwise direction.

Understanding the Patterns

To see these patterns clearly, think about them like this:

1. Addition: You move along the complex plane, adding each part together.
2. Subtraction: You move in the opposite direction, adjusting where you land based on the second number.
3. Multiplication: You think of it as size changing and turning in a new direction.

By plotting these operations and watching how the points change in the complex plane, you’ll get a better feel for how complex numbers work together!