In Grade 12 Algebra II, it's really important to learn how to see addition and subtraction of complex numbers on the complex plane.

A complex number looks like this: $a + bi$.

Here, $a$ is the real part, and $b$ is the imaginary part. The complex plane is like a two-dimensional graph.

• The horizontal line (real axis) shows the real part, $a$.
• The vertical line (imaginary axis) shows the imaginary part, $b$.

Adding Complex Numbers

When you add two complex numbers, like $z_1 = a_1 + b_1i$ and $z_2 = a_2 + b_2i$, here’s how you can picture it:

1. Plotting Points:

   • Plot $z_1$ at the point $(a_1, b_1)$.
   • Plot $z_2$ at the point $(a_2, b_2)$.

2. Using Vectors:

   • Think of each complex number as a vector starting from the center point $(0,0)$ to where the point $(a_i, b_i)$ is.

3. Vector Addition:

   • To add them, draw a straight line from point $(a_1, b_1)$ to $(a_2, b_2)$. The end of this line shows where the sum lands at the point $(a_1 + a_2, b_1 + b_2)$.

4. Resulting Complex Number:

   • The added number, $z_1 + z_2$, shows up as the point $(a_1 + a_2, b_1 + b_2)$, or in simpler terms, $(a_1 + a_2) + (b_1 + b_2)i$.

Subtracting Complex Numbers

For subtraction, like $z_1 - z_2$, the steps are pretty similar:

1. Plot the Points:

   • Plot $z_1$ at $(a_1, b_1)$ and $z_2$ at $(a_2, b_2)$ again.

2. Vector Representation:

   • Draw the vectors for each complex number just like you did for addition.

3. Vector Subtraction:

   • To find $z_1 - z_2$, look for the line that goes from the end of $z_2$ to the end of $z_1$. This tells you how to get from one to the other.

4. Resulting Complex Number:

   • The difference is shown by the point $(a_1 - a_2, b_1 - b_2)$, which is $(a_1 - a_2) + (b_1 - b_2)i$.

Summary

In short, you can see both adding and subtracting complex numbers using the shapes on the complex plane.

When you add, you're combining the parts of the complex numbers.

When you subtract, you're finding the difference between them.

Understanding this can help you get a better grasp of how complex numbers work!