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Complex Numbers

Complex Plane

Similar to graphing real-number coordinates or functions on the Cartesian plane with an x-axis and y-axis, we can graph complex numbers on the complex plane. The complex plane can be helpful in representing complex numbers visually by using the real axis (Re) and imaginary axis (Im), which are the horitzontal and vertical axes, respectively. The complex plane looks like this:

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Plotting Complex Numbers

Graphing complex numbers on the complex plane is very similar to graphing (x,y)-coordinates on the Cartesian plane. By convention, the complex number \(z=x+yi\) corresponds to the coordinates (Re,Im) on the complex plane.

Example: Plot \(3i-4\) on the complex plane.

1. Find the corresponding (Re,Im)-coordinates of the complex number based on its real and imaginary parts. For \(-4+3i\), the corresponding (Re,Im)-coordinates are \((-4,3)\).

2. Plot point on the complex plane representing the complex number as if you were plotting it on the Cartesian plane:

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Representing Complex Numbers as Vectors

Now that we know how to plot complex numbers on the complex plane, we can represent these numbers using vectors. Presenting complex numbers on the complex plane as vectors can allow us to use vector addition, subtraction, etc. to evaluate complex expressions with operations.

To represent complex numbers as vectors, all we need to do is plot it as a point on the complex plane, then draw a vector connecting the origin \((0,0)\) to the plotted point. The head of the vector should be at the plotted point and the tail should be at the origin.

Example: Represent \(3i-4\) as a vector on the complex plane.

Continuing from our previous example, where we demonstrated how to plot \(3i-4\) on the complex plane as follows:

                                                            

To finish, we draw a vector between the origin and the plotted point, representing \(3i-4\):

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