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3. Graphical Representation of Complex Numbers

by M. Bourne

We can represent complex numbers in the complex plane.

We use the horizontal axis for the real part and the vertical axis for the imaginary part.

Example 1

The number `3 + 2j` (where `j=sqrt(-1)`) is represented by:

1 2 3 4 1 2 3 R j
3 + 2j
A

The complex number 3 + 2j.

The point A is the representation of the complex number `3 + 2j.`

The horizontal axis is marked R (for the "real" numbered-component), and the vertical axis is marked j (for the imaginary component of the complex number).

Adding Complex Numbers Graphically

We can add complex numbers graphically.

This is the same idea as adding vectors graphically. See more at Adding 2D Vectors.

Example 2

Add `1 + 2j` and `3 - j` graphically.

Answer

1 2 3 4 1 2 3 -1 R j
1 + 2j
3 − j
4 + j

Graphically adding 1 + 2j to 3 −j.

We add the complex numbers by setting up a parallelogram, just as we did when adding vectors. The solution is `(4 + j)`.

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Exercise

Perform graphically: `(3 - 2j) + (-1 - j)`

Answer

1 2 3 4 -1 -2 1 2 -1 -2 -3 R j
−1 − j
3 − 2j
2 − 3j

Graphically adding 1 + 2j to 3 −j.

As we can see from the graph, the answer is `2 − 3j`.

See many more examples of adding vectors in an interactive applet over in the Vectors chapter.

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