Representing Complex Numbers Graphically
(+ and -)

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Due to their unique nature, complex numbers cannot be represented on a normal set of coordinate axes.

In 1806, J. R. Argand developed a method for displaying complex numbers graphically as a point in a coordinate plane.  His method, called the Argand diagram, establishes a relationship between the x-axis (real axis) with real numbers and the y-axis (imaginary axis) with imaginary numbers.

In the Argand diagram, a complex number  a + bi  is the point (a,b) or the vector from the origin to the point (a,b).

 

Graph the complex numbers:

1.  3 + 4i       (3,4)

2.  2 - 3      (2,-3)   

3.  -4 + 2    (-4,2)

4.  3 (which is really 3 + 0i (3,0)

5.  4i  (which is really 0 + 4i (0,4)

The complex number is represented by the point, or by the vector from the origin to the point.

 

 

Add 3 + 4i and -4 + 2i graphically.

Graph the two complex numbers 3 + 4i and -4 + 2i as vectors.

Create a parallelogram using these two vectors as adjacent sides.

The answer to the addition is the vector forming the diagonal of the parallelogram (read from the origin).

This new vector is called the resultant vector.

 

Subtract 3 + 4i from -2 + 2i

Subtraction is the process of adding the additive inverse.
(-2 + 2i)  -  (3 + 4i)
= (-2 + 2i) + (-3 - 4i)
= (-5 - 2i)

Graph the two complex numbers as vectors.

Graph the additive inverse of the number being subtracted.

Create a parallelogram using the first number and the additive inverse.  The answer is the vector forming the diagonal of the parallelogram.