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Complex numbers cannot be represented on a normal set of coordinate axes.

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In 1806, J. R. Argand developed a method for displaying complex numbers graphically as a point in a special coordinate plane. This method, called the Argand diagram or complex plane, 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 represented by the point (a,b), as shown at the left.

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Graph the following complex numbers:

1.   3 + 4i
          (3,4)

2.   -4 + 2i       (-4,2)

3.   2 - 3i          (2,-3)

4.   3 (which is really 3+ 0i)       (3,0)

5.   4i (which is really 0 + 4i)     (0,4)

 

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Geometrically, the concept of "absolute value" of a real number, is depicted as its distance from 0 on a number line. The "absolute value" of a complex number, is depicted as its distance from 0 in the complex plane.

The absolute value of a complex number
z = a + bi  is written as | z | or | a + bi |.
It is a non-negative real number defined as:
absoluteformula

Notice the Pythagorean Theorem at work in the right triangle.

A complex number can be represented by a point, or by a vector from the origin to the point. When thinking of a complex number as a vector, the absolute value of the complex number is simply the length of the vector, called the magnitude.
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Find | z | for :

1.    z = 3 + 4i
horizontal length a = 3
vertical length b = 4
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2.    z = -4 + 2i
horizontal length | a | = 4
vertical length b = 2
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Graphical addition and subtraction of complex numbers.

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

Graph the two complex numbers as vectors.

• Create a parallelogram using these two vectors as adjacent sides. (Count off the horizontal and vertical lengths from one vector off the endpoint of the other vector.)

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

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2. Subtract 3 + 3i from -1 + 4i graphically.

Subtraction is the process of adding the additive inverse.
(-1 + 4i) - (3 + 3i)
= (-1 + 4i) + (-3 - 3i)
= -4 + i
• Graph the two complex numbers as vectors.

• Graph the additive inverse of the number being subtracted.
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• Create a parallelogram using the first number and the additive inverse.

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

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