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 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.


Graph the following complex numbers:

1.   3 + 4i

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)




The Pythagorean Theorem will be used to determine the absolute value of a complex number.

Geometrically, the concept of "absolute value" of a real number, such as 3 or -3, is depicted as its distance from 0 on a number line. Thus, | 3 | = 3 and | -3 | = 3.

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:

NOTE: Another term for "absolute value" is "modulus".
When dealing with a complex number, a + bi, the terms "absolute value", "modulus", and "magnitude" all refer to absoluteformula.

In the complex plane, a complex number may be represented by a single point, or by the point and a position vector(from the origin to the point). When referenced as a vector, the term "magnitude" is commonly used to represent the distance from the origin (absolute value).


Find | z | for :

1.    z = 3 + 4i
horizontal length a = 3
vertical length b = 4

2.    z = -4 + 2i
horizontal length | a | = 4
vertical length b = 2

The complex numbers in this Argand diagram are represented as ordered pairs with their position vectors.


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).
• This new vector is called the resultant vector.


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.

• 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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