 Graphing Complex Numbers 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          (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)   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 . 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). NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation and is not considered "fair use" for educators. Please read the "Terms of Use".

Contact Person: Donna Roberts