# 2: Complex Numbers

• • Contributed by Marcia Levitus

Chapter Objectives

• Be able to perform basic arithmetic operations with complex numbers.
• Understand the different forms used to express complex numbers (cartesian, polar and complex exponentials).
• Calculate the complex conjugate and the modulus of a number expressed in the different forms (cartesian, polar and complex exponentials).
• Be able to manipulate complex functions.
• Be able to obtain expressions for the complex conjugate and the square of the modulus of a complex function.

• 2.1: Algebra with Complex Numbers
The imaginary unit i is defined as the square root of -1.
• 2.2: Graphical Representation and Euler Relationship
Complex numbers can be represented graphically as a point in a coordinate plane. In cartesian coordinates, the x -axis is used for the real part of the number, and the y -axis is used for the imaginary component. Complex numbers can be also represented in polar form.  We can also represent complex numbers in terms of complex exponentials.
• 2.3: Complex Functions
The concepts of complex conjugate and modulus that we discussed above can also be applied to complex functions.
• 2.4: Problems

Thumbnail: A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i satisfies $$i^2 = −1$$. (CC BY-SA 3.0 unported; Wolfkeeper via Wikipedia)