Complex Numbers

It is often stated that the square root of a negative (radicand) number is "not a Real number" since no Real number when squared would produce a negative product.  While it is true that this number is not "Real," it is none the less a number.  In Algebra,   is defined to be the letter i Since the square root of negative one is now defined to be the letter i , it is now possible to simplify a radical with a negative radicand, such as .
These numbers that contain the
i which stand for the are called imaginary numbers.

Often when a Real number and an imaginary number are in the same expression the Real number is first and the imaginary number follows.  This form is commonly known as the a + bi form and the two numbers together form what is called a complex number.

When multiplying complex numbers, the result may be another complex number.  For example, consider the product of the following binomials:

Normally we would combine the like terms and arrive at the product of 

Since the complex number is left in the form a + bi, the must then be addressed.

  since    so if we substitute negative one in the place of the
i squared in the last example, then we would have

 

is now a complex number of the form a + bi.

An interesting result occurs when multiplying two Complex number conjugates.  Consider multiplying the conjugate binomials INSERT REMAINING EXPLANATION

 

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 16-Sep-2008