TABLE OF CONTENTS This appendix is a review of the algebra of complex numbers. The basic operations are defined and illustrated by several examples. Applications using Euler s identities are presented, and the exponential and polar forms are discussed and illustrated with examples.
In the language of mathematics, the square root of minus one is denoted as i, that is, 
. In the electrical engineering field, we denote i as j to avoid confusion with current i. Essentially, j is an operator that produces a 90-degree counterclockwise rotation to any vector to which it is applied as a multiplying factor. Thus, if it is given that a vector A has the direction along the right side of the x-axis as shown in Figure D.1, multiplication of this vector by the operator j will result in a new vector j A whose magnitude remains the same, but it has been rotated counterclockwise by 90 .
Also, another multiplication of the new vector j A by j will produce another 90 counterclockwise direction. In this case, the vector A has rotated 180 and its new value now is ?A. When this vector is rotated by another 90 for a total of 270 , its value becomes j( ?A)= ?JA. A fourth 90 rotation returns the vector to its original position, and thus its value is again A.
Therefore, we conclude that j 2= ?1, j 3= ?j , and j
