Exercise 20 Page 255

The Commutative and Associative Properties of Multiplication hold true for imaginary numbers. We can use them to find the powers of i. Let's consider the first four powers. Recall that any number raised to the power of 1 equals itself, so i^1= i. Moreover, by definition i=sqrt(- 1), so we know that i^2= - 1. ccccccc i^1&=& i i^2&=& - 1 i^3&=& i^2 * i &=& - 1 * i&=& - i i^4&=& i^2 * i^2&=& ( - 1)^2 * ( - 1)^2&=& 1 Let's now calculate the following four powers. To do so, we will use the results obtained above. ccccccc i^5 &=& i^4 * i &=& 1 * i &=& i i^6 &=& i^4 * i^2 &=& 1 * ( - 1) &=& - 1 i^7 &=& i^4 * i^3 &=& 1 * ( - i) &=& - i i^8 &=& i^4 * i^4 &=& 1 * 1 &=& 1 Notice that the pattern i, - 1, - i, 1, ... is repeated in that order continuously after the first four results.