Start by calculating i^1, i^2, i^3, and i^4. Then calculate i^5, i^6, i^7, and i^8. Do you notice any patterns?
i
Practice makes perfect
Generally, the value of i^n, where n is a whole number, can be calculated by dividing n by 4 and considering the remainder. Let R be the remainder when n is divided by 4.
i^n=
i, ifR=1
- 1, ifR=2
- i, ifR=3
1, ifR=0
Now, let's consider i^(41). We need to find the remainder when 41 is divided by 4.
41 Ă· 4 = 10 R1
Because the remainder is 1, we get that i^(41)=i.
Extra
Powers of i
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 one 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, ... repeats in that order continuously after the first four results.
