Cyclic Nature of the Powers of i

     To be cyclic means to be repetitive in nature.  When the imaginary unit, i, is raised to increasingly larger powers, it creates a cyclic pattern.

The powers of i repeat in a definite pattern:
( i, -1, -i, 1 )

 Powers of i ...
 Simplified form i -1 -i 1 i -1 -i 1 ...

 


You need to remember that:

Think about what happens when i  is raised to a given power:


LOOK OUT!!!

is not true when a and b are both negative.

False

TRUE:

 

Whenever the exponent is greater than or equal to 5, you can
use the fact that to simplify a power of i.

 

When raising i to any integral power, the answer is always i, -1, -i or 1.

Another way to think of this process of simplifying powers of i is to divide the exponent by 4,
- if the remainder is 0, the answer is 1  (i0).
- if the remainder is 1, the answer is i  (i1).
-if the remainder is 2, the answer is -1  (i2).
-if the remainder is 3, the answer is -i  (i3). 

 

Let's examine two ways to simplify :

Using the patterns shown in the robot table above:

Looking at remainders when dividing by 4:
with a remainder of  3,
 which means the answer is
i 3 = -i.

How to use your
TI-83+ graphing calculator  with powers of i.
Click calculator


Roberts