## Powers of i -- Imaginary Number

The powers of i is always equal to either one of these 4 numbers: 1, i , -1,-i. Here we will see why.

i0 = 1, this is because for any any non-zero number, any number with exponent 0 is equal to 1.

i1 = i, this is because  any number with exponent 1 is equal to itself, in this case i itself.

i2 = -1, this can be explained by the definition of imaginary number, which says i is defined as `sqrt(-1), so its square is -1.`

i3 = -i, since i2=-1, i3 = i2 * i = -1 * i = -i

i4 = 1 becausei4 =  i3 * i = -i * i = -`sqrt(-1)` ` `*`sqrt(-1)`   = 1

We see that i4 = i0, so this is a cycle, so by prediction i5 equals to i, i6 equals -1, and so on.... and in fact that is true.

So i with the exponent n (any whole number from 0 to above) can be determined by the following rules:

Divide the exponent by 4 and find the remainder,

if the remainder is 0 (no remainder), then in equals 1;

if the remainder is 1, then in equals i;

if the remainder is 2, then in equals -1;

if the remainder is 3, then in equals -i.

• Subject : Math
• Topic : Algebra 2
• Posted By : Jason