Imaginary numbers - Is there a shortcut? I have recently learned about imaginary numbers and I sat in front of this pattern of -1, -i, 1, i and I asked myself, isn't there an easy way to describe this pattern ? Or: isn't there a really short way to find out about imaginary numbers raised to high powers ?

(Despite the "find a factor of 4"-method) enter link description here
I am not sure if this works really ALL the time and I have only tested it for powers that are positive and negative integers, but haven't found any contradictions and algebraically this works like that power of 4 thing:

When your power divided by 2 gives an integer (so no fraction or decimal):
Just divide the power by 2 : If the result is odd, then the number is = (-1)
If it is even, then the number is =(1) BUT ONLY IF YOUR RESULT IS AN INTEGER !

What if the power of i itself is odd (result is not an integer)? Like 5 or 9999 ?
Then the result is i or -i . To figure out: go back one step and divide the (power-1) by 2 .
If the result is odd => it's -i
If the result is even => it's +i

E.g.: i^4 = ? and i^5= ?
=> 4:2 = 2 ; 2 is even ; => 1
=> 5:2 does not work (for x E Z) so it must be i or -i . So subtract 1 from the power = 4 , 4:2 = even => +i

This also works for negative powers, BUT BE AWARE that, say a power of (-5)-1 is = -6 not -4 !!

Example:
i^9999 ; 9999:2 = Not an integer -> 9999-1 = 9998:2 = 4999 (That is ODD)
so i^9999= -i

i^34; 34:2 = 17 -> Odd number and an integer => i^34=-1

Be aware ! Some numbers like 336 seem to give an odd result "something with 3 at the end" when you are in a hurry - but it's actually even. This is where I often do stupid mistakes :D

Hope you enjoy this. I can not guarantee that this is always true. I just found it out by myself (maybe somewhere on the internet this is already explained :) ))

At my original post at Khan-academy a guy named Ian has well explained why this works out. As also as I mentioned before, you can work this out by algebra since 2 is a factor of 4.Khanacademy post
I am not sure if this works really ALL the time and I have only tested it for powers that are positive and negative integers, but haven't found any contradictions and algebraically this works like that power of 4 thing:

When your power divided by 2 gives an integer (so no fraction or decimal):
Just divide the power by 2 : If the result is odd, then the number is = (-1)
If it is even, then the number is =(1) BUT ONLY IF YOUR RESULT IS AN INTEGER !

What if the power of i itself is odd (result is not an integer)? Like 5 or 9999 ?
Then the result is i or -i . To figure out: go back one step and divide the (power-1) by 2 .
If the result is odd => it's -i
If the result is even => it's +i

E.g.: i^4 = ? and i^5= ?
=> 4:2 = 2 ; 2 is even ; => 1
=> 5:2 does not work (for x E Z) so it must be i or -i . So subtract 1 from the power = 4 , 4:2 = even => +i

This also works for negative powers, BUT BE AWARE that, say a power of (-5)-1 is = -6 not -4 !!

Example:
i^9999 ; 9999:2 = Not an integer -> 9999-1 = 9998:2 = 4999 (That is ODD)
so i^9999= -i

i^34; 34:2 = 17 -> Odd number and an integer => i^34=-1

Be aware ! Some numbers like 336 seem to give an odd result "something with 3 at the end" when you are in a hurry - but it's actually even. This is where I often do stupid mistakes :D

Hope you enjoy this. I can not guarantee that this is always true. I just found it out by myself (maybe somewhere on the internet this is already explained :) ))

At my original post at Khan-academy a guy named Ian has well explained why this works out. As also as I mentioned before, you can work this out by algebra since 2 is a factor of 4.Khanacademy post