since x = x mod 4 + 4K
i^x = i^(x mod 4 + 4k)
= i^(x mod 4) * i^(4k)
= i^(x mod 4) * (i^4)^k
= i^(x mod 4) * 1^k
= i^(x mod 4)
Yes, this is always true ...
]]>I was mainly interested in the idea of multiple orders of i's, each coming from its previous form (i.e. √-1, √-i, √-√-i, etc.) and their behavior (that is if the x mod 4 power rule still works for them, or if it's slightly changed), and if these special behaviors exhibit patterns.
]]>More general theorem:
[list=*]
[*]
Even more general theorem:
[list=*]
[*]
Actual theorem:
Sometimes little things get out sight..... but I think that happens to everyone.
]]>This is really weird:
for x>0 and x is positive.
I don't get this. For example, when x is 3.
]]>We get that
Now we also know
And manipulation of that reveals
So we can reword equation 2 and obtain
Which means this rotating pattern...
...works for strange cases of roots.
Example:
]]>If this works for
, then it should work for other imaginary values.... (i.e. imaginary numbers created other ways, but still using a radical)]]>