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:

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Even more general theorem:

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[/list]Note that 1, i, i², i³ are all distinct. We say that is a primitive

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:

]]>for x>0 and x is positive.

If this works for

, then it should work for other imaginary values.... (i.e. imaginary numbers created other ways, but still using a radical)]]>