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**αβγ****Member**- Registered: 2014-09-02
- Posts: 19

This is really weird:

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)α as in alpha stands for "**a** love of math"

β as in beta stands for "**b**eing skilled in the latter"

γ as in gamma stands for "**g**etting sick of homework"

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**Olinguito****Member**- Registered: 2014-08-12
- Posts: 649

It works for negative integers as well: e.g. .

*Bassaricyon neblina*

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**αβγ****Member**- Registered: 2014-09-02
- Posts: 19

Since

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:

α as in alpha stands for "**a** love of math"

β as in beta stands for "**b**eing skilled in the latter"

γ as in gamma stands for "**g**etting sick of homework"

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,039

That is not much of a surprise, since 0.02th root of i is 50th power of i. No need for for modulo to get involved.

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Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**αβγ****Member**- Registered: 2014-09-02
- Posts: 19

It was just an example to show the (1/x)th root of i can be reduced to one of the 4 forms of i, and since roots and powers of i can be defined, logarithms of i can be defined, since logarithms are just fancy, difficult forms of powers and roots.

α as in alpha stands for "**a** love of math"

β as in beta stands for "**b**eing skilled in the latter"

γ as in gamma stands for "**g**etting sick of homework"

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αβγ wrote:

This is really weird:

for x>0 and x is positive.

I don't get this. For example, when x is 3.

*Last edited by Agnishom (2014-09-04 02:43:56)*

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I'm not crazy, my mother had me tested.

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**αβγ****Member**- Registered: 2014-09-02
- Posts: 19

Sorry! I forgot to add the ^x ...

Actual theorem:

Sometimes little things get out sight..... but I think that happens to everyone.

**a** love of math"

β as in beta stands for "**b**eing skilled in the latter"

γ as in gamma stands for "**g**etting sick of homework"

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**Olinguito****Member**- Registered: 2014-08-12
- Posts: 649

That’s because i is a root of the equation .*n*th root of unity iff are all distinct. If is the set of all primitive *n*th roots of unity, then is a primitive root if and only if for all isomorphisms .

More general theorem:

Even more general theorem:

*Last edited by Olinguito (2014-09-05 09:55:10)*

*Bassaricyon neblina*

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**αβγ****Member**- Registered: 2014-09-02
- Posts: 19

I don't understand any of that, but thank you for the input!

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.

**a** love of math"

β as in beta stands for "**b**eing skilled in the latter"

γ as in gamma stands for "**g**etting sick of homework"

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**marcuslee****Member**- Registered: 2014-09-04
- Posts: 12

Can I put it like this?

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 ...

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**αβγ****Member**- Registered: 2014-09-02
- Posts: 19

Your way seems to make sense and show how it works... thank you!

**a** love of math"

β as in beta stands for "**b**eing skilled in the latter"

γ as in gamma stands for "**g**etting sick of homework"

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