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#1 2014-09-02 07:13:37

αβγ
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Registered: 2014-09-02
Posts: 19

i = i^x mod 4

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 "being skilled in the latter"
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#2 2014-09-02 10:14:53

Olinguito
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Registered: 2014-08-12
Posts: 649

Re: i = i^x mod 4

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


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#3 2014-09-03 04:01:18

αβγ
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Registered: 2014-09-02
Posts: 19

Re: i = i^x mod 4

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 "being skilled in the latter"
γ as in gamma stands for "getting sick of homework"

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#4 2014-09-03 05:28:32

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 16,037

Re: i = i^x mod 4

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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#5 2014-09-04 02:28:40

αβγ
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Registered: 2014-09-02
Posts: 19

Re: i = i^x mod 4

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 "being skilled in the latter"
γ as in gamma stands for "getting sick of homework"

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#6 2014-09-04 02:42:34

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,838
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Re: i = i^x mod 4

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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#7 2014-09-04 02:52:39

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

Re: i = i^x mod 4

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

Actual theorem:

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


α as in alpha stands for "a love of math"
β as in beta stands for "being skilled in the latter"
γ as in gamma stands for "getting sick of homework"

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#8 2014-09-04 18:55:30

Olinguito
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Registered: 2014-08-12
Posts: 649

Re: i = i^x mod 4

That’s because i is a root of the equation
.

More general theorem:


Even more general theorem:



Note that 1, i, i², i³ are all distinct. We say that
is a primitive nth root of unity iff
are all distinct. If
is the set of all primitive nth roots of unity, then
is a primitive root if and only if
for all isomorphisms
.

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


Bassaricyon neblina

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#9 2014-09-06 13:20:55

αβγ
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Registered: 2014-09-02
Posts: 19

Re: i = i^x mod 4

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.


α as in alpha stands for "a love of math"
β as in beta stands for "being skilled in the latter"
γ as in gamma stands for "getting sick of homework"

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#10 2014-09-08 08:51:14

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

Re: i = i^x mod 4

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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#11 2014-09-23 02:23:41

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

Re: i = i^x mod 4

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


α as in alpha stands for "a love of math"
β as in beta stands for "being skilled in the latter"
γ as in gamma stands for "getting sick of homework"

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