Math Is Fun Forum

αβγ

Member

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)

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

Member

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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Bassaricyon neblina

αβγ

Member

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:

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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"

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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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αβγ

Member

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.

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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"

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 24,996

Website

Re: i = i^x mod 4

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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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
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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.

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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"

Olinguito

Member

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:

[list=*]
[*]

[/*]
[/list]

Even more general theorem:

[list=*]
[*]

[/*]
[/list]

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)

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Bassaricyon neblina

αβγ

Member

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.

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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"

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 ...

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!

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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"