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 Author Topic: Using different inequalities in the rules of complex numbers.  (Read 2196 times) Description: 0 Members and 1 Guest are viewing this topic.
TheRedshiftRider
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 « on: November 27, 2014, 08:22:59 AM »

I've been wondering this for a long time and I dont seem to be able to find the answers myself. (please correct me if my thoughts are wrong about this.)

Complex numbers follow the rule that sqrt(x)=-1. Is it possible to use a different inequation for this to change the properties of complex  numbers, for example this one: x+1=x-1

If the properties change, how do they change and how would for example the mandelbrot set or a different fractal look with the changed rules?
 « Last Edit: November 27, 2014, 10:23:34 AM by Toofgib » Logged

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claude
Fractal Bachius

Posts: 563

 « Reply #1 on: November 27, 2014, 02:04:10 PM »

You can indeed create new number systems by defining equivalence classes (quotient group).  If you take the real number line and define  x = x + 1  then you get a circle measured in turns, each time you go around a full turn you end up where you started.  This circle is still 1 dimensional.  Complex numbers are kind of special because they add a new dimension, so you get a pair of real numbers instead of just one.  You can do something similar by starting with the rational numbers and adding an irrational square root, like $z = a + b \sqrt{2}$ where a and b are rationals - multiplying or adding two numbers of this form gives a third number of the same form.  An important difference from this construction to complex numbers is that rationals extended with a square root can be mapped to the totally ordered real line, while complex numbers can't be ordered in a natural way.

https://en.wikipedia.org/wiki/Quotient_group
https://en.wikipedia.org/wiki/Field_extension
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claude
Fractal Bachius

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 « Reply #2 on: November 27, 2014, 03:13:51 PM »

Another variation is to use a different distance metric for the complex number space, going to hyperbolic space (with many parallels to a given line through a given point) instead of flat Euclidean space (unique parallel through a given point):

http://www.itpa.lt/~acus/Knygos/Clifford_articles/Varia/10.1007_s00006-010-0265-1.pdf Generating Fractals Using Geometric Algebra
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TheRedshiftRider
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 « Reply #3 on: November 27, 2014, 04:53:36 PM »

An important difference from this construction to complex numbers is that rationals extended with a square root can be mapped to the totally ordered real line, while complex numbers can't be ordered in a natural way.
But an inequality also cannot have an answer in a natural way, right?
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kram1032
Fractal Senior

Posts: 1863

 « Reply #4 on: November 27, 2014, 05:18:35 PM »

I'm not quite sure what you mean by inequality.

$x+1 = x-1$ isn't an inequality. It's an equation with a variable. This particlar equation doesn't have a solution in $\mathbb{R}$ but it has a solution in $\mathbb{R}$ modulo 2 if I'm not mistaken:

$x+1 = x-1 \Rightarrow x=x+2$

http://en.wikipedia.org/wiki/Modular_arithmetic
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TheRedshiftRider
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 « Reply #5 on: November 27, 2014, 08:19:48 PM »

I'm not quite sure what you mean by inequality.

<Quoted Image Removed> isn't an inequality. It's an equation with a variable. This particlar equation doesn't have a solution in <Quoted Image Removed> but it has a solution in <Quoted Image Removed> modulo 2 if I'm not mistaken:

<Quoted Image Removed>

http://en.wikipedia.org/wiki/Modular_arithmetic
You're right about that. It is even solvable with simple maths. I'll have to find a better example.
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youhn
Fractal Molossus

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Shapes only exists in our heads.

 « Reply #6 on: November 27, 2014, 08:28:17 PM »

I'm no math wiz, but when I see x+1=x-1 I would do:

x - x = -1 - 1
0 = -2

This looks like nonsense in math context.

Why are you looking for alternatives in the first place?
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kram1032
Fractal Senior

Posts: 1863

 « Reply #7 on: November 27, 2014, 08:52:44 PM »

as said, 2 = 0 is a totally valid result in modular arithmetic. - Also called Clock Arithmetic.

Like, on the clock, $9+4=1$ or $5 \times 12=0$. This sort of math is actually rather important for, for instance, cryptography, because it's rather easy to perform some tasks in this space but almost impossible to invert. If I'm not mistaken, elliptic curve protocols are essentially based on that kind of math.
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youhn
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Shapes only exists in our heads.

 « Reply #8 on: November 27, 2014, 09:16:12 PM »

So it just shorthand for something like (9 + 4) % 12 = 1 but written exactly the same as a normal addition? Scanning through the modular wikipedia article I do understand the basic. I just don't get the notation which seem to case a lot of confusion.
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kram1032
Fractal Senior

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 « Reply #9 on: November 27, 2014, 09:36:08 PM »

What notation, the one in the wiki or the one I'm using here?
The one in the wiki gives more information and is thus clearer, so it's better in that way. However, if you explicitly say you are going to use modular arithmetic, and you specify to what you want to have it modulo, you can just use straight-up notation from normal math as always. You can also define stuff like division, exponentiation or logarithms over such spaces. Taking the logarithm is extremely hard while exponentiation is reasonable in computational complexity as far as I understood. IIRC, calculating a modular logarithm is essentially what's necessary to crack an elliptic curve encryption.

So in the below examples, when I wrote $0=2$, that was in modular arithmetic, working modulo 2: $2\equiv 0 \text{ mod } 2$
And the example on the clock: $5\times12=0$ means $5 \times 12 \equiv 0 \text{ mod } 12$.

It really is exactly equivalent though. Like, what you'll typically do is take a function from integers to integers and then take the result of that modulo whatever number you want to work with. However, that doesn't work as easily for inverse functions. Those need a special treatment. (That's why exponentiation is relatively easy but taking a logarithm isn't) $2^3 \equiv 3 \text{ mod } 5$ so the inverse of this should be $\log_2{3} \equiv 2 \text{ mod } 5$. Obviously, had I taken the "usual" logarithm, this would not have held. Finding this inverse in general is rather tricky to do.
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TheRedshiftRider
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 « Reply #10 on: November 28, 2014, 08:10:55 AM »

Why are you looking for alternatives in the first place?
In the first place I was just wondering if It was possible and what the effect would be. The second thing is that it would be a different way of changing the way a fractal is projected instead of changing the plane.

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jehovajah
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 « Reply #11 on: November 28, 2014, 11:42:10 AM »

Complex numbers follow the rule that sqrt(x)=-1. Is it possible to use a different inequation for this to change the properties of complex  numbers, for example this one: x+1=x-1

These rules are not commonly used in complex numbers. Rather x2 = -1 is the usual form . Assuming that was a typo ( and by god I make them all the time! ) the next part of your statement is not properly specified.

A modulo arithmetic may be what you meant originally or you may be using x in place of z, which is a conventional indicator of a complex term.

In any case keep wondering , and keep trying to express as clearly as possible what you are wondering about, because that is how you will tease out what is really of interest to you.
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TheRedshiftRider
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 « Reply #12 on: November 28, 2014, 12:24:14 PM »

These rules are not commonly used in complex numbers. Rather x2 = -1 is the usual form . Assuming that was a typo ( and by god I make them all the time! ) the next part of your statement is not properly specified.

A modulo arithmetic may be what you meant originally or you may be using x in place of z, which is a conventional indicator of a complex term.

In any case keep wondering , and keep trying to express as clearly as possible what you are wondering about, because that is how you will tease out what is really of interest to you.

Complex numbers follow the rule that x^2=-1. What happens if that rule would be different for example:

x^(x-1)=-1 (this would probably be a better example than x+1=x-1)

Will the properties change? If so, how? And how would certain fractals (for example the mandelbrot set) look when using this number system instead of complex numbers?
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Alef
Fractal Supremo

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 « Reply #13 on: November 28, 2014, 05:14:35 PM »

It's y*y=-1 not x. But if it would be x*x=1 and y*y=1 with x><y you'll get square mandelbrot set and a bitt more interesting julia sets if you use inside colouring. Like this:
Quadratic General formula in Ultra Fractal.
If you could make general formula for y^(y-1)=-1 with x*x=1 you could see a result;)
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TheRedshiftRider
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 « Reply #14 on: November 28, 2014, 05:23:22 PM »

It's y*y=-1 not x. But if it would be x*x=1 and y*y=1 with x><y you'll get square mandelbrot set and a bitt more interesting julia sets if you use inside colouring. Like this: