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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?

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 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

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

But an inequality also cannot have an answer in a natural way, right?

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

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



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.

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?

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

Like, on the clock, or . 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.

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.

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 , that was in modular arithmetic, working modulo 2:
And the example on the clock: means .

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) so the inverse of this should be . Obviously, had I taken the "usual" logarithm, this would not have held. Finding this inverse in general is rather tricky to do.

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.

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! :embarrass:) 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?

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:
http://nocache-nocookies.digitalgott.com/gallery/10/thumb_5956_21_03_12_6_28_15.jpeg (http://nocache-nocookies.digitalgott.com/gallery/10/thumb_5956_21_03_12_6_28_15.jpeg)
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;)

Very nice to know that. Unfortunately I dont use UF and dont know how to construct fomulas with it. But Ill have to give it a try.  

there are a bunch of variations. You can get a working 2D algebra by having any "non-real" number squaring to any real number:

.

You obviously get the usual complex numbers by choosing but similarly, you can choose which gives you the so-called split complex numbers, also known as hyperbolic numbers. A choice of will give you dual numbers which have interesting properties in that they allow automatic differentiation:

- If I recall correctly (I haven't played around with these in a while and am kind of rusty) by choosing , applying will always yield . That might not be correct but I distinctly remember this choice being closely linked to differentiation either way.

The other thing you can do is add multiple different non-real square roots of real numbers. Doing so and putting them all together with certain straight forward rules leads to Geometric Algebras or Clifford Algebras (two names for the same thing) - essentially an extension of complex numbers to arbitrary numbers of dimensions. A subset of such Algebras is the perhaps familiar Quarternions.

There is a rather big forum topic on Geometric Algebra if you are interested: http://www.fractalforums.com/new-theories-and-research/geometric-algebra-geometric-calculus/

Other extensions will not lead to higher-dimensional structures but rather will cause completely different things to happen. For that, for instance, look up p-adic numbers which are a sort of alternate structure to real numbers, resulting from generalizing the rational numbers in a different manner.

Maybe a strange question but I watched some videos on the subjects x/0 and 0^0. These arent possible to solve with natural numbers and with complex numbers. But what would happend if we would use one of these rules:

x=1/0

x=0^0

Would it help to solve problems? Or not?
I think the shape will not be much different from a plane with natural numbers because 1/0 is not commonly used.

If you set ( being any finite number) you'll end up with projective numbers. Read up on the projective number line or the riemann sphere for example. There even are more complex variations where you have multiple such values corresponding to, for instance, rolling up the complex plane into a torus etc.

The case with seems different to me. Basically,  what is depends on what is as approaches a common zero-point .
Same goes for
I don't think those two values would lend themselves to such a treatment.

What might work is a definition like - something not typically true for any value (except,  as mentioned, for certain cases of but not generally for all 0s)
Not sure what would happen then. Never tried it.

EDIT: actually is fullfilled for , so if I had to guess, this would just be your typical projective treatment. There may be some significant differences in the details though.

https://youtu.be/BBp0bEczCNg
https://youtu.be/FVZqPaH94qU

Just a question. What if we use these instead?

The biggest difference in this case would be that you get infinitesimals, as far as I can tell. One thing that this still wouldn't give you, for instance, is division by 0. (Unless you are in a projective space where is an unambiguous statement with both 0 and being undirected values of your system, as I already mentioned before.)
But I'm not sure, this stuff has a lot of subtleties.

Norman Wildbergers Maths foundation series starting with around MathsFoundation 150 tackles this issue for me . The difficulty is with mathematicians believing their own jargon, and a lot of mystique. Both Justus and Hermann Grassmann are from an era when revolution was the spirit of the time, the Zeitgeist, and that spirit was to see and think clearly with an unblinking uncompromising Eye!

That sums up Normans point of view for Today's Mathematics.

It is also my point of view.

The symbols and symbolic traditions in mathematics have been given a Divine Status? Why?
You have to go back to an ancient era, where the Pythagorean school captured the spirit of the times, the spirit of the Musai or Muses, being the source of all Culture : the arts and crafts, skills and philosophy of being.

That school was so influential despite its small size that it naturally attracted powerful enemies, asvwellmas great and Rich admirers. Plato was such a young rich idealist. He portrayed the school in his writings, the only source we have that describes the workings of that group and its founder, Pythagoras.

And that was and is the problem: we the next generations always get a romanticised version of what actually happened in the Raw!

The French Revolution defined a European wide intellectual renaissance that eventually culminated in the Prussian Spring and the dissolution of the Prussian Holy Roman Empire. Many ideas, like functions, and infinite sets arose in that context, many reflecting the christology of the originators .

It is that confusion of Philosophy and Theosophy or Christology that created the divine nonsense that mathematics is the language of God or the universe.

http://m.youtube.com/watch?v=uw6bpPldp2A
This is understanding .
Newtonian triples fulfil this 3 dimensional requirement Hamilton sought for. However topologically we have to relinquish orthogonal l axes. This was what he was loathe to do!!