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 Author Topic: Using different inequalities in the rules of complex numbers.  (Read 1746 times) Description: 0 Members and 1 Guest are viewing this topic.
kram1032
Fractal Senior

Posts: 1863

 « Reply #15 on: November 29, 2014, 08:52:21 PM »

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

$x\notin \mathbb{R} \: \wedge \: x^2 = a \in \mathbb{R}$.

You obviously get the usual complex numbers by choosing $a=-1$ but similarly, you can choose $a=1$ which gives you the so-called split complex numbers, also known as hyperbolic numbers. A choice of $a=0$ will give you dual numbers which have interesting properties in that they allow automatic differentiation:

$\left(x+a y\right)^2=x^2+a^2 y^2 + x a y+ a y x=x^2+2 a x y$ - If I recall correctly (I haven't played around with these in a while and am kind of rusty) by choosing $x=1$, applying $f\left(1+a y\right)$ will always yield $f(1)+a f'(y)$. 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.
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TheRedshiftRider
Fractalist Chemist
Global Moderator
Fractal Iambus

Posts: 854

 « Reply #16 on: December 03, 2014, 08:29:44 AM »

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.
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Motivation is like a salt, once it has been dissolved it can react with things it comes into contact with to form something interesting.
kram1032
Fractal Senior

Posts: 1863

 « Reply #17 on: December 03, 2014, 08:57:42 AM »

If you set $x=\frac{a}{0}$ ($a$ 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 $0^0$ seems different to me. Basically,  what $0^0$ is depends on what ${f\left(x\right)}^{g\left(x\right)}$ is as $x$ approaches a common zero-point $x_0$.
Same goes for $\frac{0}{0}$
I don't think those two values would lend themselves to such a treatment.

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

EDIT: actually $x^x=0$ is fullfilled for $x\to-\infty$, so if I had to guess, this would just be your typical projective treatment. There may be some significant differences in the details though.
 « Last Edit: December 04, 2014, 02:06:01 PM by kram1032 » Logged
TheRedshiftRider
Fractalist Chemist
Global Moderator
Fractal Iambus

Posts: 854

 « Reply #18 on: September 07, 2015, 07:08:02 PM »

Just a question. What if we use these instead?
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Motivation is like a salt, once it has been dissolved it can react with things it comes into contact with to form something interesting.
kram1032
Fractal Senior

Posts: 1863

 « Reply #19 on: September 07, 2015, 10:45:15 PM »

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 $\frac{1}{0}=\infty$ is an unambiguous statement with both 0 and $\infty$ being undirected values of your system, as I already mentioned before.)
But I'm not sure, this stuff has a lot of subtleties.
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jehovajah
Global Moderator
Fractal Senior

Posts: 2749

May a trochoid in the void bring you peace

 « Reply #20 on: September 15, 2015, 11:10:24 AM »

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.
 « Last Edit: July 16, 2016, 10:34:50 AM by jehovajah » Logged

May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!
jehovajah
Global Moderator
Fractal Senior

Posts: 2749

May a trochoid in the void bring you peace

 « Reply #21 on: July 15, 2016, 10:16:44 AM »