Solving the square root of -1 (nonimaginary)

[pioneer]

Here is something that popped into my head a few month ago that I would like to share with the community. This simple system allows one to solve the square roots of negative numbers without imaginary numbers.

 

What it involves is using a tetrahedral coordinate system that has 4 axis a,b,c and d, instead of x,y and z. What this does is avoid the need for negative numbers since the negative of A can be expressed as positive values of b, c and d. It just takes movement between the two systems to get rid of those pesky negative square roots.

[the tree]

What? How can (for instance) [math]\begin{bmatrix} -4 \\ 0 \\ 0 \\ 0 \end{bmatrix}[/math] be expressed in a form [math]\begin{bmatrix} 0 \\ b \\ c \\ d \end{bmatrix}[/math] at all? Let alone them all being positive?

[John Cuthber]

I'm noot sure but I think he means use 4 coordinates.

Take a point in the middle of a tetrahedron and draw lines to each of the points.You get something that looks like a caltrop

http://en.wikipedia.org/wiki/Caltrop

label the 4 points a,b,c,and d.

Now you can say that the vector corresponding to -a is composed of b+c+d (with some constant, possibly even 1, to account for scaling.

 

I don't doubt that this coordinate system could be used, but I'd much rather use 3 orthogonal ones. The "resolution of forces" in mechanics would be ugly, to say the least, in this system.

[the tree]

Ah, so not normal orthogonal axis then. This seems an awful lot of faff for not much gain.

[pioneer]

I am not suggesting moving away from x.y.z, but this system requires using negative numbers to populate the entire coordinate system. The four axis of the tetrahedral system are not perpendicular, but are 104 degrees to each other. As such, there is no need for negative numbers, since what should be the negative of say A can be expressed with the other 3 positive axis due to their 104 degree angle. I am not a mathematician and my skills are very atrophied. But this grid does lead to some very interesting implications like avoiding square root of -1.

 

Here is an interesting thought. Say the fathers of mathematics had chosen this tetrahedral system instead of cartesian coordinates, the theory of negative and positive charge would have never occurred. The positive charge may have been defined as (a,0,0,0) and the negative charge as (0,b,c,d). It is interesting to speculate that physics would now be so very different when it comes to EM force. Science would be looking for the three or four parameters of charge and may actually find them. With the current system plus and minus is good enough. They would make use of the system and have to work with what they had.

 

The tetrahedral system is interesting for speculation, but would not be easy to use after centuries of convention that uses 3-D. It could open many cans of worms that may be better left in the can. For example, space-time is 4-D, would that simply fit the tetrahedral axis or still need to be treated as an extended system with time out of the loop?

 

Philosophically it is interesting in that without plus and minus there is no hard philosophical polarization of postive and negative, good-evil. Rather polarization is simply the opposing ratio of otherwise postive parameters. This sort of like the subjectivity of right and wrong that is the study philosophy.

[Bignose]

I guess I just don't see the advantage to using 4-D when 3-D would do just as well. The real trouble with using 4-D when the situation is really 3-D is that one of the dimensions of the 4-D system will be dependent on the other 3. And, a really big one would be like John C said, resolution of forces or any other vector mathematics would become significantly more difficult, and it isn't all that easy in the first place.

 

On a little bit broader note, what is wrong with [math]\sqrt{-1}[/math] anyway? It is well defined, it obeys all the logical rules, the mathematics of it are pretty well established and developed (as demonstrated by the size of the average Complex Analysis text). I personally just don't see a problem.

[Dave]

On a little bit broader note, what is wrong with [math]\sqrt{-1}[/math] anyway? It is well defined, it obeys all the logical rules, the mathematics of it are pretty well established and developed (as demonstrated by the size of the average Complex Analysis text). I personally just don't see a problem.

 

There's nothing wrong with the complex numbers, but finding different ways to transform them can help to simplify problems. Whilst I'm not exactly sure what the OP intended, you could almost certainly connect the two spaces through a homeomorphism making them effectively two representations of the same thing.

[John Cuthber]

Incidentally, when you have finished rewriting all the vector geometry books to make them more difficult, please tell me what will be the value of x that solves the equation

x squared plus one equals zero.

[pioneer]

I hope I was clear that the 4 axis are equally separted in 3-D space using the tetrhedral angle of 104 degrees between all four axis. I don't recommend converting the existing math into this system. But specialty tools somethings can be used in unique situations.

 

Part of the reason I presented this was connected to using math to support theory in science. Math is clean and pure, but it is also a horse that can be led anywhere one so desires depending on the theory. In the tetrahedral system since there is no negative, than negative charge would have to be modelled with postive parameters. It would have three or four aspects which will differ from positive charge (a,0,0,0) and (0,b,c,d), respectively. If we let the math lead the theory than either the x,y,z or the a,b,c,d system is leading knowldge down the path of an illusion. This problem is especially common in phyics where dimensions are increasing. The math is a faithful workhorse but theory can lead it anywhere. So if the math adds up, it is still not certain if the result reflects reality.

 

In western religous tradition, God is considered a trinity. If we assume a type of symbolic parallel between God and reality, this would imply that x,y,z is the correct system for modelling reality. The tetrahedral will leave one hanging in a state of suspension. You'll gain wisdom but pay a price. One needs to look at the tetrahedral system as a specialty tool.

[insane_alien]

why use 4 axis when 2 will do? it also seems easier to write a+ib than [0 b c d]^t

 

sorry i don't know how to use LaTeX

[John Cuthber]

"One needs to look at the tetrahedral system as a specialty tool." One thing this tool seems unable to do is answer the equation I posted earlier, i.e. to find a square root of minus 1. Since that's the thread's topic, this failing wouuld seem to be pretty catastrophic.

 

 

"If we assume a type of symbolic parallel between God and reality"

I have a better (or at least, more scientific) idea, lets' not make totally unjustified assumptions about the existence of God, the accuracy of the trinity and the idea that Western religion has got the will of God right, even if He does exist.

[the tree]

Also not confusing maths with something that is obliged to correlate with anything the "real world".

 

Despite getting it more than I did at the beginning of the thread, I still don't see this as useful.