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 October 28th, 2017, 12:32 AM #11 Senior Member   Joined: Nov 2015 From: USA Posts: 103 Thanks: 6 $\displaystyle R_4$ I have no idea what this is supposed to be. But seriously, I'm exploring possibilities that I think are awesome. Just looking at what everyone else has done and accepting that as the best or greatest or only valid way is just not my thing, and I believe that truth only comes from questioning these things, and indeed questioning everything, especially the most basic assumptions we don't even realize we make. Last edited by MystMage; October 28th, 2017 at 12:35 AM.
October 28th, 2017, 10:08 AM   #12
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 Originally Posted by MystMage $\displaystyle R_4$ I have no idea what this is supposed to be. But seriously, I'm exploring possibilities that I think are awesome. Just looking at what everyone else has done and accepting that as the best or greatest or only valid way is just not my thing, and I believe that truth only comes from questioning these things, and indeed questioning everything, especially the most basic assumptions we don't even realize we make.
Consider $\displaystyle R_2$ for a moment. This is the group with members $\displaystyle \{ R(0), R( \pi ) \}$ where R is a rotation about the origin and multiplication is adding rotations. This group rotates a point by either no angle or 180 degrees. That is to say, given P = (x, y), $\displaystyle R(0) P = (x, y)$ and $\displaystyle R( \pi )P = (-x, -y)$. Setting $\displaystyle x = a^2, ~ y = 0$; then the square root of P is either $\displaystyle R(0)(a, 0) = (a, 0)$ or $\displaystyle R( \pi ) (a, 0) = (-a, 0)$. So the square root of $\displaystyle a^2$ is either (a, 0) or (-a, 0) ie. $\displaystyle \sqrt{a^2} = \pm a$.

This is pounding on a nail using an atom bomb.

But it's easy to generalize. I'll leave the details to you but the elements of $\displaystyle R_4$ are $\displaystyle \left \{ R(0), ~ R \left ( \frac{ \pi }{2} \right ), ~ R ( \pi ), R \left ( \frac{3 \pi }{2} \right ) \right \}$. (We are rotating by 90 degrees with this one.) This group takes a point P from (x, 0) to (x, 0), (0, x), (-x, 0), (0, -x) respectively. This means you can represent your operation as taking $\displaystyle P = (a^2, 0)$ to the four equivalent solutions (a, 0), (0, a), (-a, 0), (0, -a). I'll let you extend this to (x, y) because I don't know what you would want to do with it..

Reviewing old assumptions is a good idea. But you can't erase the Historical development of a well known and useful function with impunity. You are looking at the square root function and questioning it. Fine. But you can't just throw the baby out with the bath water. Either add a new aspect to the square root function or create a new, more general square root. You can't just throw it away just because it doesn't do what you want it to. (The same goes for your comments about the complex number system.)
-Dan

Last edited by topsquark; October 28th, 2017 at 10:18 AM.

 October 28th, 2017, 05:35 PM #13 Senior Member   Joined: Sep 2016 From: USA Posts: 417 Thanks: 231 Math Focus: Dynamical systems, analytic function theory, numerics I am not sure you understand what the word "dimension" means. I'm also curious how you can justify your definition of the square root. If we would follow your procedure, we would not have $\sqrt{x}\cdot \sqrt{x} = x$ which I personally feel is a fairly important property of the square root. What with it being the only important property and all. Thanks from topsquark
October 28th, 2017, 10:31 PM   #14
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Quote:
 Originally Posted by SDK I am not sure you understand what the word "dimension" means. I'm also curious how you can justify your definition of the square root. If we would follow your procedure, we would not have $\sqrt{x}\cdot \sqrt{x} = x$ which I personally feel is a fairly important property of the square root. What with it being the only important property and all.
What makes you think that what I proposed doesn't have this property?

The magnitude of the results remains unchanged.

When using powers you are basically "squeezing" the input space into half the output space. When using powers, two different inputs yield the same output, while with the inverse, roots have one input and yield two outputs, basically stretching the input space into twice the output space.

Basically, with normal roots, root 25 = +-5, a pair of solutions. To get root 5 * root 5 to equal 25, you assume the same solution from both of the root functions. If you don't make this assumption, you get four possibilities all with the same magnitude of 25, but two of which will be negative and two will be positive.

With my idea, the above remains true for rooting positive numbers. Now root -25 = +-5, also a pair of solutions, but to get root -5 * root-5 = -5, instead you assume opposing solutions from the root functions, else, without making an assumption about what solutions to use you would then get the same four possibilities for +-25 as a result.

In the end it comes out the exact same for rooting positive numbers as with current methods, but mine simply recognizes the assumptions being made and uses that to get simpler results.

Either way, an assumption is made. This idea probably fits better in a system that handles polarity independent of magnitude because then it would match up with the rest of the system instead of being unusual, but it still works.

If you really want to generalize though, then remove the assumptions.

Root X * Root X = Y.
Each Root X gives two solutions,
each of the two solutions multiplied by the two solutions of the other Root X gives four combinations and thus four solutions for Y.

In my method, all four of those solutions for Y, when rooted, get two of the solutions from the left side of the equation, but the current method of roots, adding in complex numbers, means that two of the solutions for Y can't be rooted back to two of the solutions on the left, unbalancing the equation.

Basically, when assumptions are removed, my method still loops all four solutions for rooting Y back to two of the four possibilities from left side, while current methods only equate two solutions and redefines the other two.

My way
root X = +x & -x
root X * root X = Y
(-x & +x) * (-x & +x) = Y
+x * +x = Y[1]
+x * -x = Y[2]
-x * +x = Y[3]
-x * -x = Y[4]
Y[1] = Y[4]
Y[2] = Y[3]
Y[1] = Y[2]*-1

Current way,
root X = +x & -x
root X * root X = Y
(-x & +x) * (-x & +x) = Y
+x * +x = Y[1]
-x * -x = Y[4]
Y[1] = Y[4]
Y[2] = Y[3]
Y[1] = Y[2]*-1
These are different from my method,
+x * -x =/= Y[2]
-x * +x =/= Y[3]
Y[2] = xi
Y[3] = xi ;Of course, now i needs to be defined somewhere

Does this show better why I think my method is simpler and cleaner?

P.S. How does everyone always get the fancy notation? I presume that you guys don't have fancy keyboards with a bunch special keys, so is there a reference somewhere for whatever code or whatever you guys use to do it?

October 29th, 2017, 04:40 AM   #15
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 Originally Posted by MystMage How does everyone always get the fancy notation? I presume that you guys don't have fancy keyboards with a bunch special keys, so is there a reference somewhere for whatever code or whatever you guys use to do it?
It's called Latex. Enclose your code in $...$ tags. For a quick reference to some commonly used symbols, google "latex for beginners".

$\displaystyle \LaTeX: \frac{2}{3}$

To see the code I used to generate the above, quote this post.

October 29th, 2017, 05:15 AM   #16
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 Originally Posted by MystMage Yea, and they seem rather ridiculous. All they are is the realization that a two dimensional number will result in two dimensional possibilities, and then did a rather clunky solution. Why have this ordered pair when you can simply have a solution that result in positive or negative numbers? The imaginary numbers thing is a construct. It is people deciding on certain rules then discovering the emergent consequences. If you've read my stuff before, you'll notice one of the things I like trying to show, is circumstances where the emergent consequences, like imaginary numbers, is purely the result of the rules chosen and that other mathematical systems will have different emergent results. This seems to me like a valid alternative to imaginary numbers. Though honestly, I'm curious how people might try to prove or disprove it's validity/usefulness, or just what people think of the concept. Sometimes I wonder if no one wants to discuss anything that questions the basic assumptions of math, or even question if there are assumptions (which is an assumption itself).
What 'stuff' have you 'shown' exactly? What is a valid alternative to imaginary numbers? Prove or disprove the validity of what?!

I suppose you don't approve of any mathematics at all since, it too, is a 'construct'.

Also, why the attitude? You seem to have some sort of chip on your shoulder about 'truthfulness' of mathematics. As though every mathematician is a zombie, mindlessly following rules and obeying orders without ever questioning things.

Ridiculous, indeed.

October 29th, 2017, 05:36 AM   #17
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Quote:
 Originally Posted by MystMage The imaginary numbers thing is a construct. It is people deciding on certain rules then discovering the emergent consequences.
This is literally the definition of mathematics.

Standard mathematics uses the rules that have proved most useful.

October 29th, 2017, 06:04 AM   #18
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 Originally Posted by MystMage An analogy would be number bases. Are number bases somehow true and natural? Is base 10 somehow more true and natural than base 2? No. But a number in either base can easily be converted to the other. There is a significant difference though. Base 10 is a whole lot easier for us to handle and interpret than base 2, and different bases can be more difficult and imprecise in certain circumstances. Just like how 1/3 in base 10 is more easily represented accurately in the more complex form of a fraction in base ten, but swap to base 12 and a fraction is no longer required to get the same result with the same level of accuracy. The same applies to roots. Imaginary numbers is a more complicated way to show something, but that complication is a result of how we have defined our functions. Use a different definition and you can simplify the representation required. It might give us a whole new way to view the same thing we've been seeing all along, or it might do nought but simplify the set of symbols required to show it.
Mathematcians rarely use bases because they focus on the properties of numbers rather than the properties of representations of numbers. The differences between the bases are essentially trivial in most applications.

When people rail against a "more complicated way" of doing something, it's usually a tacit admission that they don't understand. In particular, they don't understand the power and benefits of what they rail against.

If one wishes to design a replacement for a tool, it is first necessary to understand what that tool does. Otherwise one has no hope of making sure that the replacement is even fit for purpose as a replacement, let alone better.

Of course, one can always design a new tool and then see what uses can be found for it, but then it isn't a replacement for anything else. It just is.

At a glance, one reason that your idea wouldn't replace complex numbers is the arbitrary nature of your pairing of signs (might one get different results for different pairings?). You talked about how the concept of number involves squeezing 2 dimensions into 1 and then proceeded to squeeze three dimensions into 2. By extension, your idea requires an arbitrarily large number of dimensions. And how will it handle $\sqrt[\pi]{x}$? You seem to need to introduce more and more arbitrary ideas.

The complex numbers, by contrast, introduce one single answer (not a technique) and the rest all falls into place more or less naturally.

 October 29th, 2017, 10:12 AM #19 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,344 Thanks: 2466 Math Focus: Mainly analysis and algebra I could also add that the imaginary unit $i$ is perhaps the best example of one of the most powerful techniques in mathematics. What happens is that we simply say the suppose there were a number that satisfied the equation $x^2 + 1=0$. Let us call such a number $i$. What can we then determine about $i$? And what can we do with it as a result? In other words, the whole process starts not by decreeing that $i$ is a new type of number with various properties, but rather by assuming that the answer we are looking for exists and then finding what properties emerge. It is a massively more beautiful approach than attempting to rewrite a function in order to force it to give the result that you wish it had given before.
October 29th, 2017, 04:15 PM   #20
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Quote:
 Originally Posted by v8archie This is literally the definition of mathematics. Standard mathematics uses the rules that have proved most useful.
True, but that doesn't mean those rules are the only one worthwhile ones, nor that we shouldn't question them and see if entirely alternative rules sets have value in some cases.

Right now, most of the time everyone talks about math like it is some inherent aspect of nature that can't be changed or done differently, as though even the order of operations is natural and that no other order of operations could ever be possible.

Actually, this forum is the only place where anyone I've interacted with even acknowledges the fact that math is a system constructed by people.

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