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October 27th, 2017, 02:44 AM   #1
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Interesting thoughts on square roots

One thing I always figured was really strange was roots of negative numbers and that was handled in designing math language.

One thing I've figured out recently is that roots are set up to ignore a simple fact, they are two dimensional numbers being reduced to one dimension. Well two dimensional numbers have four quadrants they can be from, while a single dimensional number has two polarities (arguably the same thing). So it naturally follows, that trying to make a 2d number fit into a 1d number will be a problem sometimes. But what if we alter how we are looking at the number and root process, as right now, it seems like current math methodology only cares about magnitude and the one dimensional polarity.

If you take a graph, and plot a line Z (Z=X*Y), then you'll get a curve that is asymptotic to the X and Y axi. Rooting Z is nothing more than finding the spot on the line closest to the origin, which also happens to be the spot where X=Y.

But a 2d graph has four quadrants, ++, +-, -+, and --. Plot the same line, rotated, in each quadrant, and each line has the same magnitude. That means there are 3 other lines of the same magnitude.

Now with rooting, both the ++ and the -- quadrant give the same result, as they both have positive results from multiplying X and Y. So really, squaring Z actually has two solutions, the points on either the ++ or the -- line that are closest to the origin. Given that the results are identical, it goes unnoticed, and isn't even important, but, what about the other two quadrants?

This is where thinking about rooting in this way alters things. Basically, the other two quadrants have the same magnitude but are negative instead, thus their two solutions are -X*X. With the X and Y axi interchangeable, the results of both of these quadrants are also identical. With this we basically can see that rooting should work on all quadrants of a two dimensional number, which, functionally, corresponds to positive and negative results.

Thus, we take a closer look. +Z=X*X=X*X. -Z=-X*X=X*-X. Functionally, therefore, rooting needs to be defined with two cases despite four quadrants. The simple thing to realize is that the curves all map to Z or -Z. Well, that is quite simple, have the sign remain unchanged from rooting, thus root of -Z equals -X, and root of Z equals X.

So why did they not define rooting numbers in this way?
Anybody have thoughts on this? Good, bad, potentially valid/invalid for some reason?
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October 27th, 2017, 04:55 AM   #2
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Haven't you heard of complex numbers, which can be represented as ordered pairs of reals?
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October 27th, 2017, 10:36 PM   #3
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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).
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October 27th, 2017, 10:55 PM   #4
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Are you saying you would prefer an operation that is similar to the square root but will give four possible solutions to represent the four quadrants? ie. Label the square root of $\displaystyle x^2$ as something like Qx, where Q takes on the symbol +, -, and two others? As skipjack says, you can represent them as complex numbers: $\displaystyle \pm a \pm i b$ where the two $\displaystyle \pm$'s are independent. You can even generalize these to 3D...one example is the quaternion system.

Physics is rife with complex numbers. They are very useful so I don't see why you are disparaging them.

-Dan

Last edited by topsquark; October 27th, 2017 at 10:57 PM.
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October 27th, 2017, 11:30 PM   #5
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Quote:
Originally Posted by MystMage View Post
The imaginary numbers thing is a construct. It is people deciding on certain rules then discovering the emergent consequences.
Tell that to the quantum systems that have complex numbers, along with their algebraic rules, at their heart.

Mankind didn't invent those systems.
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October 28th, 2017, 12:41 AM   #6
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Another way to put it.

If rooting now means A where A equals X*Y when X and Y are equal.

Then you could alternatively define square root as the magnitude of A where A equals X*Y when X and Y are equal in magnitude.

Taking a closer look at this,
Where does the idea of square roots come from?

Well, before powers and roots, there is multiplication. Powers is just a shorthand for repeated multiplication. But to define a function that shorthands multiplication, you need the function to take inputs.

Of course, give a mouse a cookie, you will get asked for a glass of milk. In this case, making a function that shorthands repeated multiplication, you now need an inverse function that takes the output of the powers function and returns the inputs.

Sounds simple, but if you try to show this with a graph, you find that there are four quadrants, so the original powers/roots concepts are incomplete as four quadrants exist in 2d space, but powers only handles two of those quadrants, and yet the inverse is defined in a way that takes all four quadrants.

Obviously, this can be fixed by changing the powers function to handle all four quadrants, then the inverse of that can handle all four quadrants without adding a bunch of silliness like imaginary numbers.

The simplest way to do so, is to make it so that the power function doesn't assume polarity is the same among the terms. If the polarities are the same, then the resulting answer will be positive, and the inverse gives the two solutions of same magnitude in quadrants 1 and 3, while if the polarities are different, then the result is negative, with the inverse function give two solutions in quadrants 2 and 4.

This is a nicer and cleaner solution, and doesn't require imaginary numbers at all and can handle any input value without contradictions requiring patches like imaginary numbers (that I've discovered so far any way. Someone else might find one, who knows).

Of course, then the question comes up about powers/roots above 2. Well, that is rather easy, since it always boils down to pairs. This identifies half of the multidimensional space as positive and half as negative. Thus there are a number of solutions equal to 2 to the (power-1).

I.E. 3rd power/root
((++)+) = +
((++)-) = -
((-+)+) = -
((-+)-) = +
((+-)+) = -
((+-)-) = +
((--)+) = +
((--)-) = -

Magnitude will always be the same, the identification of positive or negative does nothing but identify which quadrants/octants/etc the solutions are in. which is rarely important, but easy to identify from the polarity alone if required.
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October 28th, 2017, 12:57 AM   #7
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Quote:
Originally Posted by romsek View Post
Tell that to the quantum systems that have complex numbers, along with their algebraic rules, at their heart.

Mankind didn't invent those systems.
We invented how we see and handle the numbers. We designed functions, the mathematical operations. We designed what they take as inputs and how they produce outputs.

These designs are no doubt based on our understanding and what is useful at the time.

Assuming that these systems are somehow inherent in the universe is rather close-minded.

A scientist is always willing to accept the possibility of being wrong. How many times have long standing theories been debunked. More than enough to not assume anything.

Math is just a constructed system designed to have the minimal amount of subjectivity, and while the math doesn't have our subjectivity, the system that math runs on is limited by our subjectivity.

As I just posted, a description of a mathematical function that should be just as valid as current powers/roots and yet has different emergent consequences.

Emergent consequences is something you forget. They are inherent results of the system's design, but that doesn't make them somehow equal with some universal truth. They are true within the system, and mutiple systems might develop results that can easily be mapped to each other, despite a wide difference in how each system comes to those results, to say nothing of the representation of them.

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.
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October 28th, 2017, 12:59 AM   #8
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Quote:
Originally Posted by Eseh Stan View Post
The square root of negative one (-1) is equal to one (1);Proven !
Proven for given system of defined functions. Change the functions and the result may or may not be any different, but your proof will need to be re-proven for the new functions.
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October 28th, 2017, 01:06 AM   #9
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@MystMage: I don't know my History but I know that square roots go all the way back to the Greeks, maybe Phoenicians. It would make much more sense to say that you are inventing a new function instead of changing a function that has existed for more than two millennia and has done what it does extremely well.

Here's something more constructive for you. Consider the symmetry group $\displaystyle R_4$. That sounds similar to what you are reaching for.

-Dan
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October 28th, 2017, 01:31 AM   #10
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Here is a couple videos about creating math,

The first one is about an assumption that mathematicians didn't realize they were making until recently, then when on to describe a different definition of distance. So how is this some sort of of universal only one valid definition, when clearly people have being using math for a long time using a different definition. Therefore, redefining things isn't unnatural, just different from what we are used to.



This second one reveals holes in the altered functionality, but doesn't explore alternatives at all, for example, when he adds a number he adds it to the right and he assigns the negative sign towards the right instead of the left, etc, but in the end, he solves an equation according to rules different than what we use. Which means it is possible.
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