My Math Forum Interesting thoughts on square roots

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 October 27th, 2017, 03:55 AM #2 Global Moderator   Joined: Dec 2006 Posts: 20,469 Thanks: 2038 Haven't you heard of complex numbers, which can be represented as ordered pairs of reals? Thanks from topsquark
 October 27th, 2017, 09:36 PM #3 Senior Member   Joined: Nov 2015 From: USA Posts: 107 Thanks: 6 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).
 October 27th, 2017, 09:55 PM #4 Math Team     Joined: May 2013 From: The Astral plane Posts: 2,138 Thanks: 872 Math Focus: Wibbly wobbly timey-wimey stuff. 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 09:57 PM.
October 27th, 2017, 10:30 PM   #5
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 Originally Posted by MystMage 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.

October 27th, 2017, 11:57 PM   #7
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 Originally Posted by romsek 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.

October 27th, 2017, 11:59 PM   #8
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 Originally Posted by Eseh Stan 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.

 October 28th, 2017, 12:06 AM #9 Math Team     Joined: May 2013 From: The Astral plane Posts: 2,138 Thanks: 872 Math Focus: Wibbly wobbly timey-wimey stuff. @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
 October 28th, 2017, 12:31 AM #10 Senior Member   Joined: Nov 2015 From: USA Posts: 107 Thanks: 6 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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