October 29th, 2017, 06:07 PM  #21  
Senior Member Joined: Nov 2015 From: USA Posts: 107 Thanks: 6  Quote:
People don't question the use of base 10, they never ask if a different base might be better for a particular situation. They simply take what they are used to and build on it and build on it, and yet never question what they are building on. Mathematicians might, but I never see any sign of it, and just about everyone I talk to (if they are even willing to talk math) treats math as a whole the same way. They take it as an absolute given, and never question it. Quote:
I like pondering these assumptions that everyone makes without thinking about it. Then I like making a discussion about it. Usually I can get more interesting discussions, but math tends to be either people telling how stupid I am or ignoring me. But writing it out and explaining it still helps me flesh out the ideas and examine them more closely, which helps me understand things better, even when the ideas I present are terrible or broken. Sometimes, the simple act of discussing something has value, even when the thing discussed falls flat. In this, I'm making my own entire system of math from scratch, and I think my roots idea will fit that system better than the standard (my system treats polarity separately from magnitude, where as normal math treats it inherently in a number), but it still helps to discuss it and examine it more closely, or have people point things out I missed. Such as the post after yours. [/QUOTE] Quote:
As all possibilities were shown, there are no different pairings to give different results. I listed them all and worked on all of them. Quote:
The number of dimensions being worked with doesn't matter. It always breaks down further and further till you have two numbers giving a third. The interaction between two dimensions, the result of which interacts with the next dimension. Always. Quote:
I don't know yet. Not like I waited till I had some ironclad impregnable idea to come discuss it. Honestly, I never found a good visualization for partial powers or roots. Partial multiplication or division has a simple solid concept for how it works that is beyond simply repeating addition, but while the techniques are there for moving powers beyond simply repeating multiplication operations, I just haven't visualized it yet, haven't seen a nonabstract way for how you get partial dimensions to interact, and that makes it harder to work with. (and no, complex numbers did not work as a visualization) It is something I've been working on and building though, but each method so far leaves multidimentional at odds with single dimensional. I.E. treating it as two axi that are not perpendicular and the area contained (similar to multiplication) being the answer tends to fail when axi lay on top of each other, where it should equal addition/subtraction, but doesn't. Quote:
There are many cases of scholars being proven wrong after long periods of time. It is rather best to consider that anything you develop will be superseded by something else in the future that is considered better for some reason. Or in my case, many things in the present.  
October 29th, 2017, 06:43 PM  #22  
Senior Member Joined: Nov 2015 From: USA Posts: 107 Thanks: 6  Quote:
Quote:
Note: the paper I found describing latex doesn't mention the plus/minus symbol. My attempt to produce it didn't work. What is the code snippet for it? That isn't to say i is bad, but rooting gives two solutions, not one, i comes from trying to enforce a single solution on something that always gives two solutions. The result of those efforts works, and well, but it is still a case of adding a rule and discovering all the emergent behavior of that added rule. I didn't force it to give an answer that it previously didn't, rather when the problem comes up of rooting negative numbers, I simply looked for a different solution. Normal math found one solution, I found a different solution. Each makes sense in their own way, and each has their own set of consequences. Last edited by MystMage; October 29th, 2017 at 06:45 PM.  
October 29th, 2017, 06:48 PM  #23 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,913 Thanks: 1113 Math Focus: Elementary mathematics and beyond 
$\pm$ is \pm. $\mp$ is \mp.

October 29th, 2017, 07:06 PM  #24  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,600 Thanks: 2588 Math Focus: Mainly analysis and algebra  Quote:
The "square root" function only returns a single value (the principal value) by convention. In the reals, it is always the positive solution. In the complex plane it is always the solution with the smallest positive argument. The reason for the convention is that it makes life needlessly complicated to have a single expression represent two (or more) values. When we need both, we just say $\pm \sqrt{a}$ (although in the complex plane, it's not always as simple as that, but you can still specify the solutions explicitly if necessary). Quote:
Mathematicians of the past weren't stupid. There are usually very good reasons why conventions have become conventional.  
October 29th, 2017, 07:24 PM  #25  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,600 Thanks: 2588 Math Focus: Mainly analysis and algebra  Quote:
Again, the order of operations is a conventional thing. It minimises the number of parenthesis we need  because parenthesis are only there to go against the usual order of operations.  
October 29th, 2017, 09:16 PM  #26 
Senior Member Joined: Nov 2015 From: USA Posts: 107 Thanks: 6 
Two posts referencing that I claimed something. I'm not sure why everyone seems to think I am claiming it is better overall. I have a bit of a distaste for complex numbers. The entire concept just rubs me wrong, but I don't disregard it or believe it it shouldn't be used. I was only pointing out what I believe to be a particular advantage of my idea over normal. I believe it is better in a particular way, but not necessarily better overall. I'm sorry if it sounded like that is what I meant. (I swear, communication is far more confusing than any form of math. 
October 29th, 2017, 10:48 PM  #27 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,600 Thanks: 2588 Math Focus: Mainly analysis and algebra 
Well, this post: Interesting thoughts on square roots did quite a good job of it.

August 6th, 2018, 12:56 AM  #28 
Senior Member Joined: Nov 2015 From: USA Posts: 107 Thanks: 6 
Been reading some of my old stuff, and there is one question on this thread that never got answered, why complex numbers? This thread went to interesting places, but I never got an answer on why complex numbers was chosen as the solution to square rooting negative numbers instead of something else. It was an explicit choice, so why that choice? 
August 6th, 2018, 02:14 AM  #29  
Senior Member Joined: Oct 2009 Posts: 733 Thanks: 247  Quote:
So why complex numbers? Because they work awesomely well. The unify a lot of math, they simplify a lot of math. They're really easy to work with, and give powerful results. Even if you only work with real numbers, the notion of complex numbers kind of is forced on you. Let me explain. First of all, the notion of an imaginary number first was considered when solving the cubic polynomial. It kind of naturally appears. Even if your cubic polynomial has three real roots, the method of finding them will pass through the complex domain. Complex numbers can be seen as numbers to encode very simple geometrical properties such as rotations, reflections and scalings. Like multiplying with a complex number is just a rotation/scaling, adding a complex number is a translation. So in this sense, complex numbers are just natural geometrical entities. Even if you only work with real numbers, the notion of a complex number is forced on you. Let's take a look at series to make this clearer. We have the series expansion $$\frac{1}{1x} = 1 +x + x^2 + x^3 + ...$$ But this doesn't hold everywhere. This series expansion only holds for $x<1$. It is clear geometrically where this condition comes from, since if you allow $x = 1$, you might allow the number $x=1$, and thus get an undefined (divide by 0) error in the left hand side. Now take the following series expansion $$\frac{1}{1+x^2} = 1  x^2 + x^4  x^6 + ...$$ Again, you can check when this relation holds. It turns out to hold for $x<1$ yet again. But now it is not clear why this is so at all. Indeed, the function on the left hand side is defined everywhere, so there is never a division by 0 problem. There is just no natural way geometrically to see where the condition $x<1$ comes from. The mystery gets bigger if we decide to expand this series around another point, say $x=1$, we get $$\frac{1}{1+x^2} = \frac{1}{2}  \frac{x1}{2} + \frac{(x1)^2}{4}  ... $$ But apparently this only holds for $x1 \leq \sqrt{2}$. How could we possibly interpret this result? Apparently, there is an obstruction to the convergence of this series which lies a distance of 1 from 0, and a distance of $\sqrt{2}$ from 1. With complex numbers, the reason is immediately clear: the obstruction is the number $\pm i$, which forces the denominator of $\frac{1}{1+x^2}$ to disappear. If you didn't know complex numbers, this would be totally inexplicably, there is no way you could possibly intuit this result. With complex numbers, such things become trivialities. Sure, complex numbers are an arbitrary human CHOICE. But one that works damn well. Last edited by skipjack; August 6th, 2018 at 02:49 AM.  
August 6th, 2018, 03:49 AM  #30 
Senior Member Joined: Nov 2015 From: USA Posts: 107 Thanks: 6 
I think that is one of the best answers I've ever gotten on a math question. Sure you don't want to help me craft my own version of math? 

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