My Math Forum Interesting thoughts on square roots

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October 29th, 2017, 06:07 PM   #21
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Quote:
 Originally Posted by v8archie 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.
It was an analogy. To show that a system can be used and even become the standard and yet have alternatives that are equally good in general and have advantages in some cases while having disadvantages in other cases.

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:
 Originally Posted by v8archie 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.
A major part of me making discussions is that it provides a new viewpoint that questions the very things people don't even realize they've been ignoring. You can take most any person and explain number bases, and they'll get the idea, but if you ask them if 10 is a good base, they will not have even considered that it could be a different base, before you asked them even when they understand the concept, it never occurs to them to question it.

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.

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Quote:
 Originally Posted by v8archie 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?).
How is this arbitrary? I took the full set of possible solutions, and those solutions could clearly be grouped into those with a positive result (and thus matching normal math) and those with a negative result (which normal math assigns to i). There was nothing arbitrary about it. Not that I can see anyway.

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:
 Originally Posted by v8archie 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.
You probably missed the shift in topic. I was talking how in 2d you have four quadrants, but the results only fill two. I then also had a side note about how it works not only for square roots, but also for cube roots. The concept of of the total input space being squeezed into half that space for the output applies to more than just squares, and that was what I was trying to show.

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:
 Originally Posted by v8archie And how will it handle $\sqrt[\pi]{x}$? You seem to need to introduce more and more arbitrary ideas.
Good question!

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 non-abstract 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 multi-dimentional 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:
 Originally Posted by v8archie The complex numbers, by contrast, introduce one single answer (not a technique) and the rest all falls into place more or less naturally.
Complex numbers work. I still think they are a technique, but I consider all of math as a technique. But it still isn't inherently the best or only answer, just the best and only answer we currently know of.

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
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Quote:
 Originally Posted by v8archie I could also add that the imaginary unit $i$ is perhaps the best example of one of the most powerful techniques in mathematics.
Quote:
 Originally Posted by v8archie 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?
This is not a definition of i I've ever seen. It poses an interesting element. Breaking it down from this side, you get $x=\sqrt{-1}$ for which there are two solutions, meaning x = two things at once. In standard math, this pair of things is labeled as $x=i$, my idea is to simply label it as $x=+-1$.

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.

Quote:
 Originally Posted by v8archie 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.
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: 7,725 Thanks: 985 Math Focus: Elementary mathematics and beyond $\pm$ is \pm. $\mp$ is \mp.
October 29th, 2017, 07:06 PM   #24
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Quote:
 Originally Posted by MystMage Breaking it down from this side, you get $x=\sqrt{-1}$ for which there are two solutions, meaning x = two things at once. In standard math, this pair of things is labeled as $x=i$, my idea is to simply label it as $x=+-1$... $i$ comes from trying to enforce a single solution on something that always gives two solutions.
One of the emergent properties of $i$ is that it makes no difference which of the two solutions you pick for $i$. One is $i$ and the other is $-i$. Again, your attack on the conventional solution is based on a lack of understanding.

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:
 I simply looked for a different solution.
Nowt wrong with that, but see my earlier comments about claiming that yours is more right or better than an existing solution that you don't fully understand.

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
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Quote:
 Originally Posted by MystMage 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.
Of course it doesn't. Look at non-Euclidean geometry for a perfect example. There are also different number systems in use in various parts of mathematics. These systems live happily alongside each other, each being good for their own purposes. But you can't just mess with the rules and claim that your new system is better. You have to show that it is better for some application before it is likely to be taken seriously.

Quote:
 Originally Posted by MystMage as though even the order of operations is natural and that no other order of operations could ever be possible.
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 Member   Joined: Nov 2015 From: USA Posts: 96 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: 7,137 Thanks: 2381 Math Focus: Mainly analysis and algebra Well, this post: Interesting thoughts on square roots did quite a good job of it.

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