The answers to 1^(1/2) and 1/0

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exfret
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Re: The answers to 1^(1/2) and 1/0

Post by exfret »

robly18 wrote:Man, you sure have great skill at making walls of text my brain hurts when reading.
Why is your brain being mean to my poor posts? :cry:

robly18 wrote:As for the not a number thing, here's why it isn't a number:
(You don't have to read this part. Summary: Great reason to say 1/0 is impossible, even though your reason is wrong). Ah, now I see. So you're nailing it down for not following the basic axioms of mathematics. Just saying it wasn't a number wasn't a very precise definition (*cough* Andy *cough*) or very valid argument, but saying it can't be because it violates basic properties of arithmetic seems like a more solid argument. It's just that words like 'number' can be varied in their meaning, and it isn't very obvious or provable (at least I haven't seen it proven) what 'not being a number' would even mean, much less why it would imply the impossibility of taking an operation. Besides, i technically isn't a 'real' number, and you could define it as 'not a number' at all, so that would mean you could never take sqrt(-1), but i still follows basic laws of arithmetic. Saying an operation is impossible due to its violation of the axioms that are the foundations of mathematics makes much more sense.

robly18 wrote:The axioms apply for everything. They're the basis of mathematics. They are literally what defines our system of counting, adding, subtracting... If something doesn't obey these axioms, then it isn't really a thing.
Once again, may I say you offer a very clear explanation, but as I hope to do the same, you should hopefully see that the logic of these axioms needs to be revised when discussing ∞, and that ∞ actually follows these axioms anyways.

robly18 wrote:As for the "how can it not be a number thing" let me make an analogy.

Say, n = n + 1. Is this possible?
As my good old friend Jim would say, this would mean 0 = 1. It's not possible as you can see.
Actually, n=n+1->0=1 is not exactly correct. You see, when you write the whole thing out, something pops up:

  n=n+1
-n  -n
n-n=n-n+1

Now, you might say: "Hey! n-n=0, right?" Well, that's right, but remember, we're taking an operation here, and operations can return multiple values. This may hurt your head, but it happens a lot when it comes to ∞, and it happens all the time with square roots, too, so it isn't an entirely new concept. Now, remember, since n=n+1, we can plug in n+1 in as n, and we get this:

n-n=n-(n+1)+1
0=n-n-1+1
0=0+0
0=0

Now, before you start telling me how this violates the basic laws of mathematics, I never said n-n wasn't equal to 0. It just could be equal to other things, too. This doesn't mean 0 isn't one of its 'solutions'. You have to remember that some operations have multiple values, but only one of them is the 'right' one (remember that discussion about 0/0?), and sometimes we can even deduce which one is right, but you can't just pick out one of those values and plug it in, because it'll get contradictory things, like you've shown. In this case, n-n actually can't be 0 for both sides of the equation. It's kind of like derivatives. A derivative is actually just 0/0, but by doing some cool tricks, you can find out which of the values returned by the operation 0/0 was right. n-n is the same way: it's indeterminate, and that means you can't just assume one of it's values is correct.

robly18 wrote:Jim isn't a real person. Anymore, at least.
:(

robly18 wrote:Anyway, as for the n = 1/0, here's why it's wrong:
Multiply both sides by zero
0n = 1
Use axiom proofs to figure out this:
0n = 0
And then ,you get 0 = 1.
Remember, 1/0 can be multiple values, and 0*1/0 can be multiple values, too. Think of it this way: You take both square roots of 1, and then you set it equal to both the square roots of 1. -1 is a square root of one, right? So is 1. So shouldn't that mean that -1=1 because they both equal 1^(1/2)? The thing that's wrong here is that you're picking out one value equal to 1^(1/2) and setting it equal to another value equal to 1^(1/2) and saying they're equal, which they aren't. The exact same thing is happening here. Yes, 1/0*0=1, yes 1/0*0=0, but both these equations aren't completely correct. A better answer would be 1/0*0={0,1}. You can then choose an answer based on context (that is, if you have context). Even so, this wouldn't be the best answer. In fact, the best answer is indeterminate, because 1/0*0=0*1/0=0/0*1=0/0, so 1/0*0 is actually 0/0. You can argue 0/0 isn't a number either, because n/n=1, but because it has a 0 in the numerator, it's the same as 0*1/0, and therefore should be 0 because 0*n=0, but we call it indeterminate, NOT undefined, and it's because it actually equals both/either of these numbers, and we should at least give some such name to 1/0, be it :H (which could even stand for a set of numbers if 1/0 in fact returns more than one value) or whatever.

robly18 wrote:Therefore, the existence of a number that is equal to one over zero is impossible, just like the existence of a number equal to itself plus one. They do not obey the axioms and cause inconsistencies.
As I have shown, there may not be a number equal to 1/0, but that doesn't mean there are no numbers equal to 1/0. Also, your disproof of a number equal to itself plus one overlooked the possibility that n-n could have more than one possible value. Finally, I would like to introduce the notion that the axioms aren't even extendable to this realm. The notion that 0 of anything is 0 creates an exception. Just make up a number, :H, where 0 * :H = 1, and there's no reason why this number can't be. It would be like saying there's no number that is the sqrt(-1), because it just can't be. Well, you know what we did? We created a number that was the sqrt(-1), and the applications of the number i have not ceased to amaze us. You see, this type of thinking limits us, and as I have shown, limits are deceiving. :P Axioms are more of a general rule. You can always make up your own exceptions. The logic that created them always has loopholes. Like the 0*n=0 logic forgets that infinite is so large that one thing out of it is literally nothing compared to infinite, so therefore, that one thing would be 0 of infinite, which although it 'violates' a 'fundamental axiom of mathematics,' has a logic of its own. And logic is what mathematics was based on, not some set of axioms created by mathematicians centuries ago.

robly18 wrote:It may sound like a case of "it doesn't obey what we know, so sweep it under the rug", but that's the thing.
You know what we've been doing all these years? Sweeping it under the rug, and now it's time for it to come out.

robly18 wrote:Their existence would make mathematics inconsistent.
So the solutions to inconsistency is to create inconsistency? Their 'nonexistence' makes math inconsistent. Also math is never completely consistent, because it's about logic, not consistence.

robly18 wrote:We call the undefined because, with our set of axioms, they cannot be defined.
But with logic, it can.
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exfret
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Re: The answers to 1^(1/2) and 1/0

Post by exfret »

Wow, exactly two weeks. That's a long time. I also want to ask what to do about the ^(1/2) thing.
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exfret
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Re: The answers to 1^(1/2) and 1/0

Post by exfret »

And yet another 4 days pass by.
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exfret
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Re: The answers to 1^(1/2) and 1/0

Post by exfret »

And yeti another week.
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exfret
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Re: The answers to 1^(1/2) and 1/0

Post by exfret »

And two days pass by. (Is anyone even noticing this?)
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exfret
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Re: The answers to 1^(1/2) and 1/0

Post by exfret »

Now 1 week and 2 days. I don't really do this w/ my other topics, but this is something that's important to me, so I'll just keep on resurfacing it until someone finds it when they happen to be in the mood of replying. I don't want to pressure you to reply to the walls of text in this topic, but if you have the time, it would be nice to get a reply to something (you don't have to reply to everything). My main points are:

-About the 1/0 stuff:
    -Is it correct? If so...
        -Is it meaningful? Basically, does it have anything to give to the field of mathematics? If so...
            -Should I try to publish it somehow? If so...
                -How would I do that?

-And about raising to the power of 1/2:
    -Does it make a difference whether you learn x^(1/2)=sqrt(x) or x^(1/2)=+/-sqrt(x)? If so...
        -Is that difference significant enough for me to try to bring attention to the fact that people are learning the wrong thing? If so...
            -How would I bring attention to this?

So those are the questions I care about. Answer them as you wish. You don't have to answer the conditions before answering indented questions (i.e. you could answer how I would publish my proof without answering whether you think it's correct, meaningful, or whether I should publish it).
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robly18
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Re: The answers to 1^(1/2) and 1/0

Post by robly18 »

Look, the reason no one has been answering is because, well, there isn't much to answer to.
For the 1/0 stuff, I'd say it is not correct due to the sheer fact that, unless you're working with hyperreals, there is no number that does not obey the good old axiom of 0x = 0.
None.
So you're either redefining axioms in a dumb way (exclusion of one, and one sole number is not an okay thing to do in my book) or you're just plain breaking the rules.

As for the square roots thing, it all comes down to teaching and general consensus.
When you're teaching kids about square roots, the concept is confusing enough for most. I've seen kids do square roots by dividing the number by itself, thinking that if a square is multiplying by itself, the opposite would be division.
So, teachers at least START by asking and giving only the positive result.
However, as you move up in school, you eventually get to a point where, if given a square root, unless implicit like in the case of distances for example, you NEED to give the positive AND negative result.
Besides, kids can be one of two things:
Gullible enough to believe in this, in which case they don't even bother to learn and simply take mathematics as "something they'll never use in their life" and simply memorize formulae they don't even bother reading
Or they can be like us, and actually think about it. Eventually, they'll decide to ask their teacher (or the internet) about this, and probably get a response of the likes of "it's simplified for nth graders" or "it's general consensus that the square root of a number is, by default, the positive result.
My point is, don't fret about it. (pun intended) Kids, when it comes to math, are either gullible or curious. If a kid doesn't ever question square roots until they are taught the subject, they weren't going to be using that knowledge in the time between learning square roots and their dual result property anyway.
Convincing people that 0.9999... = 1 since 2012
exfret
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Re: The answers to 1^(1/2) and 1/0

Post by exfret »

robly18 wrote:Look, the reason no one has been answering is because, well, there isn't much to answer to.
For the 1/0 stuff, I'd say it is not correct due to the sheer fact that, unless you're working with hyperreals, there is no number that does not obey the good old axiom of 0x = 0.
None.
So you're either redefining axioms in a dumb way (exclusion of one, and one sole number is not an okay thing to do in my book) or you're just plain breaking the rules.
robly18 wrote:there is no number that does not obey the good old axiom of 0x = 0
First of all, I never said 0 * :H != 0. I said that it was indeterminate (which means one of its 'solutions' is 0, so it does in fact satisfy that axiom). In fact, there's a way I can show ∞ itself doesn't just equal 0 when you multiply 0 by it. ∞ is the number that ( ∞ + 1 ) = ∞, so ( ∞ + 1 ) / ∞ = 1, meaning ( ∞ / ∞ ) + ( 1 / ∞ ) = 1, and therefore 1 + ( 1 / ∞ ) = 1, leading to the result that ( 1 / ∞ ) = 0. Then, multiply infinity on both sides and you clearly get the result that 0 * ∞ = 1. Keep in mind that 0 * ∞ = {x|x is real}, and 1 satisfies this because 1 is a real number, but this doesn't mean 0 won't satisfy it as well (meaning 0 * ∞ = 0). (I'm just saying real here to be cautious, and I don't want to have to defend why it could be equal to imaginaries as well.) Hey, that also means 1 / 0 = ∞. Yay, now I don't have to use :H anymore! (Of course, I still have to prove the non-controversial-ness of that division by zero step, but a foothold is a foothold nonetheless). A less number-y way to think about it is that if you add 1 to infinity (the alt+5 was getting too annoying), it doesn't get you any change, so 1 is zero of infinity, because it causes zero relative change, just like 2 is 1/2 of 4, because it gets you 50% (or 1/2) change in the quantity (e.g. When you add 2 to 4, you get six, which is 50% more than four, and 50% = 1/2). Finally, "the rules" you speak of aren't axioms, they're logic, and you can only disprove my work through the laws of logic, not the axioms that are derived from them.


robly18 wrote:(exclusion of one, and one sole number is not an okay thing to do in my book)
All numbers are either non-real, nonnegative, or nonpositive. Oops, except for zero. BOOM! Exception of one sole number!!!! ;) It's okay to exclude a number. That's what you do in solving. There are reasons for exclusion. And, besides, the real exclusion would be us excluding 1/0 from 'being a number'. Of course it's a number! It's a mathematical object!!!


robly18 wrote:As for the square roots thing, it all comes down to teaching and general consensus.
When you're teaching kids about square roots, the concept is confusing enough for most. I've seen kids do square roots by dividing the number by itself, thinking that if a square is multiplying by itself, the opposite would be division.
So, teachers at least START by asking and giving only the positive result.
However, as you move up in school, you eventually get to a point where, if given a square root, unless implicit like in the case of distances for example, you NEED to give the positive AND negative result.
Besides, kids can be one of two things:
Gullible enough to believe in this, in which case they don't even bother to learn and simply take mathematics as "something they'll never use in their life" and simply memorize formulae they don't even bother reading
Or they can be like us, and actually think about it. Eventually, they'll decide to ask their teacher (or the internet) about this, and probably get a response of the likes of "it's simplified for nth graders" or "it's general consensus that the square root of a number is, by default, the positive result.
My point is, don't fret about it. (pun intended) Kids, when it comes to math, are either gullible or curious. If a kid doesn't ever question square roots until they are taught the subject, they weren't going to be using that knowledge in the time between learning square roots and their dual result property anyway.
Good thinking. My only concern was that this had grown to be the general consensus in mathematics. I'm also a little concerned that my Algebra II w/ Trig. teacher actually believed in this stuff, even after I presented her with a ~5 page proof, using the reason that the power of a power property only holds true for integers (e.g. (x^2)^(1/2) != x), but I guess I'll just have to let that pass.



Thank you for replying. :D I'm sorry if I post too many things for you guys. I'll try not to post just to see my name all over the fora or because I'm bored and I just want something to think about besides work.
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testtubegames
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Re: The answers to 1^(1/2) and 1/0

Post by testtubegames »

robly18 wrote:So you're either redefining axioms... or you're just plain breaking the rules.
This.

In the standard mathematical framework, infinity is not a number. It's a more abstract concept than that... and is nearly always used in conjunction with limits. So the expression infinity*0 has no meaning as it sits. Since infinity is not a number. And multiplication, division, etc, are axiomatic frameworks designed for numbers.

*But*, it seems as if you want to say... hey, what if I defined a new term... the 'exfretNumber'. And exfretNumbers are all the regular old 'numbers' *plus* exfretInfinity (which is kinda like infinity, but it's got the properties of an exfretNumber).

...k.

And I want to define new operations, like 'exfretDivision'. Which is kinda like division, but it works for all exfretNumbers, not just for numbers.

...k.

And I'll say that exfretInfinity/exfretInfinity = 1 in this new exfretDivision.

...alright.

Now these are all things you can choose to define. But we need to remember, this isn't division and numbers. This is exfretDivision and exfretNumbers. So what you're getting at is that 1 exfretDivided by 0 is exfretInfinity. Which is a result drawn from the definitions you've chosen. But not terribly applicable when we're talking about division, numbers, and infinity.

And in math, it's wonderful to devise new systems and new axioms and play around with them. (And ask questions like: is exfretMath self consistent? Or are there any axioms that conflict with each other?) Sometimes it even leads to new discoveries. (Hey, what if parallel lines could cross? Boom: non-euclidean geometry is born!) But at the end of the day, we want to be clear whenever we step outside the accepted axioms of math. Your argument is based in exfretian Arithmetic. And, hey, maybe such a system could be really cool. Maybe it would even have some uses! (Like non-euclidean geometry, say) But remember, you aren't showing that 1/0 = infinity. You're showing that 1 exfretDivided by 0 is exfretInfinity.
exfret
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Re: The answers to 1^(1/2) and 1/0

Post by exfret »

testtubegames wrote:
robly18 wrote:So you're either redefining axioms... or you're just plain breaking the rules.
This.

In the standard mathematical framework, infinity is not a number. It's a more abstract concept than that... and is nearly always used in conjunction with limits. So the expression infinity*0 has no meaning as it sits. Since infinity is not a number. And multiplication, division, etc, are axiomatic frameworks designed for numbers.

*But*, it seems as if you want to say... hey, what if I defined a new term... the 'exfretNumber'. And exfretNumbers are all the regular old 'numbers' *plus* exfretInfinity (which is kinda like infinity, but it's got the properties of an exfretNumber).

...k.

And I want to define new operations, like 'exfretDivision'. Which is kinda like division, but it works for all exfretNumbers, not just for numbers.

...k.

And I'll say that exfretInfinity/exfretInfinity = 1 in this new exfretDivision.

...alright.

Now these are all things you can choose to define. But we need to remember, this isn't division and numbers. This is exfretDivision and exfretNumbers. So what you're getting at is that 1 exfretDivided by 0 is exfretInfinity. Which is a result drawn from the definitions you've chosen. But not terribly applicable when we're talking about division, numbers, and infinity.

And in math, it's wonderful to devise new systems and new axioms and play around with them. (And ask questions like: is exfretMath self consistent? Or are there any axioms that conflict with each other?) Sometimes it even leads to new discoveries. (Hey, what if parallel lines could cross? Boom: non-euclidean geometry is born!) But at the end of the day, we want to be clear whenever we step outside the accepted axioms of math. Your argument is based in exfretian Arithmetic. And, hey, maybe such a system could be really cool. Maybe it would even have some uses! (Like non-euclidean geometry, say) But remember, you aren't showing that 1/0 = infinity. You're showing that 1 exfretDivided by 0 is exfretInfinity.
I see your point, and exfretinfinity was what I was using the :H symbol for before, but I'm still using the same exact operations. If you really have to say that infinity isn't a number, then fine, I'll define the number :H, such that :H + 1 = :H, then :H * 0 = indeterminate, which as I state once more, includes zero and one, meaning if violates neither the multiplicative/division inverse property thingy nor the 0 multiplied by any number property. If you must limit division to only the numbers we are familiar with, then fine, call that "restrictedivision," but my division is better, because it has no restrictions. I don't want to rewrite my whole proof again to support my claims, so it would be better if you could just point out the specific part of my logic that isn't correct, making so I'd have to use "exfretdivision."
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