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September 18th, 2014, 05:54 AM   #21
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Quote:
Originally Posted by v8archie View Post
I'd actuallly be interested in finding out about respectable theories of infinitesimals.
Maybe you should start a new thread, lest it be lost in the detritus here.
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September 18th, 2014, 07:23 AM   #22
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I agree with v8archie. You are exhibiting a fundamental misunderstanding of the concept of infinity. You are trying to treat infinite and infinitesimal numbers like real numbers. There is no difference between 0, +0, and -0 for instance. Surreal numbers can not be expressed in the traditional way (don't ask me what the proper way of expressing them is, because I know very little about surreal numbers other than that they are a theoretical possibility).

You also show a fundamental misunderstanding of limits. Your statement about the limit of zero being something other than zero is meaningless. If you want to truly understand limits, I suggest you look up the Cauchy convergence test. You might want to look up the Cauchy test for convergent sequences first, since it uses fewer variables and is somewhat easier to understand; it's a good intermediate step towards understanding limits of functions.
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September 18th, 2014, 09:17 AM   #23
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Let me take this in a different direction. There are lots of systems other than the real numbers out there. For example, there is the free monoid on, say, ASCII characters. In that system "0" is different from "+0" since the latter is two letters and the former is only one, and "-0" is different from both since the first letters differ.

Or take sequences of real numbers relative to some fixed nonprincipal ultrafilter U. You could define $\alpha=(1,1/2,1/3,1/4,\ldots)$ and (as usual) identify the real number $x$ with the sequence $(x,x,x,\ldots)$. Then you have $-\alpha<0<\alpha,$ just as you like (though you can't use the symbol +0 for what I have called $\alpha$ since +0 has its own meaning in that system), regardless of your choice of U. Note that for any real number $\varepsilon>0$ you have $0<\alpha<\varepsilon$ so this $\alpha$ is an actual infinitesimal. But it doesn't make sense to ask for its decimal expansion -- it's a hyperreal, not a real. (You could ask for the decimal expansion of its shadow, which is the real number 0 = 0.000000000....)
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September 20th, 2014, 12:40 AM   #24
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Quote:
Originally Posted by CRGreathouse View Post
I suspect that you could not write a formal proof, though, so the process is not as easy as I make it sound.
I think I have never written a formal proof. I have proofs, what makes them formal I don't know, I could call them just proofs, normal proofs. I have also what I could call "prophetic proofs".
My method has not been mainstream. I have my own way. I need to believe
in what I do, because I don't have always proofs at my disposal.
I have noticed that it is possible to arrive at right answers in two way:
even a wrong method can lead to a correct answer, as well as a right method
of course.
All my "prophetic proofs" have been systematically denied in all forums
where I have written them. When I drop them, and when I choose to write
instead my "normal" proofs which I consider as facts, these are also the same way systematically denied. Might it be better not to write any proofs at all?
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September 20th, 2014, 03:49 AM   #25
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If you want to have a proper discussion about your mathematical ideas, you need proofs to show why your ideas aren't just wishful thinking and hand-waving that are based on a misunderstanding of theories that are well-known, rigorouslesslly tested, formally proved and completely understood by the mathematical community.

I don't recall reading any proof of yours in this thread.
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September 20th, 2014, 04:48 AM   #26
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Another Dimension

What lies beyond infinity is another dimension.

1/3 is done by infinite binary divisions in 1 dimensional space (a number line).



But can easily be accomplished in 2 dimensions.



Angle trisection can be done by infinite angle bisection.

But can easily be done in 3 dimensional space.



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September 20th, 2014, 08:26 AM   #27
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Quote:
Originally Posted by long_quach View Post
What lies beyond infinity is another dimension.
1/3 is done by infinite binary divisions in 1 dimensional space (a number line).
But can easily be accomplished in 2 dimensions.
Angle trisection can be done by infinite angle bisection.
But can easily be done in 3 dimensional space.
I think you are demonstrating that 1/3=0.3333............first by infinite steps of binary divisions, something like a never ending long division. Then you show some plots which show how to divide a line and an angle into three parts. The last one of your pictures is low quality and it is hard to see exactly what is going on there.

I can also tell about my approach to how to divide by three, and what happens if the division is continued for ever. It is an example which is posted on sciforums, I avoid writing all again so I will write the link here

It is also an example of how I try to prove what I say so that my ideas aren't just wishful thinking and hand-waving. I am writing that dividing
by three leaves a nonzero infinitesimal, but usually it is "swept under the rug" because it is so small that no-one knows or sees it, but dividing for example by 2 does not leave an infinitesimal. The decimal number
system allows an infinitely small decimal number, which is nonzero.
The infinitesimal x has no exact value, therefore it is not equal to 0,
it has a limit which is 0. Limit of x is 0. The mathematical community
has not accepted my ideas because they think that the infinitesimals
are always equal to zero, therefore they don't exist.
My ideas are not accepted, but I don't see what is wrong with my
ideas. I might as well stop writing on these forums if I realized what is wrong. This is after all just my hobby. Life would be much easier for
me if I did not need face the flamewars into which my ideas lead. I am not trying to teach anyone to accept my ideas, I am not a teacher, if my ideas are not accepted, it does not matter.

Last edited by TwoTwo; September 20th, 2014 at 08:33 AM.
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September 20th, 2014, 09:37 AM   #28
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The most important thing I can say about that post is that 0.333333... (and the whole of the decimal system) is no more than a representation of numbers.

For your argument to hold (that division by three leaves some some small remainder) you will have to explain $\frac{1}{3}$. Where is the remainder now? How is this different to $\frac{1}{2}$.

I rather suspect that the only difference is that 3 is coprime to 10, which means that your infinitesimals depend entirely on the arbitrary choice of base. If we chose to use base 3, we'd have $\frac12 = 0.111\cdots$. Does that mean that division by 2 both leaves an infinitesimal and that it doesn't?

Your idea seems to be either inconsistent or to introduce a quantity that can be made to disappear whenever we want by changing our notation. And since changing the notation doesn't change the object, I'm forced to conclude that it never existed anyway.

As a mathematician, I have always disliked using decimals, because of the loss of accuracy that comes from using finite representations. I always prefer to use fractions and symbols that represent irrationals such as $\pi$ and $\mathbb e$.
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September 20th, 2014, 10:44 AM   #29
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Quote:
Originally Posted by v8archie View Post
The most important thing I can say about that post is that 0.333333... (and the whole of the decimal system) is no more than a representation of numbers.
True.

Quote:
Originally Posted by v8archie View Post
For your argument to hold (that division by three leaves some some small remainder) you will have to explain $\frac{1}{3}$. Where is the remainder now?
There would be no remainder if $\frac{1}{3}$ had no decimal representation. Maybe it is wrong to ask what is the decimal representation of
$\frac{1}{3}$, because the question assumes a priori that $\frac{1}{3}$
has a decimal representation. The right question is "Does $\frac{1}{3}$ have a decimal representation?"

Quote:
Originally Posted by v8archie View Post
How is this different to $\frac{1}{2}$.
$\frac{1}{2}$ is not different than $\frac{1}{3}$ if they both had no decimal representation.


Quote:
Originally Posted by v8archie View Post
I rather suspect that the only difference is that 3 is coprime to 10, which means that your infinitesimals depend entirely on the arbitrary choice of base. If we chose to use base 3, we'd have $\frac12 = 0.111\cdots$. Does that mean that division by 2 both leaves an infinitesimal and that it doesn't?


I did not think what happens to infinitesimals if we chose to use a different base. It is possible that they disappear, their existence depends on the base.

The base-10 system that we use, seems to allow the existence of infinitely
small decimal numbers. They seem to exist only because of the decimal system that we have chosen to use, the base-10 with its decimals.

Quote:
Originally Posted by v8archie View Post
If we chose to use base 3, we'd have $\frac12 = 0.111\cdots$. Does that mean that division by 2 both leaves an infinitesimal and that it doesn't?
Again, the problem can be avoided by asking only right questions.
Don't ask what is the decimal represention of $\frac{1}{2}$, don't
ask what is the decimal represention of $\frac{1}{2}$ in base 3.
Ask only "does $\frac{1}{2}$ have a decimal representation?"


Quote:
Originally Posted by v8archie View Post
Your idea seems to be either inconsistent or to introduce a quantity that can be made to disappear whenever we want by changing our notation. And since changing the notation doesn't change the object, I'm forced to conclude that it never existed anyway.
Yes, it is possible that they don't exist at all as real objects, it would be very hard to think that there would be an elementary particle whose size
was infinitesimally small. Unlikely is also that matter is infinitely divisible. It is unthinkable that matter disappears if it is divided into infinitely small pieces. Therefore there must be a smallest size of everything, even though the smallest size was not infinitesimally small.The number system that we have chosen to use just seems to allow infinitesimals. They
are probably just products of our choice of number system.
If we used only fractions like $\frac{1}{3}$ we would not need to deal
with infinitesimals.

Quote:
Originally Posted by v8archie View Post
As a mathematician, I have always disliked using decimals, because of the loss of accuracy that comes from using finite representations. I always prefer to use fractions and symbols that represent irrationals such as $\pi$ and $\mathbb e$.
I understand. Decimal numbers belong more to physics than mathematics.
Physicists need usually deal with approximations and decimal numbers
suit for writing approximations, for example value of $\pi$ . There
are fractional representations for $\pi$ but maybe no-one uses them anymore. I have read that they were used in antiquity.
Maybe we should abandon the decimals.
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September 20th, 2014, 10:58 AM   #30
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To add to v8archie's point: one of the most important meta-skills for a mathematician to learn is the ability to switch between different representations of a problem. For example, when working on my paper on odd perfect numbers, I often had to switch between thinking of numbers in an 'Archimedian' sense (real numbers which can be compared with < and >) and what I now call a 'supernatural' sense (integers are a particular case of possibly infinite products of primes raised to possibly infinite exponents). In the first case the natural order is total while in the second case the natural order (divisibility) is partial.

For real numbers there are many forms. Binary (or decimal) expansion, continued fraction, the Engel or Pierce expansion, etc. If the number is rational you can also write it as a fraction of integers; if it's an integer the Zekendorff expansion; and so forth.

There's nothing special about the decimal expansion. In fact, it's poorly suited to most purposes. Use it when it works, but please don't mistake it for anything more than it is.
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