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YONG



Joined: 16 Mar 2005
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YONG
tthsqe wrote:
yong - the roots of the general quintic can be described by formulas. See section 8.2 in http://math.uiuc.edu/~schult25/ModFormNotes.pdf. There is one obvious and trivial typo in the statement of Proposition 8.3.1, but other than that, it looks correct.
Really? I am shocked!

Could you please show us the relevant formulas directly? The link you provided does not seem to work:

"Page not found

Our webserver cannot find the page you requested. Please check the URL to ensure the path is correct and that the URL is spelled correctly.

The page you are looking for may have been removed or moved to a different location. Please go to our homepage to find the new location.

If you still can't find what you are looking for, or to report a broken link, please contact the webmaster."

Besides, please take a look at the following theorem:

Abel–Ruffini theorem
https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem

I sincerely hope that you are actually correct on this matter.

Wink
Post 11 Aug 2015, 03:48
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revolution
When all else fails, read the source


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revolution
The URL has an errant trailing dot.
Post 11 Aug 2015, 04:05
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MHajduk



Joined: 30 Mar 2006
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MHajduk
I think you guys misunderstood each other.

Let's recall what YONG has written here
YONG wrote:
There is no explicit formula for general quintic (or higher degree) equations over the rationals in terms of radicals.
whereas tthsqe has written that
tthsqe wrote:
yong - the roots of the general quintic can be described by formulas.
The term "formula" is wider than "expression as a superposition of the four base arithmetic operations and the operation of calculation of the n-th root".

You are both right but only in the context of the problems you have described.

If you are looking for general solutions for equations of degree higher than four expressed in terms of radicals - you won't find it.

If you allow another operations to describe roots of these equations, as for example theta function, you can write in expressis verbis every solution of any algebraic equation - theorem proved by Hiroshi Umemura in 1984.
Post 11 Aug 2015, 08:25
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YONG



Joined: 16 Mar 2005
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YONG
What MHajduk said actually answers revolution's original question.

The language that we are supposed/allowed to use dictates what numbers are indescribable.

Unless the language set is well-defined, it is pointless to ask what numbers are indescribable.

So, let's see if revolution can give us a well-defined language set to work with.

Wink
Post 11 Aug 2015, 10:05
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revolution
When all else fails, read the source


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revolution
YONG wrote:
So, let's see if revolution can give us a well-defined language set to work with.
Curious. I never realised the set of indescribable numbers was dependant upon the language chosen, but it makes perfect sense of course. Anyhow, what I asked for was an example of an indescribable number. So how about this: In any language of your choosing, give an example of an indescribable number.
Post 11 Aug 2015, 12:36
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MHajduk



Joined: 30 Mar 2006
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MHajduk
revolution wrote:
So how about this: In any language of your choosing, give an example of an indescribable number.
I beg your pardon, but don't you think that what you're asking for may be impossible to do?

Suppose, that I could give you an example of indescribable number. To point precisely this number among other indescribable numbers I should give you a definition of it, i.e. describe the features that identify exactly that number and no other indescribable number has got. But this way I will create using a finite number of symbols a description of it or better if you wish, I'll give you a finitely describable algorithm that will produce numeric representation of this number (as it was in the article mentioned by you). But in this case our number will automatically fall out of the set of indescribable numbers.

How could I show you an example of indescribable number without describing it?
Post 11 Aug 2015, 16:03
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sleepsleep



Joined: 05 Oct 2006
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sleepsleep
revolution wrote:
Can someone please give me an example of an indescribable number?


check quoted below

Quote:


Post 11 Aug 2015, 17:00
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revolution
When all else fails, read the source


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revolution
MHajduk wrote:
revolution wrote:
So how about this: In any language of your choosing, give an example of an indescribable number.
I beg your pardon, but don't you think that what you're asking for may be impossible to do?

Suppose, that I could give you an example of indescribable number. To point precisely this number among other indescribable numbers I should give you a definition of it, i.e. describe the features that identify exactly that number and no other indescribable number has got. But this way I will create using a finite number of symbols a description of it or better if you wish, I'll give you a finitely describable algorithm that will produce numeric representation of this number (as it was in the article mentioned by you). But in this case our number will automatically fall out of the set of indescribable numbers.

How could I show you an example of indescribable number without describing it?
Hence the numerical oddity. It is a set of numbers where not even a single example exists, but we know the set is vastly larger than all other sets of numbers combined.
sleepsleep wrote:
check quoted below

Quote:


Okay, but what number are you describing?
Post 11 Aug 2015, 19:13
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l_inc



Joined: 23 Oct 2009
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l_inc
revolution
Just read the topic and learned about the indescribable numbers. Interesting thing is that indescribable numbers can be described... at least once. Smile The point is to go beyond the scope of a language and use the surrounding world for a description. An example of that: take two coins and drop the first one so that it hits the ground, then drop the second one so that it also hits the ground. The time interval in seconds between the two hits is an indescribable number, but the very experiment is its description. The assumption here is that the time is continuous.

I'd like to comment on my "at least once" above. Again, interesting thing is that the same experiment can be repeated to try to describe the same number, but the probability of describing the same number is zero (to be read as "almost certainly won't happen"). Nonetheless, events with zero probability (just like the outcome of the first experiment) happen all the time.

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Post 11 Aug 2015, 20:59
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revolution
When all else fails, read the source


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revolution
l_inc wrote:
Just read the topic and learned about the indescribable numbers. Interesting thing is that indescribable numbers can be described... at least once. Smile The point is to go beyond the scope of a language and use the surrounding world for a description. An example of that: take two coins and drop the first one so that it hits the ground, then drop the second one so that it also hits the ground. The time interval in seconds between the two hits is an indescribable number, but the very experiment is its description. The assumption here is that the time is continuous.
I think first you would have to prove that time is not discrete. If time is discrete (Planck Time) then in principle one could describe the interval.
Post 12 Aug 2015, 00:51
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tthsqe



Joined: 20 May 2009
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tthsqe
Yong - I have given you the relevant theory (without the trailing period). If you want the actual formulas, they are attached in this mathematica notebook. you might be able to open it in some browser plugin if you don't have the mathematica program.


Description: nb extensions not allowed, so save and rename at as .nb
Download
Filename: SolvingTheQuintic.nb.txt
Filesize: 35.23 KB
Downloaded: 194 Time(s)

Post 12 Aug 2015, 01:40
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YONG



Joined: 16 Mar 2005
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YONG
@tthsqe: I could access the pdf document of your link, and thank you for the mathematica program. But actually I do not have or need the program. If I ever want to find any of the real roots of a given quintic equation, I would rather use Newton's method, which is easy-to-understand, fast, and most importantly, does not require programming. Wink
Post 12 Aug 2015, 05:39
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YONG



Joined: 16 Mar 2005
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YONG
MHajduk wrote:
How could I show you an example of indescribable number without describing it?
Well, let me put it this way:

We CAN describe, in English, what indescribable numbers we are after.

For example, we are after the roots of a general quintic equation with rational coefficients.

However, in a mathematical language of our choosing, we CANNOT state the exact values of such roots.

For example, we cannot state, IN TERMS OF RADICALS, the exact values of the roots of a general quintic equation with rational coefficients.

This makes the numbers we are after indescribable.

Would this make sense?

Wink
Post 12 Aug 2015, 05:49
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MHajduk



Joined: 30 Mar 2006
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MHajduk
YONG wrote:
For example, we are after the roots of a general quintic equation with rational coefficients.

However, in a mathematical language of our choosing, we CANNOT state the exact values of such roots.
But the Newton's method, you have mentioned above, is an example of finitely describable algorithm that allows to produce an infinite sequence of successively better approximations of those roots. Hence we have a method that allows us to produce successively digits of decimal representations of the roots and this is exactly what makes those numbers describable.
YONG wrote:
For example, we cannot state, IN TERMS OF RADICALS, the exact values of the roots of a general quintic equation with rational coefficients.
The number doesn't need to be given in an exact form, it's enough to give a description of a process that allows us to represent the number with arbitrary accuracy (again, I refer here to the article mentioned by revolution).
Post 13 Aug 2015, 04:46
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YONG



Joined: 16 Mar 2005
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Location: 22° 15' N | 114° 10' E
YONG
MHajduk wrote:
But the Newton's method, you have mentioned above, is an example of finitely describable algorithm that allows to produce an infinite sequence of successively better approximations of those roots. Hence we have a method that allows us to produce successively digits of decimal representations of the roots and this is exactly what makes those numbers describable.

The number doesn't need to be given in an exact form, it's enough to give a description of a process that allows us to represent the number with arbitrary accuracy (again, I refer here to the article mentioned by revolution).
Perhaps I should have stated my assumption again (and again):

"Assume that you(revolution) are(is) NOT talking about numerical approximations."

Besides, if you have read the comments listed at the bottom of the original article (the one that talks about the computer programming approach), you would have noticed that some of the commenters do NOT agree with the author, especially on the point that "it does not matter even if the program runs infinitely and never stops". I take the same stance as those commenters.

Finally, I am afraid that you missed the most important part of the argument: the numbers are indescribable in a language of our choosing. It seems that you have chosen ANOTHER language for MY indescribable numbers, making them describable.

Hope that this clarifies your misunderstanding.

Wink
Post 13 Aug 2015, 05:22
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revolution
When all else fails, read the source


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revolution
Is PI indescribable? No one can give every digit to infinite precision (it is proven irrational) or a closed form function (it is proven transcendental), but it is easy to state it as the ratio of circumference to diameter of a circle.
Post 13 Aug 2015, 06:14
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YONG



Joined: 16 Mar 2005
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Location: 22° 15' N | 114° 10' E
YONG
revolution wrote:
Is PI indescribable? No one can give every digit to infinite precision (it is proven irrational) or a closed form function (it is proven transcendental), but it is easy to state it as the ratio of circumference to diameter of a circle.
In English, yes. In a language of our choosing, yes or no, and debatable.

If we regard "having a finitely describable algorithm (that produces an infinite sequence of successively better approximations)" the same as "having the number(s) describable", the answer is yes. Otherwise, it is debatable.

Wink
Post 13 Aug 2015, 06:43
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revolution
When all else fails, read the source


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revolution
So your task is to give an example of an indescribable number in any language of your choosing. No cheating and using a different language to describe the number. Everything must be given in the chosen language.
Post 13 Aug 2015, 06:47
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MHajduk



Joined: 30 Mar 2006
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MHajduk
YONG

The fact that some numbers may not be expressed using the particular set of symbols and, at the same time, the same number may be expressed using another set of symbols is quite common and not especially interesting.

I can give you a few examples of such numbers.

Let A={0, 1} be the set consisting of only two symbols 0 and 1. Using these symbols you can represent any non-negative integer treating strings made of 0's and 1's as binary representations of these numbers. But such simple set of symbols (accordingly to the convention) is not enough to build representation of all real numbers, i.e. accordingly to your definition, real non-integer numbers are "indescribable" in the context of alphabet A={0, 1} if we also want to represent all non-negative integers with it. Now, let's take a new alphabet A'={0, 1, .} consisting of the symbols 0, 1 and dot that will be used to sygnalize the place where strings made of 0's and 1's are split into the integer part and mantissa. In this new alphabet you can already denote any non-negative real number, so now you can't say that all non-integer numbers are indescribable in the context of the new alphabet A' as it was in the context of the old alphabet A.

Another example refers to the well-known problem of sections that may be constructed with use of a ruler and a compass. Lengths of such sections can be expressed as a superposition of four operations (addition, subtraction, multiplication and division) and calculation of the square root of the given number. The number being the real solution of the following equation

x^3 - 2 = 0

i.e. the cube root of 2 is inexpressible in terms of the four arithmetic operations and taking the square root. BTW, it's the main reason why the Delian problem (doubling the cube problem) is not solvable using only such simple tools as a ruler and a compass. From the other hand, the solution of the aforementioned equation is obviously describable in terms of radicals (+, -, *, / and n-th roots where n is natural).

Nevertheless, both examples presented above aren't the answers to the original question asked by revolution:
revolution wrote:
Can someone please give me an example of an indescribable number?
As I understood this question, revolution referred here to the definition of an indescribable number as a number impossible to describe with any set of arbitrarily chosen symbols. You may call such numbers absolutely indescribable to distinguish them from your relatively indescribable numbers that are describable or not depending on the context of the language chosen by you.
Post 13 Aug 2015, 20:08
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l_inc



Joined: 23 Oct 2009
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l_inc
revolution
Quote:
I think first you would have to prove that time is not discrete.

I don't have to prove it as long as I explicitly made it an assumption. Thus under the given assumption you have an example of an indescribable number.
Quote:
If time is discrete (Planck Time)

Planck time by itself is inappropriate to mention here. It's just a value, not a time discontinuity hypothesis.

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Post 13 Aug 2015, 23:20
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