arithmetic
Posts:
106
From:
venezuela
Registered:
1/23/06
|
|
Re: cube root of a given number
Posted:
Jul 23, 2007 9:53 AM
|
|
On 22 jul, 20:38, "sttscitr...@tesco.net" <sttscitr...@tesco.net>
wrote:
> On 22 Jul, 23:47, arithmonic <djes...@gmail.com> wrote:
>
>
>
>
>
> > On 22 jul, 07:29, "sttscitr...@tesco.net" <sttscitr...@tesco.net>
> > wrote:
>
> > > On 22 Jul, 04:28, arithmonic <djes...@gmail.com> wrote:
>
> > > > On 21 jul, 17:39, "sttscitr...@tesco.net" <sttscitr...@tesco.net>
> > > > wrote:
>
> > > > > On 21 Jul, 21:38, arithmonic <djes...@gmail.com> wrote:
>
> > > > > > On 16 jul, 04:39, "sttscitr...@tesco.net" <sttscitr...@tesco.net>
>
> > > > 1.- I challenge you to show such Eshbach's method in this thread,
> > > > because both of you are trying to state that my methods --based on the
> > > > Rational Mean-- are the same as the one you read in Eshbach's
> > > > work ("Handbook of Engineering Fundamentals).
>
> > > I'm sure the poster knows what he read.
>
> > Why are you so sure? Are you his friend?
> > I am sure that the poster called Grover Hughes does not know a single
> > bit of all what he was talking about, tha's why I challenged him and
> > be sure the method he mentioned is by far similar to the methods shown
> > in my web pages. The sci.math audience is also waiting and observing
> > the results of my challenges to both of you.
>
> > Now to the very specific point on ROOTS-SOLVING METHODS:
>
> > > > 2.- I challenged you in this posting to show to the sci.math audience
> > > > a very simple numerical example on your alleged GENERAL Hurwitz's
> > > > ROOT-SOLVING METHOD for computing, say, THE FIFHT ROOT OF 2.
>
> > > There are 5 fifth roots of 2. Which one do you want ?
> > > Are you saying you can find complex roots too ?
>
> > > Finding the real root of x^5-2 =0 is simple.
> > > s(x,y) is the sign of binary quntic x^5-2y^5
>
> > > The root must lie between (0,1) = 0 and (1,0) = "inf"
> > > calculate s(0,1) and s(1,0). Form the mediant (1,1)
> > > At any stage in the process s(xn,yn) will be 1 or -1.
> > > if s(xn,yn) = un The new new mediant is formed with
> > > (xk,yk) where k is the largest index <n such that un*uk = -1
>
> > > 0 1 -1
> > > 1 0 1
> > > 1 1 -1
> > > 2 1 1
> > > 3 2 1
> > > 4 3 1
> > > 5 4 1
> > > 6 5 1
> > > 7 6 1
> > > 8 7 -1
> > > 15 13 1
> > > 23 20 1
> > > 31 27 -1
> > > 54 47 1
> > > 85 74 -1
> > > 139 121 1
> > > 224 195 1
> > > 309 269 1
> > > 394 343 -1
> > > 703 612 -1
> > > 1012 881 -1
> > > 1321 1150 -1
> > > 1630 1419 -1
> > > 1939 1688 -1
> > > 2248 1957 -1
> > > 2557 2226 -1
> > > 2866 2495 -1
> > > 3175 2764 -1
> > > 3484 3033 -1
> > > 3793 3302 -1
> > > 4102 3571 -1
> > > 4411 3840 -1
> > > 4720 4109 -1
> > > 5029 4378 -1
> > > 5338 4647 -1
> > > 5647 4916 -1
> > > 5956 5185 -1
> > > 6265 5454 -1
> > > 6574 5723 -1
> > > 6883 5992 -1
> > > 7192 6261 -1
> > > 7501 6530 -1
> > > 7810 6799 -1
> > > 8119 7068 1
> > > 15929 13867 -1
> > > 24048 20935 -1
> > > 32167 28003 -1
> > > 40286 35071 -1
> > > 48405 42139 1
> > > 88691 77210 1
> > > 128977 112281 1
> > > 169263 147352 -1
> > > 298240 259633 -1
>
> > > Can you solve x^3 -2x^2 -x+1 =0 ?
>
> > > > Notice that I am challenging you and your friend Grover Hughes with two very simple inquires.
>
> > > I don't know why you think Grover Hughes is a friend of mine.
>
> > > You used to claim that you could find the best simultaneous
> > > approximations to cubrt(2), cubrt(4).
> > > As you methods are so revolutionary, I would have thought
> > > this would have been possible too.
>
> > > In fact, you can use your methods for simultaneous approximation of
> > > cubrt(2), cubrt(4), but it would be
> > > sheer luck if a best approximation was found.
> > > Shall I give you a hint ?
>
> > > Do you understand the diffference between fast convergence and best
> > > rational approximation ?
>
> > One step at the time, please.
>
> > Let us focus on the main point of this thread: The extremely simple
> > high-order arithmetical methods shown in my webpages, and your
> > allegued Hurtwitz's Root-Solving Method as being the same thing that
> > my methods, in such a way, that when talking about them you stated: "I
> > don't think the claim that these methods are in any way new stands up
> > to scrutiny."
>
> > Only after clarifying this point to the sci.math audience I will
> > procede to answer all the other
> > remarks you have made in this posting.
>
> > But... Please, If you don't mind, I want to ask you just two more
> > questions specifically related to this table you have shown on the
> > fifth root and your remark about the methods shown in my webpages: "I
> > don't think the claim that these methods are in any way new stands up
> > to scrutiny."
>
> > ARE YOU JUST PLAYING A JOKE OR WHAT?
> > HAVE YOU EVER READ A BOOK ON THE HISTORY OF MATHEMATICS?
>
> What I find strabge is that you claim to have invented
> a new method of root solving and yet seem to be incapable
> of applying it to any problem I pose.
>
What could be strange for some people is that you are not willing to
READ any single bit of my webpages. However, this is not strange to
me, at all, that is the standard reaction to my critics
on the whole history of root-solving. That is the reason I have no
intentions of sending any of these new and simple methods to any peer-
review journal, it is clear that they will not allow me to express all
those critics against the mathematics we have inherited. They will not
read any single bit of my methods in the same way as many others like
you do.
The table of values you have posted IS NOT Hurtwitz's method, that is,
Hurtwitz WAS NOT the author
of such method as you have negligently alleged. That is a FALSE
statement from yours.
The late mathematician David Fowler had the theory that Ancient Greeks
used the Mediant to do things like the table you have posted, but
there are no concrete evidences for his theory but
just probable signs.
According to concrete evidences John Wallis certainly WAS THE AUTHOR
of the method you have shown. Personally I would never assign the word
"method" to such primitive trial-&-error algorithm which
by the way, is the slowest algorithm you can find to compute anything.
So it is so ridicule your intention of comparing such primitive and
slow trial-&-error algorithm with the natural high-order arithmetical
methods shown in my book and webpages.
I say "trial-&-error" because in each step of the "method" you need to
know if your approximations are lower or higher (in this way you use:
x^5-2y^5) than the true value of the root.
Wallis used that way of operating mediants and got approximations to
number 'PI'.
Nicholas Chuquet also worked in a similar way the mediants but he
improved the method by using unit fractions. There are others from
past times who worked in such a primiteve way.
All that is fully explained in my book and briefly mentioned in my
webpages.
But...
All that has nothing to do with my extremely simple High-Order
Arithmetical methods which do not need to make any trial-&-error
checkings, and you know that, but my critics to the whole history
of root-solving really hurt you and you are not willing even to read
any single bit of my webpages.
Well, I am sorry for that, but I will continue doing so because that
is the CRUDE TRUTH, mathematicians of ancient times could have easily
used Newton's, Halley's, Householder's methods by agency of the most
simple arithmetic as shown in my book and webpages, however, they
didn't. that is a real shame, worst when considering that the root-
solving issue is the very spine of the whole history of mathematics.
I showed to you in this thread a TRIVIAL method --based on the
Rational Mean-- to find all the Householder's iterative functions for
computing the square root and urged you to ask to your math teachers
the reasons they didn't taught you such trivial stuff.
Have you asked your teachers why they didn't taught you such trivial
stuff at school?
Of course, not, because you are not willing even to read any single
bit of the square root sample I posted to you and the sci.math
audience in this thread.
The very important question here is that:
Why math teachers didn't taught you such trivial stuff at school?
Why?
The answer is very simple: Because they didn't know about the new
methods shown at:
http://mipagina.cantv.net/arithmetic/rmdef.htm
And it is striking to realize that math teachers didn't know about
these trivial methods since Babylonian times up to now. Just striking.
> Can you solve x^3 -2x^2 -x+1 =0 ?
If you want to apply the high-order root-solving methods shown in my
webpages and book, for solving algebraic equations then I suggest you
to look at (as well as my book):
MATHEMATICAL SPECTRUM. Bob Bertuello. The Rational Mean. Vol. 39, No.
2, 2006/2007. UK.
An example on solving a polynomial equation by agency of the Rational
Mean.
The author of the article used some of my methods to solve polynomial
equations.
A link showing such reference appears at:
http://mipagina.cantv.net/arithmetic
Of course, it is clear you have never visited any single bit of my
webpages.
> Why is it that I can apply your method to simultaneously
> approximate cubrt(2), cubrt(4) but you can't. ?- Ocultar texto de la cita -
>
You have not tried anything, you have not even read any single bit of
the methods shown in my webpages. You do not have purchased my book.
You have not even take a look at the links and references shown at:
http://mipagina.cantv.net/arithmetic
And you are not interested, at all, in developing anything related to
the methods shown in my webpages. That is fairly clear to me and the
sci.math audience.
You are just reacting to my hard critics to the history of mathematics
which you seems to admire so much. Sorry for that, I do not admire the
mathematics we have inherited.
This statement from yours is as false as all the other statements you
have posted in this thread, I mean, all your allegations on the
Hurtwitz being the author of using mediants to find roots and your
attempts to falsely state that my methods are the same as the one you
negligently attibuted to Hurtwitz.
You have responded to my second challenge with a false statement on
Hurtwitz's method and its non-existent connection to my methods. I
have shown you that Hurtwitz IS NOT the author, and that such
primitive trial-&-error method is by far related to the methods shown
in my webpages.
All what you have said about your alleged Hurtwitz's method cannot be
considered as a response to a challenge but just a very bad joke from
yours, sci.math is not a place for joking but for doing mathematics.
So it is clear you have an X on the second challenge.
There are other false statements to have attibuted to me in this
thread, but I am waiting for the
response of you and your friend Grover Hughes to my first challenge:
My first challenge to you and your friend Grover Hughes was:
1.- I challenge you to show such Eshbach's method in this thread,
because both of you are trying to state that my methods --based on
the
Rational Mean-- are the same as the one you read in Eshbach's
work ("Handbook of Engineering Fundamentals).
You replied to Grover Hughes and endorsed his comments, so
considering that he is absolutely unable to prove that my methods are
the same that his alleged Eshbach's method, then you have to face your
crude negligence on this matter.
It is not my fault, it is your fault for being so negligent on this
matter and reacting that way to my hard hard critics on the history of
root-solving.
In response to your mentioning the word "square root" I showed to you
in this thread an extremely TRIVIAL method --based on the Rational
Mean-- to find all the Householder's iterative functions for computing
the square root and urged you to ask to your math teachers the reasons
they didn't taught you such trivial stuff. You have not responded
anything on that.
I am not joking, I am very serious on this matter, because our young
students deserve so much respect and a true mathematics, a true
natural science.
I will not answer any other questions from you and your friend Grover
Hughes till both challenges have been answered to me and the sci.math
audience.
|
|
