 ppauper
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### Re: what did the painter do wrong?

$\sqrt{7}=2+\overline{1,1,1,4}$ where the number under the bar is the periodic integer sequence in the continued fraction. ppauper's calculator confirms this. ppauper missed the 4.
I didn't miss anything.
Perhaps for once you could just admit that you made a mistake rather than trying to change the subject by attacking everyone else lovenatalya
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### Re: what did the painter do wrong?

You are funny. How did I attack everyone, by pointing out something is wrong? Just point out the specific part of my last post that you think is wrong if you say I have made a mistake.

As for the "missing" part, do you see the 4 in your calculator? Do you see a 4 in the painting? Yes or no? katastrofa
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### Re: what did the painter do wrong?

The $\sqrt 7$ identity is wrong. The continued fraction should be the golden ratio with the only radical being $\sqrt 5$. So the two side cannot be equal.
Could you explain what you mean by that? I posted a derivation of the continuous fraction (something they teach children at primary school, I presume) and ppauper posted a link to a continuous fraction calculator - both confirm that the painting is correct. Collector
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### Re: what did the painter do wrong?

continued fraction calculator
seems to give the same answer as the painter

collector said the "official" error was the "ln" being added after the event
yes that is the only known official error I heard about from the Gravity professor!

please triple check your calculations folks before posting here, it could lead to over-painting that are irreversible! It is not just to comment on a old painting based on mathematical emotions, like it was just some modern art! It is serious stuff in the cafe of the Physics institute!

Concerning good art, recently a series of politicians and journalists and the Norwegian court system tried to put one of our best painters in prison: Nerderum. Actually the King had to over-run the decision of the High-court, very good!. The modern art artists with no art skills where jealous of him I think.

Norwegian artist Odd Nerdrum, convicted of tax evasion in several courts and facing a year in prison, has received a royal pardon from King Harald V. No reason for the unusual pardon was given.  actually the whole court case case was ridiculous byrocratic circus, he is a painter not an accountant

"“He can continue painting as the free soul he is,” Collector
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### Re: what did the painter do wrong?

clearly the paintings made many of you emotional! cool down folks and calculate! lovenatalya
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Joined: December 10th, 2013, 5:54 pm

### Re: what did the painter do wrong?

The $\sqrt 7$ identity is wrong. The continued fraction should be the golden ratio with the only radical being $\sqrt 5$. So the two side cannot be equal.
Could you explain what you mean by that? I posted a derivation of the continuous fraction (something they teach children at primary school, I presume) and ppauper posted a link to a continuous fraction calculator - both confirm that the painting is correct.
Excuse me for the late reply as I am away on a trip. But here it is.

Why do I say 'tis wrong the continued fraction for $\sqrt7$, without any computation beyond simple arithmetic? Let me count the ways.

1. 'tis wrong for an expression with an equal sign has to be exactly equal and unique. The current expression is not.

2. 'tis wrong for a number is a quadratic surd (the root of an integral quadratic polynomial) if and only if it has a continued fraction that is eventually periodic. $\sqrt7$ is a quadratic surd. The continued fraction expression with an equal sign has to contain the FULL period to be exact and unique --- otherwise with a partial period there are infinitely many quadratic surds not to mention numbers that can be expressed with the same continued fraction expression. The only possible periodicities are $[2,\overline 1]$ and $[\overline{2,1}]$ where the top bar denotes the periodic part. The periodic $[\overline1]=x=1+\frac1x$ which is exactly the definition for the golden ratio which is in the quadratic rational field of $\mathbf Q[\sqrt5]$ which does not contain $\sqrt7$.

$[\overline{2,1,1,1}]$ is purely periodic or is period from the very beginning. It can be proved that a number is a reduced quadratic surd ($>1$ and its rational field conjugate is $\in (-1,0)$) if and only if its continued fraction is purely periodic, meaning the periodic part starts from the very first natural number. A continued fraction with period commencing from the beginning is a reduced quadratic surd. $-\sqrt d<-1$ for any natural number $d$ not a perfect square is not a reduced quadratic surd. It is impossible.

Both possibilities are wrong.

3. 'tis wrong for $x:=\sqrt d$ where $d$ is not a perfect square is not a reduced quadratic surd. However $y:=x+\lfloor x\rfloor$ is because it is of course greater than $1$ and its conjugate $-x+\lfloor x\rfloor\in(-1,0)$. Therefore $y=[\overline{\lfloor y\rfloor,a_1,a_2,...,a_k}]$. But $\lfloor y\rfloor=2\lfloor x\rfloor$. So $x=\big[2\lfloor x\rfloor, \overline{a_1,a_2,...,a_k,2\lfloor x\rfloor}\big]$. That means the continued fraction of $\sqrt d$ can always be written as a natural number followed by a period and the last natural number in the period is twice the very first natural number. It can also be proved that the length of the period is $O(\sqrt{d\ln d})$. For $d=7$, the first natural number of the continued fraction is of course $2$, and the last natural number in the period should be $4$ and the period length is around $4$. So we expect to see a $4$ appearing at the end of a short stretch of natural number in the continued fraction. We do not.

4. 'tis wrong for even if we do not know all the above theory, we should still be able to carry out a plodding computation like katastrofa attempts to do and come to the conclusion. Unfortunately, katastrofa, rote memorized all too well and only, I presume, the "primary school" mechanical steps, does not seek periodicity, stops before any of the $P_i$'s in her computation repeats, and fails to find the periodicity and the all important $4$ and thus fails to find the error.

5. 'tis wrong for the continued fraction calculator ppauper set the link to confirms that $\sqrt7=[2,\overline{1,1,1,4}]$ not $[2,\overline1]$ nor $[\overline{2,1,1,1}]$. katastrofa
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### Re: what did the painter do wrong?

You can put a strictly equal sign and use three dots. Furthermore, no one claims that sqrt 7 is not what ppauper's link (or my solution for that matter) shows. The rest of your post suggests to me that you don't understand continuous fractions... Cuchulainn
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### Re: what did the painter do wrong?

You can put a strictly equal sign and use three dots. Furthermore, no one claims that sqrt 7 is not what ppauper's link (or my solution for that matter) shows. The rest of your post suggests to me that you don't understand continuous fractions...
I see $\sqrt7$ as a special case of my extremely popular and informative $e^5$ thread (most viewed in Brainteaser). In that thread we used a bunch of methods, e.g. fixed-point contraction.

$y = \sqrt 7$

$y^2 = 7$

Solve by the sequence

$a_{n+1} = 1/2 (a_{n}+ 7/a_{n}) \enspace n >= 0$ The term $a_{0}$ is arbitrary.

Now
1. Prove { $a_{n}$ } is a Cauchy sequence (for T4A).
2. Is the space complete? Is $\sqrt 7$ a rational number?
3. Compute $y = \sqrt 7$ by hand to two decimal positions accuracy. And make it snappy.

you don't understand continuous fractions...
It's a lost art.
Last edited by Cuchulainn on June 1st, 2018, 10:01 am, edited 1 time in total.
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http://www.datasim.nl Cuchulainn
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### Re: what did the painter do wrong?

Continued Fraction for Golden Ratio and yes, $e^5$
Exercise: do CF for $\sqrt 7$
// TestContinuedFractionExp5.cpp
//
// Continued fractions for Golden Ratio and yes, you've guessed it, EXP(5).
//
// Daniel J. Duffy

#include <boost/math/tools/fraction.hpp>
#include <cmath>
#include <iostream>
#include <limits>
#include <iomanip>

using namespace  boost::math::tools;

template <class T>
struct golden_ratio_fraction
{
typedef T result_type;

result_type operator()()
{
return 1;
}
};

template <class T>
struct exp_fraction
{
private:
T a, b;
T z;
public:
exp_fraction(T v)
{
z = v;
a = z;
b = z;
}

typedef std::pair<T, T> result_type;

std::pair<T, T> operator()()
{
b += 1;
a -= z;
return std::make_pair(a, b);
}
};

template <class T>
T Exp(T a)
{
exp_fraction<T> fract(a);

boost::uintmax_t max_terms;
double cf = continued_fraction_b(fract, std::numeric_limits<T>::epsilon(), max_terms);
std::cout << ", max terms needed  " << std::setprecision(16) << max_terms << std::endl;

double val = 1.0 - (a / cf);
val = 1.0 / val;

return val;
}

int main()
{

double d = 5.0;
boost::uintmax_t max_terms;
std::cout << boost::math::tools::continued_fraction_a(golden_ratio_fraction<double>(),
std::numeric_limits<double>::epsilon(), max_terms);

std::cout << ", max terms needed " << std::setprecision(16) << max_terms << std::endl;

std::cout << "error: " << Exp<double>(5.0) /*- std::exp(5.0)*/ << std::endl;

return 0;
}
Step over the gap, not into it. Watch the space between platform and train.
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### Re: what did the painter do wrong?

@"Compute $\sqrt{7}$ by hand to two decimal positions accuracy. And make it snappy."

sqrt(64) - 1/2/sqrt(64) = 8 - 1/16 \approx 3*sqrt(7) => sqrt(7) \approx  127/16/3 Cuchulainn
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### Re: what did the painter do wrong?

@"Compute $\sqrt{7}$ by hand to two decimal positions accuracy. And make it snappy."

sqrt(64) - 1/2/sqrt(64) = 8 - 1/16 \approx 3*sqrt(7) => sqrt(7) \approx  127/16/3
And as a number?
Step over the gap, not into it. Watch the space between platform and train.
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### Re: what did the painter do wrong?

U  guys need  to take a look at the wider picture to see the real and deeper solution?

Is it not all about geometry? Divine? Last edited by Collector on June 1st, 2018, 1:30 pm, edited 1 time in total. Collector
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### Re: what did the painter do wrong?  katastrofa
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### Re: what did the painter do wrong?

@"Compute $\sqrt{7}$ by hand to two decimal positions accuracy. And make it snappy."

sqrt(64) - 1/2/sqrt(64) = 8 - 1/16 \approx 3*sqrt(7) => sqrt(7) \approx  127/16/3
And as a number?
2.646 - it's easy to obtain by hand. Paul
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### Re: what did the painter do wrong? Reminds me of that old joke: "You're a true renaissance man. And not just because you carry the pox."  