"Squaring the circle"

[Jason.Marshall]

I would just like some clarification on this subject, I am aware that this is impossible with a compass and a straight edge.
So my question is can a computer do it and if a computer can do it then can there be an exact quantity of the equivalent square root of a circle that is the exact same as the square and what is that quantity according to the computers conclusion?

 

[Dinosaur]

To square a circle you must construct or determine a square with the same area. For circle of radius r, the side of the square is r*Sqrt(pi): Approximately r*1.77245

Sqrt(pi) can be calculated to many decimal places, but it is a transcendental number.

Note that Sqrt(2) is not transcendental & can be constructed. It is the diagonal of a square whose side is one.

 

[rpenner]

With just a straightedge, a compass you can construct lines segments with lengths in any positive real ratio which is governed by the real operations of addition, subtraction, multiplication, division and square roots. Since pi can't be obtained by any finite number of those operations, you can't create line segments in the ratio $$1 \quad : \quad \sqrt{\pi}$$ which is needed to construct a square with the same area as a circle of a given radius. For similar reasons, on cannot construct line segments in the ratio $$1 \quad : \quad \sqrt[3]{2}$$ so creating a cube of double the volume of a given cube is out.

In theory a straightedge and compass can find lines in the ratio of 1 to the smaller root of 115725089776811172448 - 168977039635388531038 x + 61214926932454753193 x^2 or 1 to 72254523693324347/57348453460122131 and while these numbers may be close to $$\sqrt[3]{2}$$ (indeed, both match to over 35 decimal places) the problem as stated is not one of approximation but equality.

 

[Jason.Marshall]

So since the exact quantity cannot be reached with the straight edge how does a computer determine this exact quantity, I would assume if the machine is able to determine the exactness then there should be an exact quantity that exist in some form that is equivalent to both the square and the circle,and it should be measurable? Am I incorrect?

 

[rpenner]

Computers have two ways of dealing with pi.
1) Approximate arithmetic. Most computer programs use binary IEEE arithmetic which can only approximate pi (or 1/3 or 1/10 or $$\sqrt[3]{2}$$). But IEEE arithmetic is fast and practical for many reasons.
2) Symbolic representations. Programs like Maple or Mathematica use symbols to represent expressions exactly. Any expression of numbers as (approximate) decimals only happens as an optional last step.

Because

$$\pi = \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{\infty} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} } $$​

we have a ready scheme for approximating pi to any degree of precision.

$$\begin{array}{c|cc|c} n & \textrm{Formula} & \textrm{Value} & \textrm{Precision} \\ \hline
\\ 0 & \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{0} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} }} & \frac{2^7 \cdot 5 \cdot 23 \cdot 29 \sqrt{10005}}{13591409} & 6 \times 10^{-14}
\\ 1 & \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{1} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} }} & \frac{2^{22} \cdot 3^2 \cdot 5^3 \cdot 23^4 \cdot 29^4 \sqrt{10005}}{29735444608353174286057} & 3.1 \times 10^{-28}
\\ 2 & \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{2} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} }} & \frac{2^{40} \cdot 3^3 \cdot 5^6 \cdot 23^{7} \cdot 29^{7} \sqrt{10005}}{867407410133324147761288805130794983129} & 2 \times 10^{-42}
\\ 3 & \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{3} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} }} & \frac{2^{55} \cdot 3^8 \cdot 5^8 \cdot 23^{10} \cdot 29^{10} \sqrt{10005}}{51238551861412291819499445944171266828347098365726703413} & 1.0003 \times 10^{-56}
\end{array}$$

Likewise Newton's method gives a way of getting ever better approximations to $$\sqrt[3]{2}$$
$$a_{0} = 1, a_{n+1} = \frac{2}{3} \left( a_{n} + a_{n}^{-2} \right)$$

$$\begin{array}{c|c|c} n & \textrm{Value} & \textrm{Precision} \\ \hline
\\ 0 & 1 & 2.6 \times 10^{-1}
\\ 1 & \frac{4}{3} & 7.4 \times 10^{-2}
\\ 2 & \frac{91}{72} & 4 \times 10^{-3}
\\ 3 & \frac{1126819}{894348} & 1.3 \times 10^{-5}
\\ 4 & \frac{2146097524939083451}{1703358734191174242} & 1.3 \times 10^{-10}
\\ 5 & \frac{14826533393634610888177630132775369682221926321305163339}{11767827353048609938875466919605673393241471731471353563} & 1.2 \times 10^{-20}
\end{array}$$

 

[Jason.Marshall]

Ok so the program "Maple or Mathematica" can determine the exact quantity, so I will assume the difficulty is in how does the human translate the exact quantity in terms a human can conclude the exact quantity is rational? The approximate manual approaches seems to create only a perpetual error margin of about 1 percent or less approximately at best. So what I ask you now is can this exact quantity be extracted from the symbolic representations of these above programs because I see a conflict in the program coming to an exact quantity but all manual techniques cannot reach an exact quantity thus the machine has determined an exact quantity does in fact exist, it seems like a contradiction please clarify this?

 

[Jason.Marshall]

And I am assuming also if a human bieng had infinite amount of time they may be able to produce the exact like a computer is this logical? So is the real challenge finding a manual system that contains both perfect accuracy with realistic efficient timeframe of manageable finite steps?

 

[rpenner]

No, I'm saying Maple and Mathematica manipulate quantities symbolically in a manner closer to the way mathematicians think about equations. The approximations are truncated versions of exact infinite representations and are only used if there is a need to display an approximation.

The subject of infinite representations is covered in analysis, which is a big field of mathematics.

Your sentence "The approximate manual approaches seems to create only a perpetual error margin of about 1 percent or less approximately at best. " seems garbled.

 

[Sorcerer]

I thought the answer - assuming the question was exactly squaring the circle- was no, because pi is an infinitally recurring number and therefore not precise.

 

[Jason.Marshall]

I probably did not explain myself properly but I made that statement based on the assumption that an exact quantity existed and that percentage would be just an inference on the error margin from the approximate quantity to the exact quantity. So it would seem not as simple as I original suspected to extract the exact quantity from the machines quantity because of "Infinite representation".

 

[Jason.Marshall]

So in a way the answear is no an exact quantity cannot be determined even by a machine because the exact quantity is logically undefined. The machine is providing more of a conceptualisation of what the exact quantity should be but in reality cannot ever reach a real quantity because if it could have then pi must then be a rational number?

 

[Sorcerer]

So in a way the answear is no an exact quantity cannot be determined even by a machine because the exact quantity is logically undefined. The machine is providing more of a conceptualisation of what the exact quantity should be but in reality cannot ever reach a real quantity because if it could have then pi must then be a rational number?

That is my understanding, which is why 'squaring the circle' is used loosely as an impossible task.

 

[rpenner]

I would just like some clarification on [the subject of squaring the circle], I am aware that this is impossible with a compass and a straight edge.

Correct.
http://mathworld.wolfram.com/GeometricConstruction.html
http://en.wikipedia.org/wiki/Compass-and-straightedge_construction
http://en.wikipedia.org/wiki/Squaring_the_circle

So my question is can a computer do it and if a computer can do it

Think! What do you mean when you say "do it?"
1) Produce a list of instructions that would allow someone to construct a square with the same area of any given circle using a finite number of compass and straightedge operations executed with absolute precision? No -- that's what's been proved impossible. Computers aren't magical. Back in the 1930's, a computer was an occupation.
2) Produce a list of instructions that would allow someone to construct a square with nearly the same area of any given circle using a finite number of compass and straightedge operations executed human precision? This is likely possible, given that $$\sqrt[8]{9^2 + \frac{19^2}{22}}$$ approximates $$\sqrt{\pi}$$ to better than one part in a billion and given the finite width of pencil lead or chalk lines, this is likely better than the achievable precision on the largest physical surfaces.
3) Produce the above but in a way that would fool mathematicians? No.
4) Calculate $$\sqrt{\pi}$$ to enough digits to know the length of side of such a square to a precision finer than measured by the Large Hadron Collider even if the radius of the circle is the radius of the visible universe? Yes. That's less than 48 orders of magnitude, so if you know $$\pi$$ or $$\sqrt{\pi}$$ to 50 digits, that suffices. I did that already in [post=3176673]post #5[/post]. Using Newton's method it is rather trivial to compute $$\sqrt{10005}$$ to 56 digits, then some multiplications, one long division and one more square root get the answer to #4.
5) Something else?

then can there be an exact quantity of the equivalent square root of a circle that is the exact same as the square and what is that quantity according to the computers conclusion?

But think! Exactness in physicality or decimal representation isn't even required if we can manipulate quantities symbolically. $$\sqrt{2}$$ requires an infinite number of digits to represent as a decimal and cannot be represented at all as a finite rational number. But we can prove $$\left( 3 - 2 \sqrt{2} \right)^5 = 3363 - 2378 \sqrt{2}$$ without knowing anything about the decimal representation and we know
$$\frac{1}{6930} \lt 3363 - 2378 \sqrt{2} \lt \frac{1}{5741} $$ because we know $$ \frac{8119}{5741} \lt \sqrt{2} \lt \frac{19601}{13860}$$ and we know $$0.0000592 \lt 3363 - 2378 \sqrt{2} \lt 0.0002970$$ because we know $$ 1.4142135 \lt \sqrt{2} \lt 1.4142136$$. There is no natural property of the value $$\sqrt{2}$$ that requires us to know all digits of its unending decimal representation or to perfectly equate it to a rational number.

And we do know that we have its value located on the real number line:

$$\begin{array}{c|ccccccccc} x & 1.4142135 & \lt & \frac{8119}{5741} & \lt & \sqrt{2} & \lt & \frac{19601}{13860} & \lt & 1.4142136 \\ x^2 -2 & - 1.764177500 \times 10^{-7} & \lt & - \frac{1}{32959081} & \lt & 0 & \lt & \frac{1}{192099600} & \lt & 1.0642496 \times 10^{-7} \end{array}$$​

So since the exact quantity cannot be reached with the straight edge how does a computer determine this exact quantity, I would assume if the machine is able to determine the exactness then there should be an exact quantity that exist in some form that is equivalent to both the square and the circle,and it should be measurable? Am I incorrect?

Mathematicians have procedures which produce unending better approximations that eventually get better than any chosen precision. I listed those for $$\pi$$ and $$\sqrt[3]{2}$$ in [post=3176673]post #5[/post]. Thus knowledge of the procedure leads to the ability to work with the quantity to any level of exactness. That's how one can prove, for example, that $$\zeta(2) = \frac{\pi^2}{6}$$ or $$\Gamma(\frac{1}{2}) = \sqrt{\pi}$$, from a complete knowledge certain sequences and the ability to prove that they converge to the same number on both sides of the equals sign.

Ok so the program "Maple or Mathematica" can determine the exact quantity, so I will assume the difficulty is in how does the human translate the exact quantity in terms a human can conclude the exact quantity is rational? The approximate manual approaches seems to create only a perpetual error margin of about 1 percent or less approximately at best. So what I ask you now is can this exact quantity be extracted from the symbolic representations of these above programs because I see a conflict in the program coming to an exact quantity but all manual techniques cannot reach an exact quantity thus the machine has determined an exact quantity does in fact exist, it seems like a contradiction please clarify this?

I am pretty sure you are thinking about numbers completely the wrong way and making a fetish over decimal representations which are not the last word in exactness.

Look at 6277/20400 + 126/425 + 19/48. In decimal this is : $$0.3076 \overline{9607843137254901} \; + \; 0.29 \overline{6470588235294117} \; + \; 0.3958 \overline{3}$$ where a special notation is needed just to show that the decimals eventually repeat in certain patterns forever. It's far from obvious that 6277/20400 + 126/425 + 19/48 = 1. And the decimals for $$\sqrt{2}, \pi, \sqrt{\pi}, \; \textrm{and} \; \sqrt[3]{2}$$ never repeat. To restrict math to just arithmetic on finite decimal representations would be to rob it of everything of interest to anyone other than accountants.

And I am assuming also if a human being had infinite amount of time they may be able to produce the exact like a computer is this logical?

It is unphysical to ask what humans (or computers) can do in an infinite amount of time. But with a finite (but large) amount of time, humans can do any calculation a computer can. When a computer is restricted to doing just numerical arithmetic, however, it can never prove that $$\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}$$ which a human skilled in analysis can do on about 1/2 sheet of paper.
http://en.wikipedia.org/wiki/Basel_problem

So is the real challenge finding a manual system that contains both perfect accuracy with realistic efficient timeframe of manageable finite steps?

No, mathematics is not just about calculating values. Your goal is not shared with mathematicians and is provably not possible. There is no writing system that can list all the numbers between 0 and 1 with just a finite number of symbols.

I probably did not explain myself properly but I made that statement based on the assumption that an exact quantity existed and that percentage would be just an inference on the error margin from the approximate quantity to the exact quantity. So it would seem not as simple as I original suspected to extract the exact quantity from the machines quantity because of "Infinite representation".

An exact quantity is formally known to mathematicians as a "real number." Mathematicians can algebraically manipulate real numbers in many ways other than writing down finite approximations.

So in a way the answer is no an exact quantity cannot be determined even by a machine because the exact quantity is logically undefined.

Untrue. Specific real numbers are by definition defined. $$\pi$$ is defined as the first positive root of the equation $$\sin x = 0$$ or as the ratio of a circumference of a perfect circle to its diameter. No limitation of physics or engineering or computers affects the exactness of $$\pi$$.

The machine is providing more of a conceptualization of what the exact quantity should be but in reality cannot ever reach a real quantity because if it could have then pi must then be a rational number?

No. Rational numbers are not the end of mathematics.

That is my understanding, which is why 'squaring the circle' is used loosely as an impossible task.

Squaring the circle is formally impossible because the rules of compass and straightedge construction limit what ratios of line segments may be constructed to quantities involved repeated square roots, addition, multiplication and division and the definition of $$\sqrt{\pi}$$ means you can never get there in any finite number of steps.

Analysis allows you to work with numbers that you could only get with an infinite number of steps of arithmetic or geometric construction. Thus it is vastly more powerful and allowed for the first time the discovery of the laws of celestial motion.
http://en.wikipedia.org/wiki/Mathematical_analysis

 

[Jason.Marshall]

Yes I do have an obsession, what can I say but thank you for your thorough reply, to me I can metaphorically compare the way I was thinking about it compare to the way you were thinking about it to the deterministic point of view vs laplacian's view...hmm although our egos will tell us "We are God" then simultaneously fail to achieve perfection doing any task.

 

[Dinosaur]

Jason.Marshall:From your Post #4:

So since the exact quantity cannot be reached with the straight edge how does a computer determine this exact quantity

From your Post #6

Ok so the program "Maple or Mathematica" can determine the exact quantity

Computers cannot provide exact values for transcendentals. Software can compute approximations accurate to hundreds or thousands of digits for pi, but cannot provide an exact value.

Transcendentals are special. They cannot be the roots of a an ordinary polynomial. While rational numbers (EG: 1/3, 2/7, 3/19 . . . . .) can be roots of polynomials & can be expressed exactly using a radix other than ten, the same cannot be said of transcendentals.

BTW: Most algebraic numbers cannot given exact analytic values. There is no analytic solution for polynomials of order 5 or more.

 

[Beaconator]

I probably did not explain myself properly but I made that statement based on the assumption that an exact quantity existed and that percentage would be just an inference on the error margin from the approximate quantity to the exact quantity. So it would seem not as simple as I original suspected to extract the exact quantity from the machines quantity because of "Infinite representation".

Genius! Every word. The way you have put it here "error akin to interference along the way to exact a percentage"

Matches the description to all the tests for probability.

I think some compasses have a top swivel that allows you to draw something close enough to a square. Architectural maybe even nautical. In actually using nautical compasses some draw both part circles and lines to make sure their bearings are correct for their approximate speed of travel under current.

 

[Beaconator]

I probably did not explain myself properly but I made that statement based on the assumption that an exact quantity existed and that percentage would be just an inference on the error margin from the approximate quantity to the exact quantity. So it would seem not as simple as I original suspected to extract the exact quantity from the machines quantity because of "Infinite representation".

Genius! Every word. The way you have put it here "error akin to interference along the way to exact a percentage"

Matches the description to all the tests for probability.

I think some compasses have a top swivel that allows you to draw something close enough to a square. Architectural maybe even nautical. In actually using nautical compasses some draw both part circles and lines to make sure their bearings are correct for their approximate speed of travel under current.

 

[rpenner]

Except probability theory is not part of Euclidean straightedge-and-compass construction.

 

[Fraggle Rocker]

And I am assuming also if a human bieng had infinite amount of time they may be able to produce the exact like a computer is this logical?

I suppose so, but as the Linguistics Moderator I would say that question is more about semantics than mathematics. To talk about a human being having "an infinite amount of time" is not a very good use of the language because the entire purpose of that statement is to conceal the fact that it is impossible for any human being to have an infinite amount of time--and in fact it is impossible for any institution, any civilization, any species, to have an infinite amount of time.

As I understand it, today's macrocosmologists will tell us that even the universe itself may not have an infinite amount of time. Depending on the total amount of dark matter and dark energy that exists (a question that has not yet been answered), it's quite possible that the universe will stop expanding and begin to collapse on itself. Eventually it would do a complete reversal of the Big Bang (the "Big Crunch" as some call it) and shrink into a point with zero size, mass and energy... i.e., vanish.

So is the real challenge finding a manual system that contains both perfect accuracy with realistic efficient timeframe of manageable finite steps?

No. It has not been proven that pi is a repeating fraction. That's the first challenge.

Using the tools of Euclid, there is, today, no way to "square the circle." However, this does not mean that some new tool invented 2,400 years later can't do it. But that tool has not been invented yet.

I thought the answer - assuming the question was exactly squaring the circle- was no, because pi is an infinitally recurring number and therefore not precise.

"Precise" is a measure, not an absolute. We can calculate pi to any desired level of precision. We usually settle for five decimal digits: 3.14159; but it has been calculated beyond 100,000 digits.

Except probability theory is not part of Euclidean straightedge-and-compass construction.

Indeed! He didn't even have calculus.

 

[CHRIS.Q]

Computers have two ways of dealing with pi.
1) Approximate arithmetic. Most computer programs use binary IEEE arithmetic which can only approximate pi (or 1/3 or 1/10 or $$\sqrt[3]{2}$$). But IEEE arithmetic is fast and practical for many reasons.
2) Symbolic representations. Programs like Maple or Mathematica use symbols to represent expressions exactly. Any expression of numbers as (approximate) decimals only happens as an optional last step.

Because

$$\pi = \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{\infty} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} } $$​

we have a ready scheme for approximating pi to any degree of precision.

$$\begin{array}{c|cc|c} n & \textrm{Formula} & \textrm{Value} & \textrm{Precision} \\ \hline
\\ 0 & \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{0} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} }} & \frac{2^7 \cdot 5 \cdot 23 \cdot 29 \sqrt{10005}}{13591409} & 6 \times 10^{-14}
\\ 1 & \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{1} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} }} & \frac{2^{22} \cdot 3^2 \cdot 5^3 \cdot 23^4 \cdot 29^4 \sqrt{10005}}{29735444608353174286057} & 3.1 \times 10^{-28}
\\ 2 & \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{2} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} }} & \frac{2^{40} \cdot 3^3 \cdot 5^6 \cdot 23^{7} \cdot 29^{7} \sqrt{10005}}{867407410133324147761288805130794983129} & 2 \times 10^{-42}
\\ 3 & \frac{640320^{\tiny \frac{3}{2}}}{163 \cdot 2^3 \cdot 3^3 \cdot 7 \cdot 11 \cdot 19 \cdot 127} \frac{1}{\sum_{k=0}^{3} \left( \frac{13591409}{163 \cdot 2 \cdot 3^2 \cdot 7 \cdot 11 \cdot 19 \cdot 127} + k \right) \frac{ (6 k)! (-1)^k }{ (3 k)! (k !)^3 640320^{3k} }} & \frac{2^{55} \cdot 3^8 \cdot 5^8 \cdot 23^{10} \cdot 29^{10} \sqrt{10005}}{51238551861412291819499445944171266828347098365726703413} & 1.0003 \times 10^{-56}
\end{array}$$

Likewise Newton's method gives a way of getting ever better approximations to $$\sqrt[3]{2}$$
$$a_{0} = 1, a_{n+1} = \frac{2}{3} \left( a_{n} + a_{n}^{-2} \right)$$

$$\begin{array}{c|c|c} n & \textrm{Value} & \textrm{Precision} \\ \hline
\\ 0 & 1 & 2.6 \times 10^{-1}
\\ 1 & \frac{4}{3} & 7.4 \times 10^{-2}
\\ 2 & \frac{91}{72} & 4 \times 10^{-3}
\\ 3 & \frac{1126819}{894348} & 1.3 \times 10^{-5}
\\ 4 & \frac{2146097524939083451}{1703358734191174242} & 1.3 \times 10^{-10}
\\ 5 & \frac{14826533393634610888177630132775369682221926321305163339}{11767827353048609938875466919605673393241471731471353563} & 1.2 \times 10^{-20}
\end{array}$$

Wow, this is really huge calculation