Preprint Article Version 2 Preserved in Portico This version is not peer-reviewed

On 'Best' Rational Approximations to $\pi$ and $\pi+e$

Version 1 : Received: 15 May 2020 / Approved: 16 May 2020 / Online: 16 May 2020 (16:32:49 CEST)
Version 2 : Received: 31 May 2020 / Approved: 31 May 2020 / Online: 31 May 2020 (16:21:52 CEST)

How to cite: Teguia Tabuguia, B. On 'Best' Rational Approximations to $\pi$ and $\pi+e$. Preprints 2020, 2020050268 (doi: 10.20944/preprints202005.0268.v2). Teguia Tabuguia, B. On 'Best' Rational Approximations to $\pi$ and $\pi+e$. Preprints 2020, 2020050268 (doi: 10.20944/preprints202005.0268.v2).


Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known convergents of the continued fraction of $\pi$, $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, the convergent $920/157$ of the continued fraction of $\pi+e$ turns out to be the only rational number of this type.


$\pi$ (read Pi); $e$ (the natural logarithm base); continued fraction; convergent infinite series



Comments (2)

Comment 1
Received: 31 May 2020
Commenter: Bertrand Teguia
Commenter's Conflict of Interests: Author
Comment: Changes in the title; some clarity added concerning the aim of the paper, especially the advantage of the used approach: see the emphasis on continued fractions in the abstract, the introduction last sentence, the third section last sentence, and the discussion about the structure of $\pi+e$ in the last section.
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Comment 2
Received: 26 January 2022
The commenter has declared there is no conflict of interests.
Comment: Pi at the core of its creation is the curvature that create the area or circumfrence of a circle and sphere. No difference is present in the term circumfrence and the Curve that is its volume. When measured to the atomic level they are the same.

We first begin to define Pi at the integer of 360. This is a proper circle and sphere of infinite curvature. Logic dictates we begin at 360.
360 = pi = 1
Now we currently use a irrational and unusable digit to define this curvature. We will now begin using Divisional Segmentation to rationalize our 360 degree curve.

We begin with what we know. 355/113(china) & .31605(egypt). Each of these when used beside 360 produce a segmentation interval at 114.
1 divided by 360 = 0.00277777777777777777 x 114 = 0.31666666666666666578 (move decimal over one) x 114 = 360.9999999999999989892 we are actually over the designated number.
Our calculators have problems with percision and division. More accurate with multiplication and addition.
So lets do this the right.
1 divided by 360 = 0.0027777777777777777777
but when calculated back to 360 we fall short.
0.0027777777777777777777 x 360 = 0.9999999999999999972
Note the calculator flaw.
Now we do this 0.0027777777777777777777 and we give it the end point. 0.00277777777777777777779 x 360 = 1
This number defines the accuracy of the curve. To exist between at this segmentation as 77 - 79.
Now let us apply our triangulated divisional segmentation to this number.
360 divided by 114 = 3.15789473684210526315
Now we know we have a issue going back to the 360 interval so lets correct this before we continue.
3.15789473684210526315 has identified that the curve exists between 78 and 79. so lets add our 79 and see what happens.
3.1578947368421052631579 x 114 = 360 divided by 114 = 3.15789473684210526315
Take note of the calculator flaw when using division.

Here is another method reducing the remainder.

So 3.16 x 114 =360.24
360.2400 divided by 114 = 3.16 when entered into a calculator.

So math says we remove this from the remainder through even segmentation.

2400 divided by 114 =0.002105263157894737
subtract from 3.16 and you get 3.15789473684210526
You use this method te refine the number. Placing the remainding integer back into the radial curve through the 114 divisional segmentation.

To verify accuracy we use multiplication as the calculator is more accurate.
3.15789473684210526315 x 114 = 359.9999999999999999991
3.157894736842105263157 x 114 = 359.99999999999999999989
3.1578947368421052631578 x 114 = 359.99999999999999999998
3.1578947368421052631579 x 114 = 360
compared to 3.14-
3.1415926535897932384626 x 114 = 358.14156250923642918473
3.1415926535897932384627 x 114 = 358.14156250923642918474
3.1415926535897932384628 x 114 = 358.14156250923642918475
3.1415926535897932384629 x 114 = 358.14156250923642918477
3.1415926535897932384630 x 114 = 358.14156250923642918478

While this may appear more accurate to most it is not. The accuracy comes from the exact digit from start to finish, when this digit never varies you achieve perfect curvature which is found by divisional segmentation at 360 by 114 to produce the most accurate rational integer to use as the mathematical value of the completed 360 degree object. Allowing for no data loss during area calculations between any formula of area such as a square or triangle.
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