Pi and Golden Number: not random occurrences of the ten digits.

[Jean-Yves BOULAY]

Abstract of the paper "Pi an Golden Number: not random occurrences of the ten digits".

 

Number Pi and the Golden Section as well as the inverse of these numbers are made up of a series of apparently random decimal places. This paper is on the occurrence order of the 10 digits of decimal system in these fundamental mathematic numbers. It is in fact that the ten digits of decimal system does not appear randomly in the sequence of Pi and in Golden Section. Also, same phenomena operate in many other constants of which the square roots of numbers 2, 3 and 5, the first three prime numbers.

 

1. Introduction.

 

The number Pi (p) and the Golden Number (φ) and the inverse of these numbers are made up of a seemingly random digits. This article is about order of the first appearance of the ten figures of decimal system in these fundamental numbers of mathematics. There turns out that the ten digits decimal system (combined here with their respective numbers: figure 1 = number 1, figure 2 = number 2, etc..) do not appear randomly in the digits sequence of Pi (p) and the digits sequence of Golden Number (φ). The same phenomenon is also observed for the inverse of these two numbers (1/p et 1/φ).

 

 

1.1. Method.

 

This article studies the order of the first appearance of the ten figures of the decimal system in the decimals of constants (or numbers). After location of these ten digits merged then in numbers (figure 1 = number 1, etc), an arithmetical study of these is introduced...

 

(excerpt from the paper)...In constants π, 1/π and φ (a), the occurrence order of ten digits of the decimal system © compared to the rank of appearance (b) is organized into identical arithmetical arrangements (d)

 

 

(excerpt from the paper)...Into the occurrence order of digits of their decimals, the constants 1/π and 1/φ have the same ratio to 3/2 (probability to 1/11. 66).

 

In this division, there are the same first six and last four digits (probability to 1/210).

 

Both split their figures to form the same four areas multiples of 9 (probability to 1/420).

 

It appears finally that, for these two fundamental constants, the same digits appear in the same four areas of 1, 2, 3 and 4 figures (probability to 1/12600)…

 

 

(excerpt from the paper)...A formula, derived from the continued fraction of Rogers-Ramanujan including the 4 fundamental numbers π, φ, e and i, isorganized into same four zones of 1, 2, 3 and 4 digits and with the same first 6 and last 4 digits as the constant 1/π, and the constant 1/φ.

 

 

 

Complet paper here (and in attached file): http://jean-yves.boulay.pagesperso-orange.fr/pi/index.htm

 

Pi and Golden Number not random occurrences of the ten digits.pdf

Others some examples of phenomena introducted in the paper.

 

Edited May 20, 2017 by Jean-Yves BOULAY

[imatfaal]

!

Moderator Note

 

This is barely mathematics - and definitely not linear algebra nor group theory.

 

Moved to speculations.

 

[John Cuthber]

When you were looking at the occurrence of different digits, did you look at the whole of pi, or some truncated approximation?

[KipIngram]

When you were looking at the occurrence of different digits, did you look at the whole of pi, or some truncated approximation?

 

He's just looking at the initial appearance of each digit, which only seems to require a truncated approximation. But I don't seen any claimed significance, in the post or in the cited paper. What's the point supposed to be?

 

Also, my very first thought was that any supposed point shouldn't depend on choice of the decimal number system; you'd want to see the same thing manifest regardless of radix choice.

 

Without some claim of what this is meant to tell us, it seems like nothing more than a cute observation.

[John Cuthber]

A spot of googling indicates that pu looks normal for the first rather a lot of digits

https://arxiv.org/abs/1612.00489v1

So, unless the OP has looked at more digits, there's nothing to see here.

[KipIngram]

He did point out a pattern having to do with the order in which each digit's first appearance came, and then sums of sections of those digits. He showed that the same pattern occurred in several cases of "interesting" numbers and their reciprocals and so on. I could see for myself that his arithmetic was correct. What was missing was any statement of what that meant, and also any proof that it wasn't just coincidental. Also missing was any evidence that the same thing could be shown to be intrinsic to the numbers themselves, in the sense that you'd see the same thing regardless of representation radix.

 

I don't think there's anything there beyond coincidence, but the OP didn't indicate whether he was asking if it was significant, claiming that it was significant, or what exactly.

[Endy0816]

Neither is random to begin with.

[Daecon]

Would this be "numerography"?

[Jean-Yves BOULAY]

Hello everybody! This is the approach: just the study of the first appearance of the ten digits. And these are real facts and only singular findings without, at this time, explanations of why these curious phenomena are.

Many studies focus on Pi. Pi is a ratio (perimeter / diameter) and the number 1 / Pi (diameter / perimeter) is equally interesting to study.

[Strange]

Have you tried this with totally random numbers? I'm sure you can find patterns like this in almost any sequence.

[Jean-Yves BOULAY]

There are the statistics announced throughout the article and in the appendix.