Arithmetic Transform and Turning e into \(\pi\)

I investigate an arithmetic transform where digits in a number can be swapped, calculations performed and then swapped back to give sensible results. The errors in such a process are considered.

Consider the number \(\pi\) to (some) decimal places. If we take every pair of digits and swap them we have a transformation \(\pi\to\overline{\pi}\) \[\pi\approx3.141592653589793238462643383279502884197169399375105 \\ \overline{\pi}\approx1.314956235857939324826463338725920881479613999731550\]

Then if we take this from \(\pi\) and perform the transform again \[\pi-\overline{\pi}\approx1.826636417731853913636180044553582002717555399643556 \\ \overline{(\pi-\bar{\pi})}\approx8.162364671378135196363810044555328007271553599465365\]

Repeating this elaborate process \[\overline{(\pi-\bar{\pi})}-\pi\approx5.020772017788341957901166661275825123074384200090259 \\ \overline{(\overline{(\pi-\bar{\pi})}-\pi)}\approx0.502770271873214599710616616728552210347832400902095\]

Until finally \[\pi-\overline{(\overline{(\pi-\bar{\pi})}-\pi)}\approx2.638822381716578638752026766550950673849336998473011 \\ \overline{(\pi-\overline{(\overline{(\pi-\bar{\pi})}-\pi)})}\approx6.283283218175687367825207666559005768394339689740311\]

\[\overline{(\pi-\overline{(\overline{(\pi-\bar{\pi})}-\pi)})}-\pi\approx \\ 3.141690564585894129362564283279502884197170290365205 \approx \pi\]

With the error \[\epsilon=0.000097910996100890899920900000000000000000890990099\]

Which is a strange number indeed! Many zeroes.

The introduction calculation was quite long, let’s try something simpler...Using a nice calculation (no carries) results in a perfect answer \[\alpha=1.23456 \to \overline{\alpha}=2.14365 \\ \beta=0.10342 \to \overline{\beta}=1.03024 \\ \alpha+\beta=1.33798 \to \overline{(\alpha+\beta)}=3.17389 \\ \overline{\alpha}+\overline{\beta}=3.17389 \to \overline{(\overline{\alpha}+\overline{\beta})}=1.33798\]

Obviously any error would come from carrying forward and updating the wrong power of the base.

The digits of \(\pi\) and \(e\) were laid alongside each other. Of course in certain places the digits between the two match. These points were fixed. Then the minimum systematic list of transformations to make \(e\) into \(\pi\) was found up to \(~30\) terms. The method was such, starting with \(e^*=e\)

-Read along \(e^*\) until we find a digit that is different to that digit of \(\pi\). -Find the smallest digit that when swapped with this digit will also be the same as it’s \(\pi\) equivalent. -Swap them. Repeat.

The list so far is \[\begin{array}{|c|c|c|} \hline Move & Digits\;Swapped & Transform \\ \hline 1. & e029 <-> e001 & 2 \;became\; 3 \\ 2. & e121 <-> e002 & 7 \;became\; 1 \\ 3. & e104 <-> e003 & 1 \;became\; 4 \\ 4. & e085 <-> e004 & 8 \;became\; 1 \\ 5. & e077 <-> e005 & 2 \;became\; 5 \\ 6. & e036 <-> e006 & 8 \;became\; 9 \\ 7. & e041 <-> e007 & 1 \;became\; 2 \\ 8. & e106 <-> e008 & 8 \;became\; 6 \\ 9. & e094 <-> e009 & 2 \;became\; 5 \\ 10.& e162 <-> e010 & 8 \;became\; 3 \\ 11.& e152 <-> e011 & 4 \;became\; 5 \\ 12.& e091 <-> e012 & 5 \;became\; 8 \\ - & e013---e013 & 9 \;remain\; 9 \\ 13. & e265 <-> e014 & 0 \;became\; 7 \\ 14. & e184 <-> e015 & 4 \;became\; 9 \\ 15. & e144 <-> e016 & 5 \;became\; 3 \\ - & e017---e017 & 2 \;remain\; 2 \\ - & e018---e018 & 3 \;remain\; 3 \\ 16. & e257 <-> e019 & 5 \;became\; 8 \\ 17. & e026 <-> e020 & 3 \;became\; 4 \\ - & e021---e021 & 6 \;remain\; 6 \\ 18. & e066 <-> e022 & 0 \;became\; 2 \\ 19. & e150 <-> e023 & 2 \;became\; 6 \\ 20. & e035 <-> e024 & 8 \;became\; 4 \\ 21. & e <-> e025 & 7 \;became\; 3 \\ \hline \end{array}\]

Giving \(_{10}S^{e}_{\pi}=\)29,121,104,85,77,36,41,106,94,162,152,

91,13,265,184,144,17,18,257,26,21,66,150,35,... Where the bottom left index is the base, and the right two indices are the numbers that transform between each other. This exact transform can then be applied to other irrational numbers to map them to other constants. We have picked the transformation such that when applied to \(e\) we get \(\pi\) and when applied to \(\pi\) we get \(e\). When applied to \(\sqrt{2}\) for example we get the number \[\sqrt{1}\to 0.0000000000000000000000000001 \\\sqrt{2}\to 2.400176372943... \\
\sqrt{3}\to 5.961868679368... \\\sqrt{4}\to 0.0000000000000000000000000002 \\
\sqrt{5}\to 7.454836887489... \\\sqrt{6}\to 7.459399441233... \\\sqrt{7}\to 6.500222445164... \\\sqrt{8}\to 4.900253754886... \\
\sqrt{9}\to 0.0000000000000000000000000003 \\\sqrt{10}\to4.834235792188... \\
\sqrt{11}\to6.208080287795...\\\sqrt{12}\to0.922636248737...\\
\sqrt{13}\to4.654042489723...\\
\sqrt{14}\to3.59 \\ \sqrt{15}\to7.33
\sqrt{16}\to0.0000...04 \\ \sqrt{17}\to9.5576214029 \\
\sqrt{18}\to6.4013301 \\ \sqrt{19}\to8.96615081 \\
\sqrt{20}\to4.8086726
\sqrt{21}\to7.3647840
%...........s.dsstddd
\\
a=0.000000000001\cdot n \to a \cdot n , \; n \in [0,9]\]

An interesting test would be the base \(10\) Champernowne constant, however it is not clear which digit is \(1\) the \(0\) or the \(1\), so two possibilities are \[C_{10} = 0.1234567891011213... \\ C_{10}\to 9.557325558801 \\ C_{10}\to 1.664422715051\]

Try \[C=0.\overbar{123456789} \\ C \to 1.343484638793... \\ C \to 2.454595749814... \\ C=0.\overbar{01} \\ C \to\]

Noting that sqrt(5)/sqrt(3) is quite close to 7.454.../5.961... with a difference of \(0.040574942971...\). Noting that the transformations of sqrt(5) and sqrt(6) are close.

There will be infinite numbers that are invarient to the transform, but less that all numbers clearly. The transform when applied twice undoes itself.

There must exist in some sense, an argument over the liklihood, that in an infinite string of digits, i.e an irrational number, the probability \(P_d\) of a given digit \(d\) occuring in any base \(\beta\) tends to \[P_d=\frac{1}{\beta-1}\] that is, a uniform distribution.

However this is a naieve, assumption still. However, if this were not true then the much later digits, and therefore much less significant digits would hold some order beyond first assumptions. Is it significant that \(\sqrt{2}\),\(\sqrt{7}\) and \(\sqrt{8}\) each have a double zero after thier mapping at digits \(3\) and \(4\). Probably not, however, if it were the case it would suggest correlation in digits \(104\) and \(85\) of the original numbers. The fact this feature is in base \(10\) only (I’m guessing) dismissed it almost immediatly.

Let us conduct some experiments. Analysing the first \(10000000\) digits of pi (from http://pi.karmona.com/) gives \[\begin{array}{|c|c|c|} \hline 0's& 9.9944 \%& 999440 \\ 1's& 9.9933 \%& 999333 \\ 2's& 10.0031 \%& 1000306 \\ 3's& 9.9996 \%& 999965 \\ 4's& 10.0109 \%& 1001093 \\ 5's& 10.0047 \%& 1000466 \\ 6's& 9.9934 \%& 999337 \\ 7's& 10.0021 \%& 1000207 \\ 8's& 9.9981 \%& 999814 \\ 9's& 10.0004 \%& 1000040 \\ \hline \end{array}\]

This is a fairly flat distribution, with the largest fluctuation of \(1093/10000000\) roughly one ten-thousandth.

However for Champernowne constant we have a distinctly different situation! for \(5888890\) digits we have an exact sharing of digits between non-zero digits! But the zeros are distinctly less. \[\begin{array}{|c|c|c|} \hline 0's & 8.3019 \%& 488890 \\ 1's & 10.1887 \%& 600000 \\ 2's & 10.1887 \%& 600000 \\ 3's & 10.1887 \%& 600000 \\ 4's & 10.1887 \%& 600000 \\ 5's & 10.1887 \%& 600000 \\ 6's & 10.1887 \%& 600000 \\ 7's & 10.1887 \%& 600000 \\ 8's & 10.1887 \%& 600000 \\ 9's & 10.1887 \%& 600000 \\ \hline \end{array}\]

Perhaps there is some convergence to a constant which is the ratio of the number of zeroes in the base 10 Champernowne constant to the number of another digit. The best guess from this is \(0.814816667\) so far. Actually it is a necessarcy con

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