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 wit

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