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.

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