# Abstract

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.

# Introduction

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.

# Main

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.

# Turning e into $$\pi$$

The digits of $$\pi$$ and