P-Adic Expansion of Rational Number (Reverse Direction)5−1+1+2⋅5+52+(4⋅53+2⋅54+3⋅55+2⋅57+58)+(4⋅59+2⋅510+3⋅511+2⋅513+514)+(4⋅515+2⋅516+3⋅517+2⋅519+520)+⋯

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Colonninisxi · Accepted Answer

Step 1Given:&amp;nbsp;x=…120324¯121.15and then add/subtract the appropriate finite base-5 fraction to make it a pure repeating number ending at the ones:x-1.15=…120324¯1205=…324120¯5Then the I can convert that to a fraction essentially like one does for an ordinary decimal fraction, except that the analogue of&amp;nbsp;0.999…10=1&amp;nbsp;is now&amp;nbsp;…4445=-1x-65=3241205·…000001¯5=32412054444445·…444444¯5=−32412054444445⇒&amp;nbsp;x=65−32412054444445=65−1116015624=3794478120=1735Alternatively you could start withx-121.1553=…120324¯5and convert the right-hand side of that to a fraction in the same way.

P-Adic Expansion of Rational Number (Reverse Direction) 5^{-1} + 1 +

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