Wednesday, 12 April 2017

How can the decimal expansion of this rational number not be periodic?




I just noticed that dividing 1÷998 gives me the apparently non-periodic



0.001002004008016032064,



which is 103+2×106+4×109+8×1012+=i=02i103(i+1).



Since every rational number expands into a finite or periodic decimal expansion, does that become periodic somehow? If so, how?


Answer



It does repeat, but with a very long period: Factoring the denominator into primes gives
998=2499;

since 2 is a factor of 10 and 10 is a primitive root modulo 499, the period of repetition is 4991=498 digits long. Indeed, consulting WolframAlpha gives:



1998=0.¯00100200400801603206412825651302605210420841683366733466¯93386773547094188376753507014028056112224448897795591182¯36472945891783567134268537074148296593186372745490981963¯92785571142284569138276553106212424849699398797595190380¯76152304609218436873747494989979959919839679358717434869¯73947895791583166332665330661322645290581162324649298597¯19438877755511022044088176352705410821643286573146292585¯17034068136272545090180360721442885771543086172344689378¯7575150300601202404809619238476953907815631262525050.



(The is analogous to the repetition of the decimal digits of, e.g., 1/14: We have 14=27 and that 10 is a primitive root modulo 7, so the period of the decimal expansion of 1/14 is 7.)


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