Recently, I came across a fascinating blog and video from Mind you Decisions. It is about how a fraction
would show the Fibonacci numbers in order when looking at its decimal output.
On a spreadsheet and most standard programming languages, such output can not be attained due to the limited precision for floating point numbers. If you try this on R or Python, you will get an output of 1e-48.
Wolfram alpha,however,allows arbitrary precision.
In Julia by default we get a little better that R and Python
julia> 1/999999999999999999999998999999999999999999999999 1.000000000000000000000001000000000000000000000002000000000000000000000003000002e-48 julia> typeof(ans) BigFloat
We observe here that we are getting the first few Fibonacci numbers 
BigFloat type on demand. Of course, the higher the precision, the slower the computation and the greater the memory we use. We do that by setprecision.
julia> setprecision(BigFloat,10000) 10000
Reevaluating, we get
julia> 1/999999999999999999999998999999999999999999999999 1.00000000000000000000000100000000000000000000000200000000000000000000000300000000000000000000000500000000000000000000000800000000000000000000001300000000000000000000002100000000000000000000003400000000000000000000005500000000000000000000008900000000000000000000014400000000000000000000023300000000000000000000037700000000000000000000061000000000000000000000098700000000000000000000159700000000000000000000258400000000000000000000418100000000000000000000676500000000000000000001094600000000000000000001771100000000000000000002865700000000000000000004636800000000000000000007502500000000000000000012139300000000000000000019641800000000000000000031781100000000000000000051422900000000000000000083204000000000000000000134626900000000000000000217830900000000000000000352457800000000000000000570288700000000000000000922746500000000000000001493035200000000000000002415781700000000000000003908816900000000000000006324598600000000000000010233415500000000000000016558014100000000000000026791429600000000000000043349443700000000000000070140873300000000000000113490317000000000000000183631190300000000000000297121507300000000000000480752697600000000000000777874204900000000000001258626902500000000000002036501107400000000000003295128009900000000000005331629117300000000000008626757127200000000000013958386244500000000000022585143371700000000000036543529616200000000000059128672987900000000000095672202604100000000000154800875592000000000000250473078196100000000000405273953788100000000000655747031984200000000001061020985772300000000001716768017756500000000002777789003528800000000004494557021285300000000007272346024814100000000011766903046099400000000019039249070913500000000030806152117012900000000049845401187926400000000080651553304939300000000130496954492865700000000211148507797805000000000341645462290670700000000552793970088475700000000894439432379146400000001447233402467622100000002341672834846768500000003788906237314390600000006130579072161159100000009919485309475549700000016050064381636708800000025969549691112258500000042019614072748967300000067989163763861225800000110008777836610193100000177997941600471418900000288006719437081612000000466004661037553030900000754011380474634642900001220016041512187673800001974027421986822316700003194043463499009990500005168070885485832307200008362114348984842297700013530185234470674604900021892299583455516902600035422484817926191507500057314784401381708410100092737269219307899917600150052053620689608327700242789322839997508245300392841376460687116573000635630699300684624818301028472075761371741391301664102775062056366209602692574850823428107600904356677625885484473810507049252476708912581411411405930102594397055221918455182579303309636633329861112681897706691855248316295261201016328488578177407943098723020343826493703204299739348832404671111147398462369176231164814351698201718008635835925499096664087184867000739850794865805193502836665349891529892378369837405200686395697571872674070550577925589950242511475751264321287522115185546301842246877472357697022053e-48
That is looking much better. However it we be nice if we could extract the Fibonacci numbers that are buried in that long decimal. Using the approach in the original blog. We define a function
y(x)=one(x)-x-x^2
and calculate the decimal
a=1/y(big"1e-24")
Here we use the non-standard string literal big"..." to insure proper interpretation of our input. Using BigFloat(1e-24)) would first construct at floating point with limited precision and then do the conversion. The initial loss of precision will not be recovered in the conversion, and hence the use of big. Now we extract our Fibonacci numbers by this function
function extract_fib(a) x=string(a) l=2 fi=BigInt[] push!(fi,1) for i=1:div(length(x)-24,24) j=parse(BigInt,x[l+1+(i-1)*24:l+i*24]) push!(fi,j) end fi end
Here we first convert our very long decimal number of a string and they we exploit the fact the Fibonacci numbers occur in blocks that 24 digits in length. We get out output in an array of BigInt. We want to compare the output with exact Fibonacci numbers, we just do a quick and non-recursive implementation.
function fib(n) f=Vector{typeof(n)}(n+1) f[1]=f[2]=1; for i=3:n+1 f[i]=f[i-1]+f[i-2] end f end
Now we compare…
fib_exact=fib(200); fib_frac=extract_fib(a); for i in eachindex(fib_frac) println(fib_exact[i], " ", fib_exact[i]-fib_frac[i]) end
We get a long sequence, we just focused here on when the discrepancy happens.
... 184551825793033096366333 0 298611126818977066918552 0 483162952612010163284885 0 781774079430987230203437 -1 1264937032042997393488322 999999999999999999999998 2046711111473984623691759 1999999999999999999999997 ...
The output shows that just before the extracted Fibonacci number exceeds 24 digits, a discrepancy occurs. I am not quite sure why, but this was a fun exploration. Julia allows me to do mathematical explorations that would take one or even two orders of magnitude of effort to do in any other language.