# Finding 2014 in e, Pi, and Golden Ratio

Last year around this time John D. Cook wrote a post about finding 2013 in Pi [apparantly his daughter wanted to find the occurance of 2013 in the digits of Pi on the eve of new year]. We know that Pi is an irrational number- i.e. it can not be expressed as a ratio of two integers; and contains an infinite number of digits. Moreover these digits have no apparant pattern or in other words they are randomly distributed. Now can we locate a specific pattern or finite sequence of digits such as 2013 in these randomly behaving digits of Pi or in other famous irrational numbers such as Euler number [e] and Golden Ratio [Phi].

Since we are already in the year 2014, let us try to find the occurance of number 2014 in the decimal expressions of e, Pi, and Phi [upto 1 million digits].

In Mathematica, we can write:

ListPlot[Flatten[ Position[ Partition[First[RealDigits[N[#, 10^6]]], 4, 1], {2, 0, 1, 4}]] & /@ {E, Pi, GoldenRatio}, PlotLegends -> Placed[PointLegend[{E, Pi, GoldenRatio}, LegendMarkers -> Graphics[Disk[]]], {Right, Center}], ImageSize -> 600, AspectRatio -> 0.75, Axes -> False, Frame -> True, FrameTicks -> {{True, None}, {True, None}}, GridLines -> {Automatic, Automatic}, GridLinesStyle -> LightGray, PlotStyle -> PointSize[0.0125], PlotRange -> {{0, 120}, {0, All}}, BaseStyle -> Directive[FontFamily -> "Helvetica", 12], FrameLabel -> {Style["Occurrence of 2014", Bold, 14], Style["No. of Digits in e, Pi, and Phi", Bold, 14]}]

The first occurance of 2014 is at the 3180, 3133, and 1901th position in the of digits e, Pi, and Golden Ratio respectively. And if we keep on looking further [I have explored upto 1 million digits] we can find multiple occurances of 2014 at random positions.