Is every day tau day (or pi day) to some base?

Question

People on my facebook wall are celebrating the fact that today is tau (2 pi) day—6/28. This got me thinking: today is only tau day because we represent numbers in decimal base. Could every day of the year be the first digits of tau (or pi, I don't care) depending on what base we used?

Answer 1

If you're consistent about your use of other bases, pi day will always be in March and tau day will always be in June (since the integer part of a number doesn't depend on the base). Which day it is will vary somewhat, though. I think the most reasonable convention is to use the largest possible number of digits after the decimal point which give you a date (since base-10 pi day is March 14, not March 1). For example, since $\pi=(11.0010010000 \dots)_2$, base-2 pi day should be March $0010010_2$ (also known as March 18).

Here's a table showing the fractional parts of $\pi$ and $\tau$, and the corresponding days, for some small bases: $$ \begin{array}{ccccc} \textrm{base} & \textrm{fractional part of $\pi$} & \textrm{pi day}&\textrm{fractional part of $\tau$} &\textrm{tau day}\\ 2 & \textrm{ 0.0010010000 }& \textrm{March 18 }& \textrm{ 0.0100100001 } & \textrm{June 18 }\\ 3 & \textrm{ 0.0102110122 }& \textrm{March 11 }& \textrm{ 0.0211221022 } & \textrm{June 22 }\\ 4 & \textrm{ 0.0210033312 }& \textrm{March 9 }& \textrm{ 0.1020133231 } & \textrm{June 18 }\\ 5 & \textrm{ 0.0323221430 }& \textrm{March 17 }& \textrm{ 0.1201443411 } & \textrm{June 7 }\\ 6 & \textrm{ 0.0503300514 }& \textrm{March 30 }& \textrm{ 0.1411001432 } & \textrm{June 10 }\\ 7 & \textrm{ 0.0663651432 }& \textrm{March 6 }& \textrm{ 0.1660633164 } & \textrm{June 13 }\\ 8 & \textrm{ 0.1103755242 }& \textrm{March 9 }& \textrm{ 0.2207732504 } & \textrm{June 18 }\\ 9 & \textrm{ 0.1241881240 }& \textrm{March 11 }& \textrm{ 0.2483872481 } & \textrm{June 22 }\\ 10 & \textrm{ 0.1415926535 }& \textrm{March 14 }& \textrm{ 0.2831853071 } & \textrm{June 28 }\\ 11 & \textrm{ 0.1615070286 }& \textrm{March 17 }& \textrm{ 0.312A130562 } & \textrm{June 3 }\\ 12 & \textrm{ 0.184809493B }& \textrm{March 20 }& \textrm{ 0.349416967B } & \textrm{June 3 }\\ 13 & \textrm{ 0.1AC1049052 }& \textrm{March 23 }& \textrm{ 0.38B20950A5 } & \textrm{June 3 }\\ 14 & \textrm{ 0.1DA75CDA81 }& \textrm{March 27 }& \textrm{ 0.3D70BBD722 } & \textrm{June 3 }\\ 15 & \textrm{ 0.21CD1DC46C }& \textrm{March 31 }& \textrm{ 0.43AB3C98D9 } & \textrm{June 4 }\\ 16 & \textrm{ 0.243F6A8885 }& \textrm{March 2 }& \textrm{ 0.487ED5110B } & \textrm{June 4 }\\ 17 & \textrm{ 0.26FAG579ED }& \textrm{March 2 }& \textrm{ 0.4DE4FAF2C9 } & \textrm{June 4 }\\ 18 & \textrm{ 0.29FDEH0G77 }& \textrm{March 2 }& \textrm{ 0.51D9BG1EEE } & \textrm{June 5 }\\ 19 & \textrm{ 0.2D23982975 }& \textrm{March 2 }& \textrm{ 0.5746IG4IEB } & \textrm{June 5 }\\ 20 & \textrm{ 0.2GCEG9GBHJ }& \textrm{March 2 }& \textrm{ 0.5D59CJD3FI } & \textrm{June 5 }\\ 21 & \textrm{ 0.2K961EDI5H }& \textrm{March 2 }& \textrm{ 0.5JIC386FBD } & \textrm{June 5 }\\ 22 & \textrm{ 0.32BEK9A809 }& \textrm{March 3 }& \textrm{ 0.6517IIKG0J } & \textrm{June 6 }\\ 23 & \textrm{ 0.35KH9K813J }& \textrm{March 3 }& \textrm{ 0.6BIBJHG27G } & \textrm{June 6 }\\ 24 & \textrm{ 0.39D911BCLK }& \textrm{March 3 }& \textrm{ 0.6J2I22N1JG } & \textrm{June 6 }\\ 25 & \textrm{ 0.3DC9FINE6E }& \textrm{March 3 }& \textrm{ 0.71OJ6CM3D3 } & \textrm{June 7 }\\ 26 & \textrm{ 0.3HIGBEBOHJ }& \textrm{March 3 }& \textrm{ 0.79B6N2NN9D } & \textrm{June 7 }\\ 27 & \textrm{ 0.3M5Q3M2DCP }& \textrm{March 3 }& \textrm{ 0.7HBP7H4QPO } & \textrm{June 7 }\\ 28 & \textrm{ 0.3R06LIOJPL }& \textrm{March 3 }& \textrm{ 0.7Q0DF9LBNF } & \textrm{June 7 }\\ 29 & \textrm{ 0.4328N0CJQM }& \textrm{March 4 }& \textrm{ 0.864HH0PAOG } & \textrm{June 8 }\\ 30 & \textrm{ 0.47D01EE07R }& \textrm{March 4 }& \textrm{ 0.8EQ02SS0FO } & \textrm{June 8 }\\ 31 & \textrm{ 0.4C25OE856S }& \textrm{March 4 }& \textrm{ 0.8O4BHSGADP } & \textrm{June 8 }\\ 32 & \textrm{ 0.4GVML245KC }& \textrm{March 4 }& \textrm{ 0.91VDA48B8O } & \textrm{June 9 }\\ 33 & \textrm{ 0.4M6DN4OW9R }& \textrm{March 4 }& \textrm{ 0.9BCRD9GVJL } & \textrm{June 9 }\\ 34 & \textrm{ 0.4RN5C8IANV }& \textrm{March 4 }& \textrm{ 0.9LCAOH2LDS } & \textrm{June 9 }\\ 35 & \textrm{ 0.4XFRGMTM53 }& \textrm{March 4 }& \textrm{ 0.9VVJXAO9A7 } & \textrm{June 9 }\\ 36 & \textrm{ 0.53I5AB8P5F }& \textrm{March 5 }& \textrm{ 0.A70AKMHEAV } & \textrm{June 10 } \end{array} $$ Note that the days get later as the base increases, except when you overflow and have to start using a smaller number of digits. It seems clear from this table that you can make pi day be any day in March you want except March 1, and tau day any day in June except June 1 or 2, by using a base where you're only taking a single digit after the decimal place. On the other hand, if you habitually used a base where you could only take a single digit after the decimal point before overflowing, the idea of pi day would look a lot less natural...