Concerning Pi, again

For Pi Day this year (2018), I will provide some results related to this interesting mathematical constant. These results are mostly inspired or taken from answers I gave at about  \pi or about similar mathematical topics.

The millionth decimal digit of π is 1 (verified with Mathematica).

The 10 millionth decimal digit of π is found to be 7, and the 100 millionth decimal digit of π is 2.

The billionth decimal digit of \pi (in base 10) is 9 (verified with Mathematica).

The 2 billionth decimal digit of π is found to be 0 (this result takes a longer time to compute with Mathematica).

Here are some (repeated) number sequences or numeric strings found among the first 2 billion decimal digits of π.

The numeric string 777777777 appears at the 24, 658, 601 st decimal digit of π :


The numeric string 111111111 appears at the 812, 432, 526 th decimal digit of π :


Here are also two numeric strings from 1 to 9 in increasing order and decreasing order:

The numeric string 123456789 appears at the 523, 551, 502 nd decimal digit of π:


The numeric string 987654321 appears at the 719, 473, 323 rd decimal digit of π :


The numeric strings above can be calculated or found with the help of the following link or web page:

Irrational Numbers Search Engine

The numerical value of \pi^{\pi} to 1000 decimal digits is equal to:


Two expressions involving π and infinite sums:

pi infinite cums

Representation of π in continued fraction form:

pi continued frac form

The sum of π and e, the base of natural logarithms, is equal to:

\pi + e=\displaystyle  \sum _{k=0}^{\infty } \frac{(3 k)^2+1}{(3 k)!}+2 i \ln \left(\frac{1-i}{1+i}\right)i

The letter i  represents the imaginary unit of complex numbers.

Another expression involving π, e, and an infinite product:

\pi = \displaystyle 2 e \prod _{k=1}^{\infty } \left(\frac{2}{k}+1\right)^{(-1)^{k+1} k}

And here is an identity relating the Golden Ratio, π, e, and the imaginary unit i:

\displaystyle \varphi=e^{i\pi/5}+e^{-i\pi/5}=\frac{1+\sqrt{5}}{2}

The value of π can be deduced from the identity above:

\displaystyle\pi =5 i \ln \left(\frac{1}{2} \left(\varphi -\sqrt{\varphi ^2-4}\right)\right)