Ok, I have just learnt the Pigeonhole Principle(PHP) and its application with decimal expansion.
To convey my question clearly, I need to convey my understanding of PHP with regards to decimal expansion so here goes...
By the long division process, we can obtain a infinite number of remainders since $0$ is also considered a remainder(refer to Fig 1
to get what I mean) which is $>$ finite number of possible values of remainders(by quotient-remainder theorem, $0≤r<d
$)
→By PHP , we’ll definitely get a remainder whose value is same as a previous remainder’s value (In Fig 1, we score a hit when r7=r1)
→As such at some point, the sequence of remainders will infinitely repeat
→This sequence of remainders can either be zeroes or non-zeroes
→If it's a sequence of zeroes, we get terminating decimals like $2$(as in $2.\color{red}{\overline{000}}\ldots$$...$) or $3.625$(as in $3.625\color{red}{\overline{000}}\ldots$$...$)
→If it's a sequence of non-zeroes, we get a repeating decimal like $3/14$=$0.2\color{red}{\overline{142857}}\ldots$
My question: So why does PI not fall into either of these categories? Does it somehow violate PHP?
Apologies for the screenshots in advance.
Fig 1: