For those who can't get enough of long fractions, there's a whole site dedicated to them. Should you need to satisfy your withdrawal symptoms, you can get your fix at oeis.org

They have hundreds if not thousands of pages, with each of them dedicated to a different math constant or a sequence of integers. For example, if someone needs to make their own $pi2 alias identifier with a whole lot of precision, the page for $pi has links to where you can find more than you can possibly use.

https://oeis.org/A000796

So far there are so many trillions of digits of PI known, that most people don't have a hard drive that can store them all, and people are still calculating more of them.

Blowfish uses a little more than 4000 digits of the PI fraction to initialize some data. There wasn't anything special about PI that makes Blowfish magically better, it was just an arbitrary choice to find something non-zero to start out with. When people need to initialize some variables semi-randomly but don't want to explain how they came up with the numbers they ended up using, they just pick something like PI. Like when they created the hashes like $sha1, they needed to have some variables being initialized to 'something', so they just picked a bunch of square roots and used digits from each.

If you can think of a math constant, they've got a page for it, and it probably has thousands of digits of its fraction. There's lots of places in the real world that turn out to be related to some of these fraction constants, and some of them end up having unusual relationships to each other.

For exampe, If you want to cut a paper in half so that the ratio of total/long-piece is the same ratio as long-piece/short-piece, the ratio would be the golden ratio that is 1.6180339...

If you take this number and square it, it's the same as adding +1 to it. If you take the inverse 1/golden, that's same as subtracting 1 from it.

This turns out to be related to the Fibbonacci series, where each number in the series is the total of the previous 2 numbers. As you keep coming up with new numbers in the series, if you divide it by the previous number, the fraction gets closer and closer to... the golden ratio

//var -s %fib1.bf 1, %fib2.bf 2 , %i 1 | while (%i < 100) { echo -ag %i : %fib2.bf / %fib1.bf = $calc(%fib1.bf / %fib2.bf) | var %temp %fib2.bf, %fib2.bf %fib2.bf + %fib1.bf, %fib1.bf %temp | inc %i }

To get a thousand digits of this never-ending fraction without finding thousands of Fibbonacci numbers, you can instead find it another way, though not so simple either. The golden ratio is the same as $calc( ($sqrt(5) +1) /2), but of course for that you'd need a way to find a thousand digits for that square root. But they've got ways to calculate that too.

The natural logarithm 'e' 2.718... used by the $log identifier is something else that pops up in a lot of places. Like if you have compound money interest, the shorter you make the time interval, the closer the interest rate approaches a number that can be calculated using 'e'.

It also shows up in a lot of randomness. If you ask 1000 times for a random number from $rand(1,1000), you're not going to get each of the 1000 numbers showing up 1 time. On average, what fraction of the 1000 numbers will not come up during those 1000 outputs? It's close to 1/e, and the larger number you use in place of both 1000's, the closer it gets to 1/e. What portion of numbers show up exactly 1 time? Also 1/e. What portion shows up exactly 2 times? 1/2e. The formula to find the chance that something shows up exactly N times uses 'factorial, which is that number multiplied by all the numbers below it. i.e. 5 factorial is 1*2*3*4*5. So that means the chance of a number showing up exactly 3 times is 1/(1*2*3*'e'). So this SUM will eventually reach 100%

//var -s %e 2.718281828459 , %factorial 1 , %i 0, %sum.bf 0 | while (%i < 100) { inc -s %sum.bf $calc(1 / (%e * %factorial)) | inc %i | var %factorial %factorial * %i }

Oh, but it went over 100%. But that's because the value for '%e' didn't have enough digits. But if using enough digits, and have enough precision when dividing, then it would be on the underside of 1.0, and then you'd need to repeat more times to get the fraction closer to 100%.