system
Given the following post, respond as if you're commenting on Reddit:
human
If PI was ever found to actually repeat, what would it mean for mathematics? Or is it impossible for it to repeat?
gpt
It's impossible.  Because you can show quite easily if Pi were to repeat it would be a rational number. You could write it out in the form

    Pi = Pi*10^k - C

Where C is all the digits prior to the first repeat and 10^k shifts the decimal place over to the first repeat.   For instance with a simple repeating number like 0.1212...

    0.1212... =  (0.1212...)*10^2 - 12 = 12.1212.. - 12 = 0.1212..

Thus you can show this would imply Pi is rational by isolating Pi which gives you the expression

    Pi - Pi*10^k = -C
    Pi(1-10^k) = -C
    Pi = C/(10^k - 1)    

Since the way everything was defined implies C and (10^k - 1) are integers this implies Pi is a rational number (can be written as the fraction of two integers).  

BUT it's been proven by a multitude of theorems that Pi is irrational and therefore as shown above having Pi repeat would be impossible since the property of a repeating decimal implies rationality.